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Coherent Production of Light Pseudoscalars (Axions)
Inside a Magnetic Field with a Polarized Laser Beam
by
Yannis K Semertzidis
Submitted in Partial Fulfillment of the
Requirements of the Degree of
DOCTOR OF PHILOSOPHY
Supervised by Professor A C Melissinos
Department of Physics and Astronomy
University of Rochester
Rochester New York
1989
Work supported in part by US Dept of Energy Contract DE-AC02-76ER13065
Abstract
We have constructed a highly sensitive ellipsometer to
measure the effect of a transverse magnetic field on the
propagation of polarized light The total path length is
of order 7 Km in a field of 22 T and rotation angles as
low as 10-10 rad can be measured
We have used this instrument to set limits on the coupl ing
of light pseudoscalar and scalar particles to two photons
thus placing constraints on some recent theories of the
elementary particles (certain classes of supersymmetric
theories) We have also used the apparatus to measure for
the first time the Cotton-Mouton coefficients of the noble
gases Neon and Helium
In addition the apparatus has been designed so as to
measure photon-photon (Delbruck) scattering in the visible
as predicted by QED We have demonstrated that the apparatus
has adequate sensitivity to reach this goal and have
identified its present limitations
ii
Acknowledqements
The experiment was first approved (unanimously) by the
BNL committee in November 1987 It is encouraging to see
that big laboratories still support small (though elegant)
experiments Brookhaven National Lab has supported us with
facilities and personnel so there are numerous people to
thank Charlie Anderson Audrey Blake Frank Cullen Norman
Franklin Robert Gottschalk Ron Hauser Arnie Kreisberg
Jim Licari John Mazzeo Vera Mott Cynthia Murphy The
Riggers Ritchie Savoy and Diana Votruba A separate thanks
to Jim Briggs David Cattaneo Bill De Jong George Ganetis
Herb Hildebrand Rich Howard Paul Ribaudo Joe Skatudo
and Dan Wilson who were generous both with their time and
skills
This support was matched by the people at the University
of Rochester Pat Borelli Ernie Buchanan Betty Cook
Thomas Haelen Connie Jones and Judy Mack My appreciation
goes to the members of our research team Ruth Cameron
Giovanni Cantatore Henry Halama George Kostoulas Don
Lazarus Bruce Moskowitz Frank Nezrick Al Prodell Carlo
Rizzo and Joe Rogers for their remarks and contributions
Special thanks to Prof Emilio Zavattini for his
contagious enthusiasm and to my supervisor Prof Adrian
Melissinos for all his support encouragement and guidance
iii
Finally my special thanks go to my wife Georgia
Afxendiou-semertzidis for her support and understanding
but most of all for sharing with me the excitement of thesis
writing
This work was supported by the us Department of Energy
under contracts DE-AC02-76CH00016 and DE-AC02-76ER13065
iv
curriculum vitae
Yannis Semertzidis was born on September 16 1961 He
attended the Aristoteleio Panepistimio Thessalonikis
(Aristotle University of Thessaloniki) in Greece from 1979
to 1984 where he received a Bachelor of Science Degree in
Physics May 1985 he began graduate studies at the University
of Rochester In 1987 he was a recipient of the Furth
Fellowship administered by the University His research
work has included the CP Violation experiment at LEARCERN
during the summer of 1983 and from spring 1984 to summer
1984 Fermilab experiment 723 a search for anomalous forces
at highly relativistic velocities from May 1985 to December
1985i BNL experiment 805 the search for Galactic axions
from January 1986 to spring 1989 Subsequently he has worked
on BNL experiment 840 a coherent production of any
pseudoscalar (or scalar) that couples to two photons this
was from summer 1987 to the present This work was supervised
by Dr A C Melissinos
v
Table of contents
Abstract ii Acknowledgments iii Curriculum vitae v
List of Figures vii List of Tables ix
Chapter 1 Theoretical Motivation 1 11 Introduction 1 12 QED Vacuum Polarization 10 13 Axion 16
Chapter 2 Apparatus 34 21 Magnets 34 22 Laser 45 23 optics 54 24 Cavity 63 25 Ray Transfer 73 26 Telescope 78 27 Jones Vectors and Matrices 84
Chapter 3 Data Acquisition 93
31 Electronics 93
32 Misalignment Correction 101 33 Laser Power Stabilizer 104
Chapter 4 Analysis of Results 113 41 Noise Sources 113 42 Data Analysis 123
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases 144
Chapter 6 Conclusions 160
Index 168
vi
List of Fiqures
Chapter 1 11 Primakoff effect bull bull bull bull bull bull bull bull bull bull bull bull 3 12 Lowest order QED Vacuum Polarization bullbull 6 13 Axions mix with pi zero and eta bullbullbull 20 14 Axions decay through a triangle anomaly bull 22 15 Axion induced rotation bullbullbullbullbullbullbull 27 16 Axion induced ellipticity bullbullbullbullbullbull 28 17 Axion limits from assumed rotation bullbullbullbullbull 30 18 Axion limits from assumed ellipticity bullbullbull 31
Chapter 2 21 Experiment layout bull bull bull bull bull bull bull bull bull bull 35 22 optics inside the enclosure bullbullbull 36 23 Cross section of a CBA magnet bullbullbull 37 24 Lead pot end cap and the two magnets bullbullbull 38 25 Magnet quench current vs temperature bullbullbull 39 26 LM0014 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 42 27 LM0018 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbull 43 28 stray magnetic field bullbullbullbullbullbullbullbull 44 29 Atomic Ar levels 47 210 A typical Ar ion laser resonator bullbullbullbullbull 52 211 QWP and elliptically polarized light bull 57 212 Support tubes for the Faraday cell 60 213 Lissajous pattern bullbullbullbullbullbullbullbullbullbull 65 214 Time delay measurements bullbullbullbullbullbullbullbullbullbullbull 67 215 Birefringence axis setup bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 69 216 Circle for birefringence axis bullbullbullbullbullbullbullbullbullbull 70 217 First readout with one mirror rotated bullbullbull 72 218 Birefringence axes of a mirror bullbullbullbullbullbull 74 219 Parameters of a paraxial ray bullbullbullbullbullbullbullbullbullbullbullbullbull 75 220 Stable regions of mirrors resonator bullbullbullbullbull 77 221 Telescope lens setup bullbullbullbullbullbullbullbullbullbullbull 81
vii
Chapter 3 31 Electronics setup of the experiment 95 32 EGampG low noise preamplifier characteristics 98 33 Laser light spectral density bull 105 34 Nominal noise reduction vs frequency bull 108 35 Typical data with LPC OFF bullbullbullbullbull 109 36 Typical data with LPC ON bull 110 37 lf noise reduction with LPC ON bullbullbull 111
Chapter 4 41 lgain squared vs frequency bullbullbullbullbull 118 42 Noise sources vs amount of light bull 119 43 Single record rotation data bullbullbull 125 44 RMS average of 26 files 126 45 vector average of 26 files bullbullbullbullbull 127 46 Typical rotation data with magnets ON 130 47 Shunt mirror data with magnets ON bull 132 48 Cavity mirrors move due to the magnet motion 134 49 RotationEllipticity limits from E840 bullbullbullbull 142
Chapter 5 51 Magnet modulation at 78 mHz 147 52 Fourier analysis of the magnet current 148 53 Typical Nitrogen gas data 149 54 Vacuum ellipticity data 150 55 Ellipticity vs Nitrogen gas pressure 152 56 Ellipticity vs Ar gas pressure middot 153 57 Ellipticity vs Ne gas pressure middot 154 58 Ellipticity vs He gas pressure middot 155
viii
List of Tables
Chapter 2 21 Effective magnetic field length 22 Laser performance parameter specifications 23 Laser specifications (Innova 9o_5) bullbullbullbullbullbullbullbullbullbull
24 Laser output power specifications bullbullbull 25 REP and LEP light through a QWP bullbullbullbull 26 Ray transfer matrices 27 Jones vectors 28 Jones matrices
Chapter 3
31 Silicon photodiode 32 Data taking sheet
Chapter 4
41 Rotation data 42 Ellipticity data bullbullbullbullbullbull
Chapter 5 51 Cotton-Mouton constants 52 Various parameters for the gases 53 Radius of first excited state of some gases bull
41 49
50
51 58 82 85 86
96
112
137 141
157 158 158
ix
1012
Chapter 1 Theoretical Motivation
11 Introduction
One of the most important problems in physics today
is the behavior of the interactions of the elementary
particles at very high energies in the range 10 2 to 10 19
GeV However present and even future particle
accelerators will be able to explore only the first few
decades of this range Thus we will have to rely on
indirect information by studying the relics of the very
early universe where such energies existed for a very
brief time One such particle created at an energy of
GeV is the axion which is a pseudoscalar with very
light mass ma~10-5 eVe This makes it possible to search
for the axion either in the cosmic radiation1 or to attempt
to directly produce it in the laboratory This thesis
is an experimental search for axions and for analogous
light scalar particles We have not observed axions but
have set a limit on their coupling to two photons or
equivalent to two quarks
The search for axions was based on the effect that
they would have on the propagation of light in a magnetic
field in vacuum This can be understood by considering
1
the coupling of the axion to two photons as shown in figure
1lb Since a magnetic field is equivalent to a cloud
of virtual photons a real photon from a 1ase- bea can
scatter off the virtual photons to produce an axion 2 The
production of the axion will affect the polarization of
the incident beam as explained in detail in section 13
We have therefore constructed a very sensitive
polarimeter or ellipsometer and have achieved a
sensitivity in the change of polarization of order 10-10
rad The polarimeter is described in chapters 2 and 3
and I give here only a brief sketch
The photon source is a 2W (single line power) Argon-ion
laser operating at ~ = 514 nm High quality polarizers
are used to establish and to analyze the polarization and
are set for best extinction The magnetic field is as
high as possible 22 kgauss in the present investigation
and extends over -10m To enhance the effect the light
is made to travel repeatedly through the field region
This is achieved by forming an optical cavity between two
mirrors and deforming one of the mirrors we have reached
790 traversals
Since the signal is so weak it is important to measure
the change in the photon field amplitude rather than
intensity This can be done by interfering the effect
from the cavity with a deliberately introduced rotation
in our case this is done with a Faraday cell For
2
a 11M Photon
(a)
Photon 11M a
(b)80
Figure 11 Primakoff effect a) Decay ofaxions inside a magnetic field
b) Production ofaxions inside a magnetic field
3
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
Abstract
We have constructed a highly sensitive ellipsometer to
measure the effect of a transverse magnetic field on the
propagation of polarized light The total path length is
of order 7 Km in a field of 22 T and rotation angles as
low as 10-10 rad can be measured
We have used this instrument to set limits on the coupl ing
of light pseudoscalar and scalar particles to two photons
thus placing constraints on some recent theories of the
elementary particles (certain classes of supersymmetric
theories) We have also used the apparatus to measure for
the first time the Cotton-Mouton coefficients of the noble
gases Neon and Helium
In addition the apparatus has been designed so as to
measure photon-photon (Delbruck) scattering in the visible
as predicted by QED We have demonstrated that the apparatus
has adequate sensitivity to reach this goal and have
identified its present limitations
ii
Acknowledqements
The experiment was first approved (unanimously) by the
BNL committee in November 1987 It is encouraging to see
that big laboratories still support small (though elegant)
experiments Brookhaven National Lab has supported us with
facilities and personnel so there are numerous people to
thank Charlie Anderson Audrey Blake Frank Cullen Norman
Franklin Robert Gottschalk Ron Hauser Arnie Kreisberg
Jim Licari John Mazzeo Vera Mott Cynthia Murphy The
Riggers Ritchie Savoy and Diana Votruba A separate thanks
to Jim Briggs David Cattaneo Bill De Jong George Ganetis
Herb Hildebrand Rich Howard Paul Ribaudo Joe Skatudo
and Dan Wilson who were generous both with their time and
skills
This support was matched by the people at the University
of Rochester Pat Borelli Ernie Buchanan Betty Cook
Thomas Haelen Connie Jones and Judy Mack My appreciation
goes to the members of our research team Ruth Cameron
Giovanni Cantatore Henry Halama George Kostoulas Don
Lazarus Bruce Moskowitz Frank Nezrick Al Prodell Carlo
Rizzo and Joe Rogers for their remarks and contributions
Special thanks to Prof Emilio Zavattini for his
contagious enthusiasm and to my supervisor Prof Adrian
Melissinos for all his support encouragement and guidance
iii
Finally my special thanks go to my wife Georgia
Afxendiou-semertzidis for her support and understanding
but most of all for sharing with me the excitement of thesis
writing
This work was supported by the us Department of Energy
under contracts DE-AC02-76CH00016 and DE-AC02-76ER13065
iv
curriculum vitae
Yannis Semertzidis was born on September 16 1961 He
attended the Aristoteleio Panepistimio Thessalonikis
(Aristotle University of Thessaloniki) in Greece from 1979
to 1984 where he received a Bachelor of Science Degree in
Physics May 1985 he began graduate studies at the University
of Rochester In 1987 he was a recipient of the Furth
Fellowship administered by the University His research
work has included the CP Violation experiment at LEARCERN
during the summer of 1983 and from spring 1984 to summer
1984 Fermilab experiment 723 a search for anomalous forces
at highly relativistic velocities from May 1985 to December
1985i BNL experiment 805 the search for Galactic axions
from January 1986 to spring 1989 Subsequently he has worked
on BNL experiment 840 a coherent production of any
pseudoscalar (or scalar) that couples to two photons this
was from summer 1987 to the present This work was supervised
by Dr A C Melissinos
v
Table of contents
Abstract ii Acknowledgments iii Curriculum vitae v
List of Figures vii List of Tables ix
Chapter 1 Theoretical Motivation 1 11 Introduction 1 12 QED Vacuum Polarization 10 13 Axion 16
Chapter 2 Apparatus 34 21 Magnets 34 22 Laser 45 23 optics 54 24 Cavity 63 25 Ray Transfer 73 26 Telescope 78 27 Jones Vectors and Matrices 84
Chapter 3 Data Acquisition 93
31 Electronics 93
32 Misalignment Correction 101 33 Laser Power Stabilizer 104
Chapter 4 Analysis of Results 113 41 Noise Sources 113 42 Data Analysis 123
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases 144
Chapter 6 Conclusions 160
Index 168
vi
List of Fiqures
Chapter 1 11 Primakoff effect bull bull bull bull bull bull bull bull bull bull bull bull 3 12 Lowest order QED Vacuum Polarization bullbull 6 13 Axions mix with pi zero and eta bullbullbull 20 14 Axions decay through a triangle anomaly bull 22 15 Axion induced rotation bullbullbullbullbullbullbull 27 16 Axion induced ellipticity bullbullbullbullbullbull 28 17 Axion limits from assumed rotation bullbullbullbullbull 30 18 Axion limits from assumed ellipticity bullbullbull 31
Chapter 2 21 Experiment layout bull bull bull bull bull bull bull bull bull bull 35 22 optics inside the enclosure bullbullbull 36 23 Cross section of a CBA magnet bullbullbull 37 24 Lead pot end cap and the two magnets bullbullbull 38 25 Magnet quench current vs temperature bullbullbull 39 26 LM0014 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 42 27 LM0018 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbull 43 28 stray magnetic field bullbullbullbullbullbullbullbull 44 29 Atomic Ar levels 47 210 A typical Ar ion laser resonator bullbullbullbullbull 52 211 QWP and elliptically polarized light bull 57 212 Support tubes for the Faraday cell 60 213 Lissajous pattern bullbullbullbullbullbullbullbullbullbull 65 214 Time delay measurements bullbullbullbullbullbullbullbullbullbullbull 67 215 Birefringence axis setup bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 69 216 Circle for birefringence axis bullbullbullbullbullbullbullbullbullbull 70 217 First readout with one mirror rotated bullbullbull 72 218 Birefringence axes of a mirror bullbullbullbullbullbull 74 219 Parameters of a paraxial ray bullbullbullbullbullbullbullbullbullbullbullbullbull 75 220 Stable regions of mirrors resonator bullbullbullbullbull 77 221 Telescope lens setup bullbullbullbullbullbullbullbullbullbullbull 81
vii
Chapter 3 31 Electronics setup of the experiment 95 32 EGampG low noise preamplifier characteristics 98 33 Laser light spectral density bull 105 34 Nominal noise reduction vs frequency bull 108 35 Typical data with LPC OFF bullbullbullbullbull 109 36 Typical data with LPC ON bull 110 37 lf noise reduction with LPC ON bullbullbull 111
Chapter 4 41 lgain squared vs frequency bullbullbullbullbull 118 42 Noise sources vs amount of light bull 119 43 Single record rotation data bullbullbull 125 44 RMS average of 26 files 126 45 vector average of 26 files bullbullbullbullbull 127 46 Typical rotation data with magnets ON 130 47 Shunt mirror data with magnets ON bull 132 48 Cavity mirrors move due to the magnet motion 134 49 RotationEllipticity limits from E840 bullbullbullbull 142
Chapter 5 51 Magnet modulation at 78 mHz 147 52 Fourier analysis of the magnet current 148 53 Typical Nitrogen gas data 149 54 Vacuum ellipticity data 150 55 Ellipticity vs Nitrogen gas pressure 152 56 Ellipticity vs Ar gas pressure middot 153 57 Ellipticity vs Ne gas pressure middot 154 58 Ellipticity vs He gas pressure middot 155
viii
List of Tables
Chapter 2 21 Effective magnetic field length 22 Laser performance parameter specifications 23 Laser specifications (Innova 9o_5) bullbullbullbullbullbullbullbullbullbull
24 Laser output power specifications bullbullbull 25 REP and LEP light through a QWP bullbullbullbull 26 Ray transfer matrices 27 Jones vectors 28 Jones matrices
Chapter 3
31 Silicon photodiode 32 Data taking sheet
Chapter 4
41 Rotation data 42 Ellipticity data bullbullbullbullbullbull
Chapter 5 51 Cotton-Mouton constants 52 Various parameters for the gases 53 Radius of first excited state of some gases bull
41 49
50
51 58 82 85 86
96
112
137 141
157 158 158
ix
1012
Chapter 1 Theoretical Motivation
11 Introduction
One of the most important problems in physics today
is the behavior of the interactions of the elementary
particles at very high energies in the range 10 2 to 10 19
GeV However present and even future particle
accelerators will be able to explore only the first few
decades of this range Thus we will have to rely on
indirect information by studying the relics of the very
early universe where such energies existed for a very
brief time One such particle created at an energy of
GeV is the axion which is a pseudoscalar with very
light mass ma~10-5 eVe This makes it possible to search
for the axion either in the cosmic radiation1 or to attempt
to directly produce it in the laboratory This thesis
is an experimental search for axions and for analogous
light scalar particles We have not observed axions but
have set a limit on their coupling to two photons or
equivalent to two quarks
The search for axions was based on the effect that
they would have on the propagation of light in a magnetic
field in vacuum This can be understood by considering
1
the coupling of the axion to two photons as shown in figure
1lb Since a magnetic field is equivalent to a cloud
of virtual photons a real photon from a 1ase- bea can
scatter off the virtual photons to produce an axion 2 The
production of the axion will affect the polarization of
the incident beam as explained in detail in section 13
We have therefore constructed a very sensitive
polarimeter or ellipsometer and have achieved a
sensitivity in the change of polarization of order 10-10
rad The polarimeter is described in chapters 2 and 3
and I give here only a brief sketch
The photon source is a 2W (single line power) Argon-ion
laser operating at ~ = 514 nm High quality polarizers
are used to establish and to analyze the polarization and
are set for best extinction The magnetic field is as
high as possible 22 kgauss in the present investigation
and extends over -10m To enhance the effect the light
is made to travel repeatedly through the field region
This is achieved by forming an optical cavity between two
mirrors and deforming one of the mirrors we have reached
790 traversals
Since the signal is so weak it is important to measure
the change in the photon field amplitude rather than
intensity This can be done by interfering the effect
from the cavity with a deliberately introduced rotation
in our case this is done with a Faraday cell For
2
a 11M Photon
(a)
Photon 11M a
(b)80
Figure 11 Primakoff effect a) Decay ofaxions inside a magnetic field
b) Production ofaxions inside a magnetic field
3
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
Acknowledqements
The experiment was first approved (unanimously) by the
BNL committee in November 1987 It is encouraging to see
that big laboratories still support small (though elegant)
experiments Brookhaven National Lab has supported us with
facilities and personnel so there are numerous people to
thank Charlie Anderson Audrey Blake Frank Cullen Norman
Franklin Robert Gottschalk Ron Hauser Arnie Kreisberg
Jim Licari John Mazzeo Vera Mott Cynthia Murphy The
Riggers Ritchie Savoy and Diana Votruba A separate thanks
to Jim Briggs David Cattaneo Bill De Jong George Ganetis
Herb Hildebrand Rich Howard Paul Ribaudo Joe Skatudo
and Dan Wilson who were generous both with their time and
skills
This support was matched by the people at the University
of Rochester Pat Borelli Ernie Buchanan Betty Cook
Thomas Haelen Connie Jones and Judy Mack My appreciation
goes to the members of our research team Ruth Cameron
Giovanni Cantatore Henry Halama George Kostoulas Don
Lazarus Bruce Moskowitz Frank Nezrick Al Prodell Carlo
Rizzo and Joe Rogers for their remarks and contributions
Special thanks to Prof Emilio Zavattini for his
contagious enthusiasm and to my supervisor Prof Adrian
Melissinos for all his support encouragement and guidance
iii
Finally my special thanks go to my wife Georgia
Afxendiou-semertzidis for her support and understanding
but most of all for sharing with me the excitement of thesis
writing
This work was supported by the us Department of Energy
under contracts DE-AC02-76CH00016 and DE-AC02-76ER13065
iv
curriculum vitae
Yannis Semertzidis was born on September 16 1961 He
attended the Aristoteleio Panepistimio Thessalonikis
(Aristotle University of Thessaloniki) in Greece from 1979
to 1984 where he received a Bachelor of Science Degree in
Physics May 1985 he began graduate studies at the University
of Rochester In 1987 he was a recipient of the Furth
Fellowship administered by the University His research
work has included the CP Violation experiment at LEARCERN
during the summer of 1983 and from spring 1984 to summer
1984 Fermilab experiment 723 a search for anomalous forces
at highly relativistic velocities from May 1985 to December
1985i BNL experiment 805 the search for Galactic axions
from January 1986 to spring 1989 Subsequently he has worked
on BNL experiment 840 a coherent production of any
pseudoscalar (or scalar) that couples to two photons this
was from summer 1987 to the present This work was supervised
by Dr A C Melissinos
v
Table of contents
Abstract ii Acknowledgments iii Curriculum vitae v
List of Figures vii List of Tables ix
Chapter 1 Theoretical Motivation 1 11 Introduction 1 12 QED Vacuum Polarization 10 13 Axion 16
Chapter 2 Apparatus 34 21 Magnets 34 22 Laser 45 23 optics 54 24 Cavity 63 25 Ray Transfer 73 26 Telescope 78 27 Jones Vectors and Matrices 84
Chapter 3 Data Acquisition 93
31 Electronics 93
32 Misalignment Correction 101 33 Laser Power Stabilizer 104
Chapter 4 Analysis of Results 113 41 Noise Sources 113 42 Data Analysis 123
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases 144
Chapter 6 Conclusions 160
Index 168
vi
List of Fiqures
Chapter 1 11 Primakoff effect bull bull bull bull bull bull bull bull bull bull bull bull 3 12 Lowest order QED Vacuum Polarization bullbull 6 13 Axions mix with pi zero and eta bullbullbull 20 14 Axions decay through a triangle anomaly bull 22 15 Axion induced rotation bullbullbullbullbullbullbull 27 16 Axion induced ellipticity bullbullbullbullbullbull 28 17 Axion limits from assumed rotation bullbullbullbullbull 30 18 Axion limits from assumed ellipticity bullbullbull 31
Chapter 2 21 Experiment layout bull bull bull bull bull bull bull bull bull bull 35 22 optics inside the enclosure bullbullbull 36 23 Cross section of a CBA magnet bullbullbull 37 24 Lead pot end cap and the two magnets bullbullbull 38 25 Magnet quench current vs temperature bullbullbull 39 26 LM0014 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 42 27 LM0018 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbull 43 28 stray magnetic field bullbullbullbullbullbullbullbull 44 29 Atomic Ar levels 47 210 A typical Ar ion laser resonator bullbullbullbullbull 52 211 QWP and elliptically polarized light bull 57 212 Support tubes for the Faraday cell 60 213 Lissajous pattern bullbullbullbullbullbullbullbullbullbull 65 214 Time delay measurements bullbullbullbullbullbullbullbullbullbullbull 67 215 Birefringence axis setup bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 69 216 Circle for birefringence axis bullbullbullbullbullbullbullbullbullbull 70 217 First readout with one mirror rotated bullbullbull 72 218 Birefringence axes of a mirror bullbullbullbullbullbull 74 219 Parameters of a paraxial ray bullbullbullbullbullbullbullbullbullbullbullbullbull 75 220 Stable regions of mirrors resonator bullbullbullbullbull 77 221 Telescope lens setup bullbullbullbullbullbullbullbullbullbullbull 81
vii
Chapter 3 31 Electronics setup of the experiment 95 32 EGampG low noise preamplifier characteristics 98 33 Laser light spectral density bull 105 34 Nominal noise reduction vs frequency bull 108 35 Typical data with LPC OFF bullbullbullbullbull 109 36 Typical data with LPC ON bull 110 37 lf noise reduction with LPC ON bullbullbull 111
Chapter 4 41 lgain squared vs frequency bullbullbullbullbull 118 42 Noise sources vs amount of light bull 119 43 Single record rotation data bullbullbull 125 44 RMS average of 26 files 126 45 vector average of 26 files bullbullbullbullbull 127 46 Typical rotation data with magnets ON 130 47 Shunt mirror data with magnets ON bull 132 48 Cavity mirrors move due to the magnet motion 134 49 RotationEllipticity limits from E840 bullbullbullbull 142
Chapter 5 51 Magnet modulation at 78 mHz 147 52 Fourier analysis of the magnet current 148 53 Typical Nitrogen gas data 149 54 Vacuum ellipticity data 150 55 Ellipticity vs Nitrogen gas pressure 152 56 Ellipticity vs Ar gas pressure middot 153 57 Ellipticity vs Ne gas pressure middot 154 58 Ellipticity vs He gas pressure middot 155
viii
List of Tables
Chapter 2 21 Effective magnetic field length 22 Laser performance parameter specifications 23 Laser specifications (Innova 9o_5) bullbullbullbullbullbullbullbullbullbull
24 Laser output power specifications bullbullbull 25 REP and LEP light through a QWP bullbullbullbull 26 Ray transfer matrices 27 Jones vectors 28 Jones matrices
Chapter 3
31 Silicon photodiode 32 Data taking sheet
Chapter 4
41 Rotation data 42 Ellipticity data bullbullbullbullbullbull
Chapter 5 51 Cotton-Mouton constants 52 Various parameters for the gases 53 Radius of first excited state of some gases bull
41 49
50
51 58 82 85 86
96
112
137 141
157 158 158
ix
1012
Chapter 1 Theoretical Motivation
11 Introduction
One of the most important problems in physics today
is the behavior of the interactions of the elementary
particles at very high energies in the range 10 2 to 10 19
GeV However present and even future particle
accelerators will be able to explore only the first few
decades of this range Thus we will have to rely on
indirect information by studying the relics of the very
early universe where such energies existed for a very
brief time One such particle created at an energy of
GeV is the axion which is a pseudoscalar with very
light mass ma~10-5 eVe This makes it possible to search
for the axion either in the cosmic radiation1 or to attempt
to directly produce it in the laboratory This thesis
is an experimental search for axions and for analogous
light scalar particles We have not observed axions but
have set a limit on their coupling to two photons or
equivalent to two quarks
The search for axions was based on the effect that
they would have on the propagation of light in a magnetic
field in vacuum This can be understood by considering
1
the coupling of the axion to two photons as shown in figure
1lb Since a magnetic field is equivalent to a cloud
of virtual photons a real photon from a 1ase- bea can
scatter off the virtual photons to produce an axion 2 The
production of the axion will affect the polarization of
the incident beam as explained in detail in section 13
We have therefore constructed a very sensitive
polarimeter or ellipsometer and have achieved a
sensitivity in the change of polarization of order 10-10
rad The polarimeter is described in chapters 2 and 3
and I give here only a brief sketch
The photon source is a 2W (single line power) Argon-ion
laser operating at ~ = 514 nm High quality polarizers
are used to establish and to analyze the polarization and
are set for best extinction The magnetic field is as
high as possible 22 kgauss in the present investigation
and extends over -10m To enhance the effect the light
is made to travel repeatedly through the field region
This is achieved by forming an optical cavity between two
mirrors and deforming one of the mirrors we have reached
790 traversals
Since the signal is so weak it is important to measure
the change in the photon field amplitude rather than
intensity This can be done by interfering the effect
from the cavity with a deliberately introduced rotation
in our case this is done with a Faraday cell For
2
a 11M Photon
(a)
Photon 11M a
(b)80
Figure 11 Primakoff effect a) Decay ofaxions inside a magnetic field
b) Production ofaxions inside a magnetic field
3
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
Finally my special thanks go to my wife Georgia
Afxendiou-semertzidis for her support and understanding
but most of all for sharing with me the excitement of thesis
writing
This work was supported by the us Department of Energy
under contracts DE-AC02-76CH00016 and DE-AC02-76ER13065
iv
curriculum vitae
Yannis Semertzidis was born on September 16 1961 He
attended the Aristoteleio Panepistimio Thessalonikis
(Aristotle University of Thessaloniki) in Greece from 1979
to 1984 where he received a Bachelor of Science Degree in
Physics May 1985 he began graduate studies at the University
of Rochester In 1987 he was a recipient of the Furth
Fellowship administered by the University His research
work has included the CP Violation experiment at LEARCERN
during the summer of 1983 and from spring 1984 to summer
1984 Fermilab experiment 723 a search for anomalous forces
at highly relativistic velocities from May 1985 to December
1985i BNL experiment 805 the search for Galactic axions
from January 1986 to spring 1989 Subsequently he has worked
on BNL experiment 840 a coherent production of any
pseudoscalar (or scalar) that couples to two photons this
was from summer 1987 to the present This work was supervised
by Dr A C Melissinos
v
Table of contents
Abstract ii Acknowledgments iii Curriculum vitae v
List of Figures vii List of Tables ix
Chapter 1 Theoretical Motivation 1 11 Introduction 1 12 QED Vacuum Polarization 10 13 Axion 16
Chapter 2 Apparatus 34 21 Magnets 34 22 Laser 45 23 optics 54 24 Cavity 63 25 Ray Transfer 73 26 Telescope 78 27 Jones Vectors and Matrices 84
Chapter 3 Data Acquisition 93
31 Electronics 93
32 Misalignment Correction 101 33 Laser Power Stabilizer 104
Chapter 4 Analysis of Results 113 41 Noise Sources 113 42 Data Analysis 123
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases 144
Chapter 6 Conclusions 160
Index 168
vi
List of Fiqures
Chapter 1 11 Primakoff effect bull bull bull bull bull bull bull bull bull bull bull bull 3 12 Lowest order QED Vacuum Polarization bullbull 6 13 Axions mix with pi zero and eta bullbullbull 20 14 Axions decay through a triangle anomaly bull 22 15 Axion induced rotation bullbullbullbullbullbullbull 27 16 Axion induced ellipticity bullbullbullbullbullbull 28 17 Axion limits from assumed rotation bullbullbullbullbull 30 18 Axion limits from assumed ellipticity bullbullbull 31
Chapter 2 21 Experiment layout bull bull bull bull bull bull bull bull bull bull 35 22 optics inside the enclosure bullbullbull 36 23 Cross section of a CBA magnet bullbullbull 37 24 Lead pot end cap and the two magnets bullbullbull 38 25 Magnet quench current vs temperature bullbullbull 39 26 LM0014 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 42 27 LM0018 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbull 43 28 stray magnetic field bullbullbullbullbullbullbullbull 44 29 Atomic Ar levels 47 210 A typical Ar ion laser resonator bullbullbullbullbull 52 211 QWP and elliptically polarized light bull 57 212 Support tubes for the Faraday cell 60 213 Lissajous pattern bullbullbullbullbullbullbullbullbullbull 65 214 Time delay measurements bullbullbullbullbullbullbullbullbullbullbull 67 215 Birefringence axis setup bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 69 216 Circle for birefringence axis bullbullbullbullbullbullbullbullbullbull 70 217 First readout with one mirror rotated bullbullbull 72 218 Birefringence axes of a mirror bullbullbullbullbullbull 74 219 Parameters of a paraxial ray bullbullbullbullbullbullbullbullbullbullbullbullbull 75 220 Stable regions of mirrors resonator bullbullbullbullbull 77 221 Telescope lens setup bullbullbullbullbullbullbullbullbullbullbull 81
vii
Chapter 3 31 Electronics setup of the experiment 95 32 EGampG low noise preamplifier characteristics 98 33 Laser light spectral density bull 105 34 Nominal noise reduction vs frequency bull 108 35 Typical data with LPC OFF bullbullbullbullbull 109 36 Typical data with LPC ON bull 110 37 lf noise reduction with LPC ON bullbullbull 111
Chapter 4 41 lgain squared vs frequency bullbullbullbullbull 118 42 Noise sources vs amount of light bull 119 43 Single record rotation data bullbullbull 125 44 RMS average of 26 files 126 45 vector average of 26 files bullbullbullbullbull 127 46 Typical rotation data with magnets ON 130 47 Shunt mirror data with magnets ON bull 132 48 Cavity mirrors move due to the magnet motion 134 49 RotationEllipticity limits from E840 bullbullbullbull 142
Chapter 5 51 Magnet modulation at 78 mHz 147 52 Fourier analysis of the magnet current 148 53 Typical Nitrogen gas data 149 54 Vacuum ellipticity data 150 55 Ellipticity vs Nitrogen gas pressure 152 56 Ellipticity vs Ar gas pressure middot 153 57 Ellipticity vs Ne gas pressure middot 154 58 Ellipticity vs He gas pressure middot 155
viii
List of Tables
Chapter 2 21 Effective magnetic field length 22 Laser performance parameter specifications 23 Laser specifications (Innova 9o_5) bullbullbullbullbullbullbullbullbullbull
24 Laser output power specifications bullbullbull 25 REP and LEP light through a QWP bullbullbullbull 26 Ray transfer matrices 27 Jones vectors 28 Jones matrices
Chapter 3
31 Silicon photodiode 32 Data taking sheet
Chapter 4
41 Rotation data 42 Ellipticity data bullbullbullbullbullbull
Chapter 5 51 Cotton-Mouton constants 52 Various parameters for the gases 53 Radius of first excited state of some gases bull
41 49
50
51 58 82 85 86
96
112
137 141
157 158 158
ix
1012
Chapter 1 Theoretical Motivation
11 Introduction
One of the most important problems in physics today
is the behavior of the interactions of the elementary
particles at very high energies in the range 10 2 to 10 19
GeV However present and even future particle
accelerators will be able to explore only the first few
decades of this range Thus we will have to rely on
indirect information by studying the relics of the very
early universe where such energies existed for a very
brief time One such particle created at an energy of
GeV is the axion which is a pseudoscalar with very
light mass ma~10-5 eVe This makes it possible to search
for the axion either in the cosmic radiation1 or to attempt
to directly produce it in the laboratory This thesis
is an experimental search for axions and for analogous
light scalar particles We have not observed axions but
have set a limit on their coupling to two photons or
equivalent to two quarks
The search for axions was based on the effect that
they would have on the propagation of light in a magnetic
field in vacuum This can be understood by considering
1
the coupling of the axion to two photons as shown in figure
1lb Since a magnetic field is equivalent to a cloud
of virtual photons a real photon from a 1ase- bea can
scatter off the virtual photons to produce an axion 2 The
production of the axion will affect the polarization of
the incident beam as explained in detail in section 13
We have therefore constructed a very sensitive
polarimeter or ellipsometer and have achieved a
sensitivity in the change of polarization of order 10-10
rad The polarimeter is described in chapters 2 and 3
and I give here only a brief sketch
The photon source is a 2W (single line power) Argon-ion
laser operating at ~ = 514 nm High quality polarizers
are used to establish and to analyze the polarization and
are set for best extinction The magnetic field is as
high as possible 22 kgauss in the present investigation
and extends over -10m To enhance the effect the light
is made to travel repeatedly through the field region
This is achieved by forming an optical cavity between two
mirrors and deforming one of the mirrors we have reached
790 traversals
Since the signal is so weak it is important to measure
the change in the photon field amplitude rather than
intensity This can be done by interfering the effect
from the cavity with a deliberately introduced rotation
in our case this is done with a Faraday cell For
2
a 11M Photon
(a)
Photon 11M a
(b)80
Figure 11 Primakoff effect a) Decay ofaxions inside a magnetic field
b) Production ofaxions inside a magnetic field
3
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
curriculum vitae
Yannis Semertzidis was born on September 16 1961 He
attended the Aristoteleio Panepistimio Thessalonikis
(Aristotle University of Thessaloniki) in Greece from 1979
to 1984 where he received a Bachelor of Science Degree in
Physics May 1985 he began graduate studies at the University
of Rochester In 1987 he was a recipient of the Furth
Fellowship administered by the University His research
work has included the CP Violation experiment at LEARCERN
during the summer of 1983 and from spring 1984 to summer
1984 Fermilab experiment 723 a search for anomalous forces
at highly relativistic velocities from May 1985 to December
1985i BNL experiment 805 the search for Galactic axions
from January 1986 to spring 1989 Subsequently he has worked
on BNL experiment 840 a coherent production of any
pseudoscalar (or scalar) that couples to two photons this
was from summer 1987 to the present This work was supervised
by Dr A C Melissinos
v
Table of contents
Abstract ii Acknowledgments iii Curriculum vitae v
List of Figures vii List of Tables ix
Chapter 1 Theoretical Motivation 1 11 Introduction 1 12 QED Vacuum Polarization 10 13 Axion 16
Chapter 2 Apparatus 34 21 Magnets 34 22 Laser 45 23 optics 54 24 Cavity 63 25 Ray Transfer 73 26 Telescope 78 27 Jones Vectors and Matrices 84
Chapter 3 Data Acquisition 93
31 Electronics 93
32 Misalignment Correction 101 33 Laser Power Stabilizer 104
Chapter 4 Analysis of Results 113 41 Noise Sources 113 42 Data Analysis 123
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases 144
Chapter 6 Conclusions 160
Index 168
vi
List of Fiqures
Chapter 1 11 Primakoff effect bull bull bull bull bull bull bull bull bull bull bull bull 3 12 Lowest order QED Vacuum Polarization bullbull 6 13 Axions mix with pi zero and eta bullbullbull 20 14 Axions decay through a triangle anomaly bull 22 15 Axion induced rotation bullbullbullbullbullbullbull 27 16 Axion induced ellipticity bullbullbullbullbullbull 28 17 Axion limits from assumed rotation bullbullbullbullbull 30 18 Axion limits from assumed ellipticity bullbullbull 31
Chapter 2 21 Experiment layout bull bull bull bull bull bull bull bull bull bull 35 22 optics inside the enclosure bullbullbull 36 23 Cross section of a CBA magnet bullbullbull 37 24 Lead pot end cap and the two magnets bullbullbull 38 25 Magnet quench current vs temperature bullbullbull 39 26 LM0014 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 42 27 LM0018 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbull 43 28 stray magnetic field bullbullbullbullbullbullbullbull 44 29 Atomic Ar levels 47 210 A typical Ar ion laser resonator bullbullbullbullbull 52 211 QWP and elliptically polarized light bull 57 212 Support tubes for the Faraday cell 60 213 Lissajous pattern bullbullbullbullbullbullbullbullbullbull 65 214 Time delay measurements bullbullbullbullbullbullbullbullbullbullbull 67 215 Birefringence axis setup bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 69 216 Circle for birefringence axis bullbullbullbullbullbullbullbullbullbull 70 217 First readout with one mirror rotated bullbullbull 72 218 Birefringence axes of a mirror bullbullbullbullbullbull 74 219 Parameters of a paraxial ray bullbullbullbullbullbullbullbullbullbullbullbullbull 75 220 Stable regions of mirrors resonator bullbullbullbullbull 77 221 Telescope lens setup bullbullbullbullbullbullbullbullbullbullbull 81
vii
Chapter 3 31 Electronics setup of the experiment 95 32 EGampG low noise preamplifier characteristics 98 33 Laser light spectral density bull 105 34 Nominal noise reduction vs frequency bull 108 35 Typical data with LPC OFF bullbullbullbullbull 109 36 Typical data with LPC ON bull 110 37 lf noise reduction with LPC ON bullbullbull 111
Chapter 4 41 lgain squared vs frequency bullbullbullbullbull 118 42 Noise sources vs amount of light bull 119 43 Single record rotation data bullbullbull 125 44 RMS average of 26 files 126 45 vector average of 26 files bullbullbullbullbull 127 46 Typical rotation data with magnets ON 130 47 Shunt mirror data with magnets ON bull 132 48 Cavity mirrors move due to the magnet motion 134 49 RotationEllipticity limits from E840 bullbullbullbull 142
Chapter 5 51 Magnet modulation at 78 mHz 147 52 Fourier analysis of the magnet current 148 53 Typical Nitrogen gas data 149 54 Vacuum ellipticity data 150 55 Ellipticity vs Nitrogen gas pressure 152 56 Ellipticity vs Ar gas pressure middot 153 57 Ellipticity vs Ne gas pressure middot 154 58 Ellipticity vs He gas pressure middot 155
viii
List of Tables
Chapter 2 21 Effective magnetic field length 22 Laser performance parameter specifications 23 Laser specifications (Innova 9o_5) bullbullbullbullbullbullbullbullbullbull
24 Laser output power specifications bullbullbull 25 REP and LEP light through a QWP bullbullbullbull 26 Ray transfer matrices 27 Jones vectors 28 Jones matrices
Chapter 3
31 Silicon photodiode 32 Data taking sheet
Chapter 4
41 Rotation data 42 Ellipticity data bullbullbullbullbullbull
Chapter 5 51 Cotton-Mouton constants 52 Various parameters for the gases 53 Radius of first excited state of some gases bull
41 49
50
51 58 82 85 86
96
112
137 141
157 158 158
ix
1012
Chapter 1 Theoretical Motivation
11 Introduction
One of the most important problems in physics today
is the behavior of the interactions of the elementary
particles at very high energies in the range 10 2 to 10 19
GeV However present and even future particle
accelerators will be able to explore only the first few
decades of this range Thus we will have to rely on
indirect information by studying the relics of the very
early universe where such energies existed for a very
brief time One such particle created at an energy of
GeV is the axion which is a pseudoscalar with very
light mass ma~10-5 eVe This makes it possible to search
for the axion either in the cosmic radiation1 or to attempt
to directly produce it in the laboratory This thesis
is an experimental search for axions and for analogous
light scalar particles We have not observed axions but
have set a limit on their coupling to two photons or
equivalent to two quarks
The search for axions was based on the effect that
they would have on the propagation of light in a magnetic
field in vacuum This can be understood by considering
1
the coupling of the axion to two photons as shown in figure
1lb Since a magnetic field is equivalent to a cloud
of virtual photons a real photon from a 1ase- bea can
scatter off the virtual photons to produce an axion 2 The
production of the axion will affect the polarization of
the incident beam as explained in detail in section 13
We have therefore constructed a very sensitive
polarimeter or ellipsometer and have achieved a
sensitivity in the change of polarization of order 10-10
rad The polarimeter is described in chapters 2 and 3
and I give here only a brief sketch
The photon source is a 2W (single line power) Argon-ion
laser operating at ~ = 514 nm High quality polarizers
are used to establish and to analyze the polarization and
are set for best extinction The magnetic field is as
high as possible 22 kgauss in the present investigation
and extends over -10m To enhance the effect the light
is made to travel repeatedly through the field region
This is achieved by forming an optical cavity between two
mirrors and deforming one of the mirrors we have reached
790 traversals
Since the signal is so weak it is important to measure
the change in the photon field amplitude rather than
intensity This can be done by interfering the effect
from the cavity with a deliberately introduced rotation
in our case this is done with a Faraday cell For
2
a 11M Photon
(a)
Photon 11M a
(b)80
Figure 11 Primakoff effect a) Decay ofaxions inside a magnetic field
b) Production ofaxions inside a magnetic field
3
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
Table of contents
Abstract ii Acknowledgments iii Curriculum vitae v
List of Figures vii List of Tables ix
Chapter 1 Theoretical Motivation 1 11 Introduction 1 12 QED Vacuum Polarization 10 13 Axion 16
Chapter 2 Apparatus 34 21 Magnets 34 22 Laser 45 23 optics 54 24 Cavity 63 25 Ray Transfer 73 26 Telescope 78 27 Jones Vectors and Matrices 84
Chapter 3 Data Acquisition 93
31 Electronics 93
32 Misalignment Correction 101 33 Laser Power Stabilizer 104
Chapter 4 Analysis of Results 113 41 Noise Sources 113 42 Data Analysis 123
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases 144
Chapter 6 Conclusions 160
Index 168
vi
List of Fiqures
Chapter 1 11 Primakoff effect bull bull bull bull bull bull bull bull bull bull bull bull 3 12 Lowest order QED Vacuum Polarization bullbull 6 13 Axions mix with pi zero and eta bullbullbull 20 14 Axions decay through a triangle anomaly bull 22 15 Axion induced rotation bullbullbullbullbullbullbull 27 16 Axion induced ellipticity bullbullbullbullbullbull 28 17 Axion limits from assumed rotation bullbullbullbullbull 30 18 Axion limits from assumed ellipticity bullbullbull 31
Chapter 2 21 Experiment layout bull bull bull bull bull bull bull bull bull bull 35 22 optics inside the enclosure bullbullbull 36 23 Cross section of a CBA magnet bullbullbull 37 24 Lead pot end cap and the two magnets bullbullbull 38 25 Magnet quench current vs temperature bullbullbull 39 26 LM0014 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 42 27 LM0018 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbull 43 28 stray magnetic field bullbullbullbullbullbullbullbull 44 29 Atomic Ar levels 47 210 A typical Ar ion laser resonator bullbullbullbullbull 52 211 QWP and elliptically polarized light bull 57 212 Support tubes for the Faraday cell 60 213 Lissajous pattern bullbullbullbullbullbullbullbullbullbull 65 214 Time delay measurements bullbullbullbullbullbullbullbullbullbullbull 67 215 Birefringence axis setup bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 69 216 Circle for birefringence axis bullbullbullbullbullbullbullbullbullbull 70 217 First readout with one mirror rotated bullbullbull 72 218 Birefringence axes of a mirror bullbullbullbullbullbull 74 219 Parameters of a paraxial ray bullbullbullbullbullbullbullbullbullbullbullbullbull 75 220 Stable regions of mirrors resonator bullbullbullbullbull 77 221 Telescope lens setup bullbullbullbullbullbullbullbullbullbullbull 81
vii
Chapter 3 31 Electronics setup of the experiment 95 32 EGampG low noise preamplifier characteristics 98 33 Laser light spectral density bull 105 34 Nominal noise reduction vs frequency bull 108 35 Typical data with LPC OFF bullbullbullbullbull 109 36 Typical data with LPC ON bull 110 37 lf noise reduction with LPC ON bullbullbull 111
Chapter 4 41 lgain squared vs frequency bullbullbullbullbull 118 42 Noise sources vs amount of light bull 119 43 Single record rotation data bullbullbull 125 44 RMS average of 26 files 126 45 vector average of 26 files bullbullbullbullbull 127 46 Typical rotation data with magnets ON 130 47 Shunt mirror data with magnets ON bull 132 48 Cavity mirrors move due to the magnet motion 134 49 RotationEllipticity limits from E840 bullbullbullbull 142
Chapter 5 51 Magnet modulation at 78 mHz 147 52 Fourier analysis of the magnet current 148 53 Typical Nitrogen gas data 149 54 Vacuum ellipticity data 150 55 Ellipticity vs Nitrogen gas pressure 152 56 Ellipticity vs Ar gas pressure middot 153 57 Ellipticity vs Ne gas pressure middot 154 58 Ellipticity vs He gas pressure middot 155
viii
List of Tables
Chapter 2 21 Effective magnetic field length 22 Laser performance parameter specifications 23 Laser specifications (Innova 9o_5) bullbullbullbullbullbullbullbullbullbull
24 Laser output power specifications bullbullbull 25 REP and LEP light through a QWP bullbullbullbull 26 Ray transfer matrices 27 Jones vectors 28 Jones matrices
Chapter 3
31 Silicon photodiode 32 Data taking sheet
Chapter 4
41 Rotation data 42 Ellipticity data bullbullbullbullbullbull
Chapter 5 51 Cotton-Mouton constants 52 Various parameters for the gases 53 Radius of first excited state of some gases bull
41 49
50
51 58 82 85 86
96
112
137 141
157 158 158
ix
1012
Chapter 1 Theoretical Motivation
11 Introduction
One of the most important problems in physics today
is the behavior of the interactions of the elementary
particles at very high energies in the range 10 2 to 10 19
GeV However present and even future particle
accelerators will be able to explore only the first few
decades of this range Thus we will have to rely on
indirect information by studying the relics of the very
early universe where such energies existed for a very
brief time One such particle created at an energy of
GeV is the axion which is a pseudoscalar with very
light mass ma~10-5 eVe This makes it possible to search
for the axion either in the cosmic radiation1 or to attempt
to directly produce it in the laboratory This thesis
is an experimental search for axions and for analogous
light scalar particles We have not observed axions but
have set a limit on their coupling to two photons or
equivalent to two quarks
The search for axions was based on the effect that
they would have on the propagation of light in a magnetic
field in vacuum This can be understood by considering
1
the coupling of the axion to two photons as shown in figure
1lb Since a magnetic field is equivalent to a cloud
of virtual photons a real photon from a 1ase- bea can
scatter off the virtual photons to produce an axion 2 The
production of the axion will affect the polarization of
the incident beam as explained in detail in section 13
We have therefore constructed a very sensitive
polarimeter or ellipsometer and have achieved a
sensitivity in the change of polarization of order 10-10
rad The polarimeter is described in chapters 2 and 3
and I give here only a brief sketch
The photon source is a 2W (single line power) Argon-ion
laser operating at ~ = 514 nm High quality polarizers
are used to establish and to analyze the polarization and
are set for best extinction The magnetic field is as
high as possible 22 kgauss in the present investigation
and extends over -10m To enhance the effect the light
is made to travel repeatedly through the field region
This is achieved by forming an optical cavity between two
mirrors and deforming one of the mirrors we have reached
790 traversals
Since the signal is so weak it is important to measure
the change in the photon field amplitude rather than
intensity This can be done by interfering the effect
from the cavity with a deliberately introduced rotation
in our case this is done with a Faraday cell For
2
a 11M Photon
(a)
Photon 11M a
(b)80
Figure 11 Primakoff effect a) Decay ofaxions inside a magnetic field
b) Production ofaxions inside a magnetic field
3
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
List of Fiqures
Chapter 1 11 Primakoff effect bull bull bull bull bull bull bull bull bull bull bull bull 3 12 Lowest order QED Vacuum Polarization bullbull 6 13 Axions mix with pi zero and eta bullbullbull 20 14 Axions decay through a triangle anomaly bull 22 15 Axion induced rotation bullbullbullbullbullbullbull 27 16 Axion induced ellipticity bullbullbullbullbullbull 28 17 Axion limits from assumed rotation bullbullbullbullbull 30 18 Axion limits from assumed ellipticity bullbullbull 31
Chapter 2 21 Experiment layout bull bull bull bull bull bull bull bull bull bull 35 22 optics inside the enclosure bullbullbull 36 23 Cross section of a CBA magnet bullbullbull 37 24 Lead pot end cap and the two magnets bullbullbull 38 25 Magnet quench current vs temperature bullbullbull 39 26 LM0014 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 42 27 LM0018 transfer function bullbullbullbullbullbullbullbullbullbullbullbullbull 43 28 stray magnetic field bullbullbullbullbullbullbullbull 44 29 Atomic Ar levels 47 210 A typical Ar ion laser resonator bullbullbullbullbull 52 211 QWP and elliptically polarized light bull 57 212 Support tubes for the Faraday cell 60 213 Lissajous pattern bullbullbullbullbullbullbullbullbullbull 65 214 Time delay measurements bullbullbullbullbullbullbullbullbullbullbull 67 215 Birefringence axis setup bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull 69 216 Circle for birefringence axis bullbullbullbullbullbullbullbullbullbull 70 217 First readout with one mirror rotated bullbullbull 72 218 Birefringence axes of a mirror bullbullbullbullbullbull 74 219 Parameters of a paraxial ray bullbullbullbullbullbullbullbullbullbullbullbullbull 75 220 Stable regions of mirrors resonator bullbullbullbullbull 77 221 Telescope lens setup bullbullbullbullbullbullbullbullbullbullbull 81
vii
Chapter 3 31 Electronics setup of the experiment 95 32 EGampG low noise preamplifier characteristics 98 33 Laser light spectral density bull 105 34 Nominal noise reduction vs frequency bull 108 35 Typical data with LPC OFF bullbullbullbullbull 109 36 Typical data with LPC ON bull 110 37 lf noise reduction with LPC ON bullbullbull 111
Chapter 4 41 lgain squared vs frequency bullbullbullbullbull 118 42 Noise sources vs amount of light bull 119 43 Single record rotation data bullbullbull 125 44 RMS average of 26 files 126 45 vector average of 26 files bullbullbullbullbull 127 46 Typical rotation data with magnets ON 130 47 Shunt mirror data with magnets ON bull 132 48 Cavity mirrors move due to the magnet motion 134 49 RotationEllipticity limits from E840 bullbullbullbull 142
Chapter 5 51 Magnet modulation at 78 mHz 147 52 Fourier analysis of the magnet current 148 53 Typical Nitrogen gas data 149 54 Vacuum ellipticity data 150 55 Ellipticity vs Nitrogen gas pressure 152 56 Ellipticity vs Ar gas pressure middot 153 57 Ellipticity vs Ne gas pressure middot 154 58 Ellipticity vs He gas pressure middot 155
viii
List of Tables
Chapter 2 21 Effective magnetic field length 22 Laser performance parameter specifications 23 Laser specifications (Innova 9o_5) bullbullbullbullbullbullbullbullbullbull
24 Laser output power specifications bullbullbull 25 REP and LEP light through a QWP bullbullbullbull 26 Ray transfer matrices 27 Jones vectors 28 Jones matrices
Chapter 3
31 Silicon photodiode 32 Data taking sheet
Chapter 4
41 Rotation data 42 Ellipticity data bullbullbullbullbullbull
Chapter 5 51 Cotton-Mouton constants 52 Various parameters for the gases 53 Radius of first excited state of some gases bull
41 49
50
51 58 82 85 86
96
112
137 141
157 158 158
ix
1012
Chapter 1 Theoretical Motivation
11 Introduction
One of the most important problems in physics today
is the behavior of the interactions of the elementary
particles at very high energies in the range 10 2 to 10 19
GeV However present and even future particle
accelerators will be able to explore only the first few
decades of this range Thus we will have to rely on
indirect information by studying the relics of the very
early universe where such energies existed for a very
brief time One such particle created at an energy of
GeV is the axion which is a pseudoscalar with very
light mass ma~10-5 eVe This makes it possible to search
for the axion either in the cosmic radiation1 or to attempt
to directly produce it in the laboratory This thesis
is an experimental search for axions and for analogous
light scalar particles We have not observed axions but
have set a limit on their coupling to two photons or
equivalent to two quarks
The search for axions was based on the effect that
they would have on the propagation of light in a magnetic
field in vacuum This can be understood by considering
1
the coupling of the axion to two photons as shown in figure
1lb Since a magnetic field is equivalent to a cloud
of virtual photons a real photon from a 1ase- bea can
scatter off the virtual photons to produce an axion 2 The
production of the axion will affect the polarization of
the incident beam as explained in detail in section 13
We have therefore constructed a very sensitive
polarimeter or ellipsometer and have achieved a
sensitivity in the change of polarization of order 10-10
rad The polarimeter is described in chapters 2 and 3
and I give here only a brief sketch
The photon source is a 2W (single line power) Argon-ion
laser operating at ~ = 514 nm High quality polarizers
are used to establish and to analyze the polarization and
are set for best extinction The magnetic field is as
high as possible 22 kgauss in the present investigation
and extends over -10m To enhance the effect the light
is made to travel repeatedly through the field region
This is achieved by forming an optical cavity between two
mirrors and deforming one of the mirrors we have reached
790 traversals
Since the signal is so weak it is important to measure
the change in the photon field amplitude rather than
intensity This can be done by interfering the effect
from the cavity with a deliberately introduced rotation
in our case this is done with a Faraday cell For
2
a 11M Photon
(a)
Photon 11M a
(b)80
Figure 11 Primakoff effect a) Decay ofaxions inside a magnetic field
b) Production ofaxions inside a magnetic field
3
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
Chapter 3 31 Electronics setup of the experiment 95 32 EGampG low noise preamplifier characteristics 98 33 Laser light spectral density bull 105 34 Nominal noise reduction vs frequency bull 108 35 Typical data with LPC OFF bullbullbullbullbull 109 36 Typical data with LPC ON bull 110 37 lf noise reduction with LPC ON bullbullbull 111
Chapter 4 41 lgain squared vs frequency bullbullbullbullbull 118 42 Noise sources vs amount of light bull 119 43 Single record rotation data bullbullbull 125 44 RMS average of 26 files 126 45 vector average of 26 files bullbullbullbullbull 127 46 Typical rotation data with magnets ON 130 47 Shunt mirror data with magnets ON bull 132 48 Cavity mirrors move due to the magnet motion 134 49 RotationEllipticity limits from E840 bullbullbullbull 142
Chapter 5 51 Magnet modulation at 78 mHz 147 52 Fourier analysis of the magnet current 148 53 Typical Nitrogen gas data 149 54 Vacuum ellipticity data 150 55 Ellipticity vs Nitrogen gas pressure 152 56 Ellipticity vs Ar gas pressure middot 153 57 Ellipticity vs Ne gas pressure middot 154 58 Ellipticity vs He gas pressure middot 155
viii
List of Tables
Chapter 2 21 Effective magnetic field length 22 Laser performance parameter specifications 23 Laser specifications (Innova 9o_5) bullbullbullbullbullbullbullbullbullbull
24 Laser output power specifications bullbullbull 25 REP and LEP light through a QWP bullbullbullbull 26 Ray transfer matrices 27 Jones vectors 28 Jones matrices
Chapter 3
31 Silicon photodiode 32 Data taking sheet
Chapter 4
41 Rotation data 42 Ellipticity data bullbullbullbullbullbull
Chapter 5 51 Cotton-Mouton constants 52 Various parameters for the gases 53 Radius of first excited state of some gases bull
41 49
50
51 58 82 85 86
96
112
137 141
157 158 158
ix
1012
Chapter 1 Theoretical Motivation
11 Introduction
One of the most important problems in physics today
is the behavior of the interactions of the elementary
particles at very high energies in the range 10 2 to 10 19
GeV However present and even future particle
accelerators will be able to explore only the first few
decades of this range Thus we will have to rely on
indirect information by studying the relics of the very
early universe where such energies existed for a very
brief time One such particle created at an energy of
GeV is the axion which is a pseudoscalar with very
light mass ma~10-5 eVe This makes it possible to search
for the axion either in the cosmic radiation1 or to attempt
to directly produce it in the laboratory This thesis
is an experimental search for axions and for analogous
light scalar particles We have not observed axions but
have set a limit on their coupling to two photons or
equivalent to two quarks
The search for axions was based on the effect that
they would have on the propagation of light in a magnetic
field in vacuum This can be understood by considering
1
the coupling of the axion to two photons as shown in figure
1lb Since a magnetic field is equivalent to a cloud
of virtual photons a real photon from a 1ase- bea can
scatter off the virtual photons to produce an axion 2 The
production of the axion will affect the polarization of
the incident beam as explained in detail in section 13
We have therefore constructed a very sensitive
polarimeter or ellipsometer and have achieved a
sensitivity in the change of polarization of order 10-10
rad The polarimeter is described in chapters 2 and 3
and I give here only a brief sketch
The photon source is a 2W (single line power) Argon-ion
laser operating at ~ = 514 nm High quality polarizers
are used to establish and to analyze the polarization and
are set for best extinction The magnetic field is as
high as possible 22 kgauss in the present investigation
and extends over -10m To enhance the effect the light
is made to travel repeatedly through the field region
This is achieved by forming an optical cavity between two
mirrors and deforming one of the mirrors we have reached
790 traversals
Since the signal is so weak it is important to measure
the change in the photon field amplitude rather than
intensity This can be done by interfering the effect
from the cavity with a deliberately introduced rotation
in our case this is done with a Faraday cell For
2
a 11M Photon
(a)
Photon 11M a
(b)80
Figure 11 Primakoff effect a) Decay ofaxions inside a magnetic field
b) Production ofaxions inside a magnetic field
3
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
List of Tables
Chapter 2 21 Effective magnetic field length 22 Laser performance parameter specifications 23 Laser specifications (Innova 9o_5) bullbullbullbullbullbullbullbullbullbull
24 Laser output power specifications bullbullbull 25 REP and LEP light through a QWP bullbullbullbull 26 Ray transfer matrices 27 Jones vectors 28 Jones matrices
Chapter 3
31 Silicon photodiode 32 Data taking sheet
Chapter 4
41 Rotation data 42 Ellipticity data bullbullbullbullbullbull
Chapter 5 51 Cotton-Mouton constants 52 Various parameters for the gases 53 Radius of first excited state of some gases bull
41 49
50
51 58 82 85 86
96
112
137 141
157 158 158
ix
1012
Chapter 1 Theoretical Motivation
11 Introduction
One of the most important problems in physics today
is the behavior of the interactions of the elementary
particles at very high energies in the range 10 2 to 10 19
GeV However present and even future particle
accelerators will be able to explore only the first few
decades of this range Thus we will have to rely on
indirect information by studying the relics of the very
early universe where such energies existed for a very
brief time One such particle created at an energy of
GeV is the axion which is a pseudoscalar with very
light mass ma~10-5 eVe This makes it possible to search
for the axion either in the cosmic radiation1 or to attempt
to directly produce it in the laboratory This thesis
is an experimental search for axions and for analogous
light scalar particles We have not observed axions but
have set a limit on their coupling to two photons or
equivalent to two quarks
The search for axions was based on the effect that
they would have on the propagation of light in a magnetic
field in vacuum This can be understood by considering
1
the coupling of the axion to two photons as shown in figure
1lb Since a magnetic field is equivalent to a cloud
of virtual photons a real photon from a 1ase- bea can
scatter off the virtual photons to produce an axion 2 The
production of the axion will affect the polarization of
the incident beam as explained in detail in section 13
We have therefore constructed a very sensitive
polarimeter or ellipsometer and have achieved a
sensitivity in the change of polarization of order 10-10
rad The polarimeter is described in chapters 2 and 3
and I give here only a brief sketch
The photon source is a 2W (single line power) Argon-ion
laser operating at ~ = 514 nm High quality polarizers
are used to establish and to analyze the polarization and
are set for best extinction The magnetic field is as
high as possible 22 kgauss in the present investigation
and extends over -10m To enhance the effect the light
is made to travel repeatedly through the field region
This is achieved by forming an optical cavity between two
mirrors and deforming one of the mirrors we have reached
790 traversals
Since the signal is so weak it is important to measure
the change in the photon field amplitude rather than
intensity This can be done by interfering the effect
from the cavity with a deliberately introduced rotation
in our case this is done with a Faraday cell For
2
a 11M Photon
(a)
Photon 11M a
(b)80
Figure 11 Primakoff effect a) Decay ofaxions inside a magnetic field
b) Production ofaxions inside a magnetic field
3
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
1012
Chapter 1 Theoretical Motivation
11 Introduction
One of the most important problems in physics today
is the behavior of the interactions of the elementary
particles at very high energies in the range 10 2 to 10 19
GeV However present and even future particle
accelerators will be able to explore only the first few
decades of this range Thus we will have to rely on
indirect information by studying the relics of the very
early universe where such energies existed for a very
brief time One such particle created at an energy of
GeV is the axion which is a pseudoscalar with very
light mass ma~10-5 eVe This makes it possible to search
for the axion either in the cosmic radiation1 or to attempt
to directly produce it in the laboratory This thesis
is an experimental search for axions and for analogous
light scalar particles We have not observed axions but
have set a limit on their coupling to two photons or
equivalent to two quarks
The search for axions was based on the effect that
they would have on the propagation of light in a magnetic
field in vacuum This can be understood by considering
1
the coupling of the axion to two photons as shown in figure
1lb Since a magnetic field is equivalent to a cloud
of virtual photons a real photon from a 1ase- bea can
scatter off the virtual photons to produce an axion 2 The
production of the axion will affect the polarization of
the incident beam as explained in detail in section 13
We have therefore constructed a very sensitive
polarimeter or ellipsometer and have achieved a
sensitivity in the change of polarization of order 10-10
rad The polarimeter is described in chapters 2 and 3
and I give here only a brief sketch
The photon source is a 2W (single line power) Argon-ion
laser operating at ~ = 514 nm High quality polarizers
are used to establish and to analyze the polarization and
are set for best extinction The magnetic field is as
high as possible 22 kgauss in the present investigation
and extends over -10m To enhance the effect the light
is made to travel repeatedly through the field region
This is achieved by forming an optical cavity between two
mirrors and deforming one of the mirrors we have reached
790 traversals
Since the signal is so weak it is important to measure
the change in the photon field amplitude rather than
intensity This can be done by interfering the effect
from the cavity with a deliberately introduced rotation
in our case this is done with a Faraday cell For
2
a 11M Photon
(a)
Photon 11M a
(b)80
Figure 11 Primakoff effect a) Decay ofaxions inside a magnetic field
b) Production ofaxions inside a magnetic field
3
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
the coupling of the axion to two photons as shown in figure
1lb Since a magnetic field is equivalent to a cloud
of virtual photons a real photon from a 1ase- bea can
scatter off the virtual photons to produce an axion 2 The
production of the axion will affect the polarization of
the incident beam as explained in detail in section 13
We have therefore constructed a very sensitive
polarimeter or ellipsometer and have achieved a
sensitivity in the change of polarization of order 10-10
rad The polarimeter is described in chapters 2 and 3
and I give here only a brief sketch
The photon source is a 2W (single line power) Argon-ion
laser operating at ~ = 514 nm High quality polarizers
are used to establish and to analyze the polarization and
are set for best extinction The magnetic field is as
high as possible 22 kgauss in the present investigation
and extends over -10m To enhance the effect the light
is made to travel repeatedly through the field region
This is achieved by forming an optical cavity between two
mirrors and deforming one of the mirrors we have reached
790 traversals
Since the signal is so weak it is important to measure
the change in the photon field amplitude rather than
intensity This can be done by interfering the effect
from the cavity with a deliberately introduced rotation
in our case this is done with a Faraday cell For
2
a 11M Photon
(a)
Photon 11M a
(b)80
Figure 11 Primakoff effect a) Decay ofaxions inside a magnetic field
b) Production ofaxions inside a magnetic field
3
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
a 11M Photon
(a)
Photon 11M a
(b)80
Figure 11 Primakoff effect a) Decay ofaxions inside a magnetic field
b) Production ofaxions inside a magnetic field
3
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
ellipticity measurements the sought effect is transformed
to a rotation by a suitable A4 plate Standard practice
in such small signal measurements is the modulation of
the signal at a well known fixed frequency this allows
detection in a very narrow frequency band thus reducing
noise and eliminating static effects Unfortunately the
high magnetic fields cannot be modulated rapidly we
reached 78 mHz in this investigation whereas the Faraday
cell was driven at 260 Hz
The whole apparatus and cavity were placed in a high
vacuum ranging from 10-4 Torr to 10-7 Torr and in a
constant temperature enclosure These precautions as well
as computer feedback on the key optical components are
absolutely necessary to reach the previously mentioned
sensitivity of 10-10 rad Furthermore the signal must
be Fourier analyzed the line width being of order 1 - 2
mHz This in turn sets the stability requirements of the
corresponding driving oscillators and receiver
electronics A typical Fourier spectrum had 400 channels
for a total width of -1 Hz For our instruments this
implied an acquisition time of order of 17 minutes per
record The records were written to disk and then further
analyzed (ie combined) off line on a VAX 11780 It
was thus possible to use very long effective integration
time Since we had phase information available we formed
vector averages this reduces the noise level as 1N
4
(where N is the number of records averaged) while the
signal remains constant
An important aspect of experiments that search for
small effects is the ability to calibrate them
Fortunately there is an atomic effect that induces an
ellipticity in light traversing gases subject to a
transverse magnetic field the Cotton-Mouton effect this
is not a Faraday rotation which involves a longitudinal
magnetic field By introducing a suitable gas in the
cavity and varying the pressure in the range of a few
Torr we are able to generate a known small ellipticity
and thus calibrate our device We used Nitrogen gas and
found that our apparatus was properly calibrated In view
of our improved sensitivity we were able to measure for
the first time the C-M coefficient of Neon and Helium and
to improve on the value for Argon These measurements
and their interpretation are discussed in chapter 5
The apparatus had been designed so that when its
ultimate sensitivity is reached it would be possible to
detect an important effect predicted by Quantum Electro
Dynamics (QED) This is photon-photon scattering as shown
in figure 12 where the incident photon scatters (twice)
from the magnetic field due to a virtual electron loop
This process known as Delbruck scattering has been
predicted 50 years ago but has never been observed with
visible photons where its interpretation is unambiguous
5
k k
B
Figure 12 The lowest order graph giving rise to dispersive effects
6
For the design parameters of the apparatus the ellipticity
induced by Delbruck scattering is 5x10-12 rad This
necessitates a 5T field replacement of the Faraday cell
by an EO modulator and further refinements in noise
reduction and data acquisition The theoretical analysis
of Delbruck scattering and of its detection are given in
chapter 1
The experiment is installed at Brookhaven National
Laboratory because of the availability of high field
magnets We are using two dipoles which were built as
prototypes for the Colliding Beam Accelerator (CBA) and
have superconducting windings Thus the necessity for a
sizeable Helium refrigeration plant (200 W) and power
supply (I= 5 kA) which were provided by Brookhaven High
vacuum technology is also necessary since as already
discussed residual gas can give rise to ellipticity
induced by atomic effects In contrast the optics was
produced by the University of Rochester shops as well as
was some of the mechanical construction Such an
experiment would not have been possible without a laser
for a light source and this explains why no Delbruck
measurements have been attempted previously The laser
has been purchased commercially as were the Fast Fourier
Analyzers The experiment is a collaboration between the
University of Rochester Brookhaven Fermilab and the
University of Trieste It was approved in the Fall of
7
1987 and the installation was completed two years later
As soon as the first data were obtained we observed
a signal which exhibited the characteristic of vacuum
optical rotation as expected for an axion However by
rotating the polarization of the light from 45middot with
respect to the magnetic field to 0middot the signal remained
present indicating that it was not due to selective
absorption along the two field orientations (parallel and
orthogonal to the field) but due to a pure rotation This
background signal sets the present limit in our
experimental measurement which corresponds to ES 4 X 10-8
rad From the measured value of E we set a limit on the
axion coupling to two photons
1 1 gayylt M = 4x 10 5 GeV
This is the first laboratory measurement of the coupling
of a light pseudoscalar or scalar particle (ma s 10-3 eV)
to two photons It el iminates several models of
supersymmetric theories3 and places important constrains
on any future theory
To set our result in a broader context we note that
accelerator experiments that have searched for axions
through their decay a ~ e+e- or a -+ yy and in k -+ na set
limits at the level M c 102 GeV These limits are much
weaker than our results but are not restricted to the low
mass region On the other hand astrophysical arguments
8
from the evolution of the sun predict that M ~ 108 GeV
This level of sensitivity corresponds to a rotation of
10-11 rad which is well within reach of the apparatus
when full field will be available and its final
configuration reached
We have searched for the source of the spurious rotation
that we observe and quickly established that it originates
in the cavity and disappears when the shunt mirror is
used We also established that it is not induced by
residual gas it is not pressure dependent We then
observed that the endcaps holding the mirrors are moving
by -6 nm in phase with the magnetic field since a
multipass cavity induces a static rotation of the
polarization plane mirror motion can induce a
corresponding time-dependent signal Incidentally the
static rotation is a direct manifestation of Berrys
phase26 which is present for the classical EM field A
new version of the apparatus will provide better isolation
of the mirrors thereby eliminating this signal We are
also examining the possibility of halo scattering from
the magnetized walls of the vacuum chamber which could
induce an ellipticity and rotation
If we exclude the spurious signal the sensitivity of
the apparatus as obtained from a rotation measurement is
E = 6 x 10-10 rad which corresponds to 2 x 10-8 radJHz
Thus a measurement of 106s should yield e-2x 10- 11 rad
9
bull
Similar sensitivity is obtained for the ellipticity
Further reductions in the noise level by using an EO
modulator and a larger dynamic range in the detector will
yield the remaining factor of ten in sensitivity These
modifications of the apparatus are part of a future long
range program
12 QED Vacuum Polarization
When a photon beam is traveling in vacuum the two
polarization states have the same speed as a consequence
of the isotropy of space This is not true any more when
a magnetic field is present because of photon-photon
interactions (fig 12) The photon beam according to
Quantum Electrodynamics (QED) creates virtual e+e- pairs
which it reabsorbs in the course of traveling When the
magnetic field is present the time evolution of the e+eshy
pair depends on whether they lie in a plane parallel or
orthogonal to the magnetic field That is the vacuum
is not isotropic any more but there is an axis defined
by the external field The same is true for an electric
field but a strength of 3x l01V em would be required to
match the effect of a 10 Tesla field
This problem was first discussed in the 1930s and I
10
shall follow the formalism of the Euler-Heisenberg4
effective electromagnetic Lagrangian which contains the
classical fields and the quantum corrections due to vacuum
polarization This Lagrangian in the low frequency limit
(hwlaquo 2mec2) is
( 11 )
in unrationalized Gaussian units valid to fourth order
E and B are respectively the total eleotric and magnetic a2 (ftlmc)3
fields and A=- 2 133 x l0- 32 em 3 erg is a constant90n mc
where me is the electron mass Any fermion pair can be
used instead of the e+e- pair with the proper replacement
of the charge and mass values in the calculation of the
constant A Equation 11 holds for fields that are well
below the critical ones Le E cr 10 15 V em and
Bcr 441 x lOl3gauss
Since the average polarizations of matter are
represented by the electric displacement D and the magnetic
field H we will use them to derive the indices of
refraction in vacuum 4
oL oLD=4nshy H=-4nshy ( 12)
oE oB
For a polarized laser beam (with Eph and Bph being the
electric and magnetic vectors of the laser beam) of a few
watts and beam diameter of a few millimeters 5 propagating
11
along z inside a magnetic field Bext and assuming Eph is
in the x direction we have
E = E e ei(kz-wt) B -+- B A i(kz-wt) - B -+- B ph ph x B -
-ext pheye - Qxt ph ( 13)
Then utilizing only linear terms of Eph and Bph
D -= 4 n ( iJ ~~h ex) = [E ph - 4 A B XI E ph -+- 14 A ( E B ext) ( B axt bull ex) ]ex (14 )
and using the formula D=E E=
( 1 c- E = 1 - 4 A B xt -+- 14 A ( ex B ex ) 2 )
Similarly
iJL 2H -4n-=B-4AB B=
iJB
B - 4 A [ B xc B ext -+- B h B QXC -+- 2 ( B ext B ph ) B xl -+shy
(16)
where in the above we have used the relation H=Hext+Hph
Using the relation Hph - ~ -1 Bph we obtain
(17 )
Now let us assume that the external magnetic field is
in the plane defined by the electric and magnetic vectors
of the laser beam Then we can distinguish two cases
1) Eph parallel to Bext From (15) and (17) we have
E = 1 - 4 A B xt -+- 1 4 A B xt = 1 -+- lOA B xt 2 - J 2
~ = ( 1 - 4 A B ext) =lt 1 -+- 4 A B ext
12
The parallel index of refraction is
~ 2 2 4 12n p = V Ell = ( 1 + 14 A B Qxt + 40 A B ext ) ~
n p 1 + 7 A B xt ( 18)
2) Eph orthogonal to Bext Again from (15) and (17)
we have
E == 1 - 4 A B ~xt and
The orthogonal index of refraction is
( 19)
Therefore we have
n p = 1 + 7 A B xt
(l10)
that is the vacuum becomes birefringent for B oxt t O
In case the magnetic field does not lie in the Eph
Bph plane Bext is the projection of the external magnetic
field on that plane4 and the more general formulas
n p == 1 + 7 A B XI si n 2 e
(l11)
apply where (
k)2 2 e 1 B axt bull sin = - bull B extk
and k is the photon wave-vector
Equation 111 is accurate to better than 01 for
13
B ex SO 1B cr 6 For magnetic fields of 01 SBwBuS 10 a
detailed calculation is given in reference 6 The
polarized laser beam entering the magnetic field region
with polarization (ie electric vector) at a 45deg angle
with respect to the magnetic field will acquire an
ellipticity given by
( 112)
where L is the length of the path of the laser light inside
the magnetic field region and A the laser wavelength The
phase retardation between the parallel and orthogonal
components is cent = 21jJ (ellipticity ljJ is defined as the ratio
of the semiminor to the semimajor axis of the ellipse
traced by the electric vector) Here we have used sine =
1 In these units (natural unrationalized Gaussian) 1
gauss can be expressed as 691-10-2 eV2 or 104 (ergcm3 ) 12
and 1 cm as 5104 eV-1 For our experimental parameters
A = 5145 nm (green light from an argon ion laser)
L = 104 m
and Bext = 5 Tesla (= 50 kgauss)
we get
( 113)
As we see the induced ellipticity is very small and
this effect can safely be ignored in the usual treatment
of electromagnetism in other words it is correct to use
14
the linear Maxwell equations in everyday life) This
is not true any more when regions with high magnetic fields
are considered as for the surface of neutron stars where
magnetic fields of the order of 1012 gauss are present
In fact Novick et al 7 proposed to use the degree of
ellipticity of the X-rays emitted from a neutron star
in order to distinguish between X-rays coming from the
surface and those coming from regions well within the
surface
The QED vacuum polarization has never been observed
directly but there are indirect observations where the
=-Il-=(1165924plusmn85)X 10- 9
effect is a small part of other processes 8 One such
measurement is the 9 j1 - 2 experiment where the muon
anomaly is found to be 9
9 -2 U
exp 2
while the theoretical value is
= (1165921 plusmn 83) x 10- 9 U 1h = U QED + U hadr + U weak
where
U QED = (11658520 plusmn 19) x 10- 9
=(667plusmn81)X 10-9u hadr
=(21 plusmn06)X 10-9U weak
The photon-photon contribution to this anomaly is
15
which contributes about two parts in 104 to the whole
process and would be good to check in a more direct fashion
Another precision measurement involves forward
scattering10 of v-rays in the electric field of a nucleus
Here again the Delbruck scattering (ie photon-photon
interaction) contribution is a small part of the dominant
processes ie Thomson Rayleigh and nuclear resonant
scattering
13 AXion
The Quantum-Chromo-Dynamics (QCD) Lagrangian contains
the term
(114 )
where F QIlV is the gluon field and e is an angular parameter
This term violates parity (P) but conserves charge
conjugation (C) and therefore violates the combined CP
symmetry There is a series of vacua distinguished by
a topological number n which are the classical solutions
of the gauge equations in QCD but are not invariant under
all possible gauge transformations The state of the true
vacuum (ie the one that is gauge invariant) is
16
constructed by a linear combination of Ingt vacua and e
is usually defined as
lllG19gt = I e 1 ngt (115)
n--oo
9 is therefore a periodic variable with a period of 2~
The angle e is related to the electric dipole moment (EDM)
of neutronll which if finite would violate CP symmetry
We can show this as follows
Any electric dipole moment of the neutron must be
aligned with the spin vector (the only preferred axis)
If we apply the T operator (time reversal symmetry) the
EDM does not change sign whereas the spin does Therefore
the presence of an EDM would violate T symmetry since
the physical state of the neutron would change under T
reflection Because of the CPT theorem which requires
that every reasonable quantum field theory must be CPT
invariant the presence of an EDM implies that CP is also
violated Actually if the neutron has an EDM P symmetry
is violated as well as can be shown by arguments similar
as used for T We discuss the EDM of a neutron because
a strict experimental upper limit on it of 4x 10-25 e middot em
already exists12 this corresponds to a limit on 9 of
191lt 10-9 or 19-nllt 10-9
bull
Because the notion of fine tuning of theoretical
parameters is not popular it is believed that e must be
exactly 0 or ~ but in principle it could be anywhere
17
between 0 and 2~ Different values of e would correspond
to different theories with different coupling constants
and there would be no problem with the term of equation
114 except for the presence of strong CP violation
Peccei and Quinn13 first introduced the idea that 0 is
not a parameter of the theory but rather a dynamical
variable and that different values of e describe different
energy states of the vacua of the same theory Below the
QeD energy scale e naturally gets the value 0 because
the Lagrangian has an effective potential with minimum
at that value The Peccei-Quinn mechanism is an extension
of the standard SU(3)XSU(2)LXU(l) theory14 with two Higgs
doublets
(116)
The U(l)pQ axial symmetry corresponds to invariance
under the following transformations
iT)Vs -2iT) ltIgt ~u i ~ e u i centu e u
IT)VS e -2iT) ltIgtd i ~ e d i ltlgtd ~ d (117)
Le the Lagrangian is invariant under these phase
transformations In the above ui and di refer to the
SU(2) quarks and n is a rotation angle (some arbitrary
phase)
Weinberg15 and Wilczek16 realized that when the axial
U(l)pQ symmetry is spontaneously broken it would give
18
rise to a Goldstone boson named the axion The axion
is massless in the zero-mass limit for light quarks but
because the light quark masses are not zero the axion
mixes with the ~O and n and becomes massive (fig 13)
The axion is therefore a pseudo-Goldstone boson Since
the breaking occurs at the electroweak scale the mass
of the axion is supposed to be greater than 100 KeV and
lt B ) 7 (lOOKeV)51ts l1fet1me -c(a~e e-)_10- s -c(a~yy =0 s --- bull Suchmiddot
an axion model is excluded by beam-dump17 and
branching-ratio experiments For example theory14
predicts 1 6 x 10-S for the product of the yen an d Y branching
ratios into y+a whereas the experimental limit for this
product is 06 x 10-9 lS Another discrepancy between
theory and experiment is the branching fraction in the
K+ decay n+a theory19 predicts BR(K+~n+a)85x 10- 6
and experimentally20 this branching ratio is found to be
less than 4S x 10-S bull
But this was not the end of the axion because the mass f2f m bull
of the axion is ma = fa where f n = 94MeV ffin = 135 MeV
for the pion mass and fa is the symmetry-breaking scale
of the PQ symmetry First it was thought that fa should
coincide with the weak symmetry-breaking scale
constant) However if we assume that fa is very large
(fagt lOS GeV) then the axion mass becomes very small
19
q
a
Pi zero Eta zero
q
Figure 13 Mixing with pi zero and eta axions become massive in the non zero light quark limit
20
and its coupling very weak such axions have been named
invisible 21 In this latter axion model an extra complex
scalar Higgs field ltP is required that has a vacuum
expectation value
so that
where x is the ratio of the vacuum expectation values of
the two Higgs doublets ex = U dIU 11) can be large In that
case the axion become the preferred candidate for
accounting for the dark matter in the universe
Axions couple to two photons through a mechanism
referred to as the triangle anomaly (fig 14) the axion
lifetime is predicted to be of the order of 1050 sec for
a mass of the order 10-5 eV However I the Primakoff effect
could be used (fig lla) to enhance axion decay Two
experiments are currently using this technique to look
for cosmic axions22 23 on the assumption that 100 of
the dark matter in the local halo is made up of axions
We can flip (T-symmetry) the graph of figure lla and
also produce axions via the Primakoff effect by shining
(laser) light through a magnetic field Here a photon
21
Photon
a
Photon
Figure 14 The axions couple to two photons through the triangle anomaly
22
from the light beam with the correct polarization can
be combined with a virtual photon from the magnetic field
and produce an axion24 as shown in figure 1lb
A suitable extension of the electromagnetic Lagrargian
of equation 11 including the axion field is then (in
natural Heaviside-Lorentz units)
(118)
where a is the axion field rna its mass me the electron
mass the electromagnetic field tensor
(F IV-~IlAV_~VAI) P =~ FPC t d 1-0 Cl r llv 2EvPC 1 S ua and a 1 I 137 is
the fine structure constant M = Igayy where gayy is the
coupling constant of the axion to two photons
Now we consider a polarized laser beam entering at a
45deg angle with respect to the external magnetic field
Bext and propagating in the z direction (normal to BeXI) bull
Since the axion field is pseudoscalar the coupling term
is of the form (8 eXI EPh) namely only the polarization
component parallel to Bex can mix with the pseudoscalar
axion field 25
From the above Lagrangian using the Euler-Lagrange
equations for a (~ - ~I[~(~a) ] = 0) and for A II
23
we get the following equations of motion (written here
in matrix form)
o Qp == o (119)
BeX1WA1
where
2 ~_ a Bexwith ( 120)
( )4Sn B cr
A o A pare respectively the amplitudes of the orthogonal
and parallel photon components and a is the amplitude
of the axion field The gauge A 0 == 0 has been used and
the longitudinal photon component due to QED vacuum
polarization is neglected in the above calculations In
our case (in vacuum where the index of refraction is
such that In-lllaquol and w+k=2w because k=nw with k
the photon wavevector) we have
and then we can approximate
0 6 p 0 (121 )lW-io+l1 ~]] l]6 M
where
-m 24 7 a B oXI
6 = -w~ t =-w~ ta = 2w and t - shyo 2 p 2 M 2M
24
Because of the mixing of the A p and a fields we can
diagonalize the two lowest parts of the above equation
by rotating the original fields through a mixing angle cent
(as in any mixing problem)
[ coscent sincentJ[A p ] ( 122)
- si n cent cos cent a
AMwith tan2cent=2~ the new ts are now expressed as
p bull
tp+t a tp-t a = ( 123)
2 2cos2cent
Then by rotating back we can get the original components
Ap(Z)] [Ap(O)]= R(z)[ a(z) a(O)
with R(z) being the operator that propagates the photon
axion fields along z
COS cent si n cent ] R(z)= [
sml cos cent
Because cent is small in our case it suffices to expand R(z)
to second order in cent R(z) = RO(z) + centRl(z) + cent2R2(Z) with
Ro(~)=[~ e~tJ Rl(~)=(l-eit)[~ ~l and ( 124)
The phase shift is then given by
cent II (z) = - pound (R 11 (z)) = cent 2 ( t p - t a) Z - si n [( t p - t II) Z] ( 1 25)
and the attenuation by
25
( 126)
both of order cp2 (figures 1 5 and 1 6) bull For cent small
tan2cp2cp and for m a = 10- 5 eV (126) becomes
( 127)
Equation 127 holds for a single pass of the photon beam
through the region of the magnetic field
If a pair of mirrors is used to bounce the light back
and forth equation 127 has to be modified Instead of
Z2 we must use N [2 where N is the number of reflections
and l is the length of the magnetic-field region This
is because axions go through the mirrors while photons
are reflected and the coherence of the two fields is
therefore destroyed after every pass We then find
Similarly the phase shift is given by
cP =N ( B ext W ) 2 m l _ sin ( m l ) ( 129) a Mm~ 2w 2w
In equations 128 and 129 QED vacuum polarization is
neglected The rotation is e=(e2)sin26 where e is the
angle between the initial polarization and the direction
of the external magnetic field and the ellipticity is
1jJ=cp2 (see section 28) In these units (natural
26
x
t 2epSilOmiddot
y
Figure 15 Only the parallel component produces axions resulting to the rotation of the original vector E
27
Axion
Figure 16 The parallel component creates and reabsorbs axions and therefore lags behind the other component (ellipticity)
28
Heaviside - Lorentz) a magnetic field of 1T can be expressed
as 195 eV2 and a length of 1cm as 5 x 104 eV-1 Figures
17 and 18 show the limits that can be obtained on the
axion coupling constant as a function of its r~ss for
rotation and ellipticity limits of 10-12 rad respectively
for N=600 reflections l=880 cm ()j=241 eV and B QxF5T
These are the kinds of sensitivities we expect from our
experiment in the final configuration For an axion mass
ma~8X10-4 eV the oscillation length of the axionphoton
system equals the magnetic field length l and the
probability of creating an axion with that mass in our
system is exactly zero The same is true when the magnetic
field length is a multiple of the above oscillation length
and is demonstrated in figures 1 7 and 18 with the
oscillating feature in the limits of the coupling constant
29
1E9
0 L-J 1E8
-+- c 0
-gt- (fJ 1[7c 0 0
CJl C -
w 0
-g- -1 E6 0 u
I1E5+1-shy1E-5 1E 4 1E 3
Axion rnass [eV]
Figure 17 Limits (the lower part of the area is excluded) on axion coupling constant vs mass assuming
-1a rotation of 10 rad
1E9x------------------------------- shy
------ gt(])
C) 1E8LJ
gtshy+- c 0
+-
1E7(f)
c 0 u en c
w ~ Q lE6
J 0
U
1E5+----+--~~-middot-~-r~~H------middot-+---~--~~+-rH-~--
1E-5 1 E 4 1E 3
Axion IllOSS leV]
Figure 18 limits (the lower part of the area is exch ded) on axion coupling constant VS mass asslJming an ellipticity of 1(j 12 rad
References
1) S De Panfilis et al Phys Rev Lett 59 839 (1987)
2) H Primakoff Phys Rev 81 899 (1951)
3) T Bhattacharya and P Roy Phys Rev Lett 59 1517 (1987) the authors discuss a model with a chiralscalar s that couples to two photons with a coupling constant 9s yy 2 x IO- 3 Cey-l Our experiment excludes this coupling constant by 3 orders of magnitude
4) H Euler and Heisenberg Z Phys 98 718 (1936) V s Weisskopf Mat-Fys Medd Dan Vidensk selsk 14 6 (1936) S L Adler Ann of Phys (NY) 87 599 (1971) J Schwinger Phys Rev 82 664 (1951)
5) The equivalent magnetic field from a laser light of 1 watt and 2 rom diameter is B ph 052 gausslaquo B extshy
6) Wu-yang Tsai and Thomas Erber Phys Rev D12 1132 (1975)
7) R Novick M C Weisskopf J R P Angel and P G Sutherland Astroph J 215 Ll17 (1977)
8) E Iacopini CERN-EP84-60 1984
9) J Calmet et al Rev of Mod Phys 49 21 (1977)
10) M Delbruck z Phys 84 144 (1933) C Jarlskog Phys Rev D8 3813 (1973)
11) J E Kim Phys Rep 150 1 (1987)
12) Baluni Phys Rev D19 2227 (1979)
13) R D Peccei and H R Quinn Phys Rev Lett 38
1440 (1977) Phys Rev D16 1791 (1977)
32
14) I Antoniadis The axion problem in 1st Hellenic School on Elementary Particles Corfu 1982 ed World Scientific
15) S Weinberg Phys Rev Lett 40 223 (1978)
16) F Wilczek Phys Rev Lett 40 279 (1978)
17) E M Riordan et al Phys Rev Lett 59755 (1987)
18) C Edwards et al Phys Rev Lett 48 903 (1982) Sivertz et al Phys Rev D26 717 (1982)
19) I Antoniadis and T N Truong Phys Lett 109B 67 (1982)
20) c M Hoffmann Phys Rev D34 2167 (1986) N J Baker et al Phys Rev Lett 59 2832 (1987) Y Nagashima Proceedings of Neutrino 81 Hawaii July 1981
21) J Kim Phys Rev Lett 43 103 (1979) M Dine W Fischler and M Srednicki Phys Lett 104B 199 (1981)
22) W U Wuensch et al Phys Rev D40 3153 (1989)
23) Talk given by Chris Hagmann Proceedings of the Workshop on Cosmic Axions C Jones ed World Sientific (1989)
24) L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
25) Georg Raffelt and Leo Stodolsky Phys Rev D37 1237 (1988)
26) M V Berry Proc Roy Soc London A392 45 (1984) R Y Chiao and Yong-shi Wu Phys Rev Lett 57 933 (1986) A Tomita and R Y Chiao Phys Rev Lett 57 937 (1986)
33
Chapter 2 Apparatus
21 Magnets
An apparatus (fig 21 22) capable of measuring
optical rotation or ellipticity at the level of 10-12 rad
must be very carefully designed and must utilize state
of the art optics and electronics equipment 1 To
appreciate the smallness of the angle note that it is
the angle seen by an observer in Rochester when a point
at Brookhaven (approximately 400 miles away) moves by 1
micron On the other hand even to reach such an angle
a large magnetic field over a long effective length is
required (in terms of light travel in the region of the
field) Therefore our task is to combine the delicate
optics with the massive superconducting magnets and the
cryogenics in such a way that they can work without causing
problems to the measurable effects
In order to be able to observe the QED effect we need
to have a dipole magnetic field on the order of 5 Tesla
and be able to modulate the whole field as fast as possible
In our experiment we are using two CBA (Colliding Beam
Accelerator) magnets (fig 23 24) each 44m long
They are maintained at 47degK by supercritical helium gas
circuit Figure 25 shows the quench current versus
34
~
() U1
r---- ~-~~~~~-~~i ~ A----------- w-----0 -~--0- ---~ ~idnetlc Optical table
bull bull W FG awP
A bull ~ Vacuum box An Analyzer1--( -u- t3 -0- -1 LASER EO Electrooptic crystal Per _ rei EO Pol Fe Faraday cell
CI HWP Half wave plate Per Periscope
Optical table Pol Polarizer QWP Quarter wave plate Tel Telescope W Window
Figure 21 Apparatus for measuring small optical rotationbirefringence
Figure 22 Apparatus in the enclosure The large box in the rear to the left is the vacuum box containing most of the optics
36
CORR[CTION
EPOXY
W J
FIBERGLASS-EPOXY
COl D BORE TUBE
WITH
SPACE fOR SurER INSUlATION-
WARM 80Rpound TUBE IVACUUM CUAMBERI
EPOXY SPACERS
FIBERGlASS-EPOXY WITU HELIUM COOLING CHANNELS
Figure 23 End section of a CBA magnet
bull
Q5
5
38
CBA MAGNET QUENCH LINE
5r-------~----------------------~
4
3
2
o~----~----~------~----~----~ 4 5 6 7 8 9
T [K]
Figure 25 Quench current vs temperature
39
temperature These magnets were prototypes for the CBA
ring (their code numbers are LM0014 and LM0018) a high
intensity p-p collider consisting of two strings of
magnets Special care was taken in design to mini~~z~
the stray magnetic field
The modulation of the magnets at a high speed presents
a problem since the specifications were for a ramping rate
of 4 As We have operated the magnets at 115 As which
for full modulation (0 H 3500 A) results in a frequency
of -165 mHz
The measured magnetic field as a function of current2
is given in table 21 The magnetic field is B[TJ = TF
x I[Amps] x 10-4 where TF is the transfer function (figs
26 27) The field configuration is that of a dipole
with vertical direction and homogeneity (field variation)
better than 10-4
The stray magnetic field (fig 28) is small a fact
which made our task of shielding the optics considerably
easier The most sensitive elements are the two big
mirrors at the cavity ends the polarizeranalyzer the
Faraday glass and the A4 Plate (QWP) If we want the
background to be less than 10 at a 10-12 rad angle then
the limitations are very stringent According to reference
3 the dominant effect on the optical properties of a
dielectric mirror is the Faraday rotation with
4gtF37 X lO-IOradIG This of course refers to a single
40
TABLE 21
MAGNET Curr [A] JBdl [Tm] Effective Length [m]
LMOO18 264 1 8190 44251
LMOO18 1200 82356 44051
LMOO18 2000 43878
LMOO18 3500 217335 43566
LMOO14 264 1 8183
LMOO14 1200 82349
LMOO14 2000 43865
LMOO14 3500 21 7443
41
--
________ __________
2 LM0014 Transfer function
- v)
I
0shyE 0
Vl VlO J~ 0 01
Z 0 1-
~ J L
a IoJ ~
~C lt Q I shy
~________~__________
degOQ _ (lt00(01)0 l J
~~ A ~
bull1 60
0
0 o~
omiddot omiddot6o middot6 of
0 o~
oa I)
0 6
0 tf6
01)
6
6 6
6 6
6 6
6 6
6 6
6
1pound shyN-o
o I N o - = o
~ I~__________
o 1000 2000 3000 4000 I (amper e)
Figure 26 LM0014 Transfer function
~ ~
42
--
I
2 LM0018 Transfer function
-J) I Ie
E
~
l n J _0 shy I shy ~-0
ill 11)0 111 o shy shy
N j CI
~z I - i
I- I -= e
lt z shy -~
=
~L-________ _________L________~~________~________~ ~
a 000 2000 OOO 000 I (ampere)
Figure 27 LM001 B Transfer function
43
I
STRAY FIEDS AT r=20
160
120
Vl Vl
o MEASUREMENT I j
r C~ Leu L to T 0 ~~ (M 0 P) ( I I
shy56 KG
l 100lshyltX lt-shyc I
I w i LI - 801-
I J
Iu I I ~
J I
IIJ i lt I I ltX 60 L
I I
40
20
0 0
0 00 00 0 0
1000 2000 3000 4000 I AMPS
Figure 28 Stray Magnetic Field
44
bounce If we have about 1000 bounces and want to have
a background of less than 10-13 rad then the permissible
modulated stray field in the normal direction to the
mirror is less than 027 ~gauss
Our Faraday glass has a Verdet constant of 325 radTm
for A = 5145nm The Faraday rotation is given by centF =
VBI where V is the Verdet constant in radTesla-meter and
is linear with the wavelength of the light B is the
magnetic field and I is the length in m (see section 24
OPTICS below) For less than 10-13 rad spurious signal
we would need to have a longitudinal stray field of less
than 10-8 gauss The stray field on the polarizeranalyzer
and QWP must be of the same order (10-8 gauss)
22 Laser
Our light source is an Innova 90 5 Argon Ion laser
manufactured by Coherent It can deliver 5 watts in all
lines (but the main ones are 488nm (blue) with 15 watts
and 5145nm (green) with 2 watts) Usually the 488nm is
the dominant line of Ar+ lasers but it has a lower
saturation level than the 514 5nm and therefore the latter
becomes the dominant one at high powers
45
Figure 29 shows the levels of Ar+4 The neutral Ar
gas must first be ionized by the discharge electrons and
then the upper level of the Ar+ must be populated by a
second collision with the electrons A current density
of 700 Ampscm2 is needed to maintain continuous lasing 5
In order to suppress high order modes of oscillations
(ie TEMnn with nraquol) and because a very large current
is needed the discharge is confined within a tube whose
diameter is a few millimeters When the electrons strike
and ionize the Ar atoms the Ar+ ions sputter the walls
at high speed A solenoid magnetic field is established
around the discharge tube and serves a twofold purpose
a) to prevent the atoms from sputtering the wall thus
decreasing gas consumption and wall damage and b) to
confine the discharge along the center part of the tube
consequently increasing the current density Because
consumption of Ar gas is unavoidable our laser has an
automatic refill system which keeps the pressure stable
within 10 mTorr which in turn means that the pressure is
within 3 of the optimum condition (-1 Torr)
The ions inside the tube have a temperature of 3000 0 K
because they have been accelerated by the discharge
electric field and water cooling is needed to remove the
generated heat A minimum of 22 gallonsminute is
46
- --plaa
25
i-20
-_______ 35 Amiddot
(Ground atate
Ar
Figure 29 Atomic Ar levels
47
required for our laser We tune the laser at the green
light (5145nm) which corresponds to
(21 )
with a line width of 35 GHz
The laser resonator (fig 210) consists of a plane
high reflectivity mirror (greater than 99) and a curved
mirror with curvature of several meters and reflectivity
of 95 The Brewster windows made of crystalline quartz
select a particular polarization (in our case vertical
ie the photon electric field vector is vertical) and
the net polarization ratio is 1001 The prism at the
far end (not shown in figure 210) is used for wavelength
selection
There are two modes of operation a) current regulation
mode which tries to keep the current in the tube constant
by feedback techniques and b) light modulation where the
sample light going through a beamsplitter at the output
coupler is utilized and compared to a stable voltage
When the light changes the feedback system corrects by
supplying the necessary current to the tube Tables 22
23 and 24 show the laser specifications (as given by
the manufacturer)
Vacuum
48
TABLE 22
PERFORMANCE PARAMETER SPECIFICATIONS
Beam Diameter (at the
output coupler)
15 rom at 1e2 points
Beam waist diameter
(located 140 cm behind the
output coupler)
12 mm at 1e2 points
Long Term Power stability
(over any 30 minute period
after 2 hour warm-up)
Current Regulation plusmn3
Light Regulation plusmn05
optical Noise Current Regulation 02
rms
Light Regulation 02 rms
49
TABLE 23
LASER SPECIFICATIONS (COHERENT INNOVA 9o_5)
Bore Configuration Tungsten disk with one piece
ceramic envelope
Plasma Tube Cooling Conductively water cooled
Resonator Construction Thermally compensated Invar
rod structure
Cavity configuration Flat high reflector long
radius output coupler
Output Polarization 1001 Electric Vector
Vertical
Cavity Length 1093 mm (4303 in)
Excitation Current Regulated DC
Input Voltage 208 plusmn 10 Vac 60 Hz 3 phase
Maximum Input Current 45 Amperes per phase
Maximum Tube Discharge
Current
40 Amperes
Cooling water
Flow Rate
Incoming Temperature
Pressure
85 liters (22 gallons) per
minute (minimum)
300 C (860 F) maximum
Minimum 1 76 kgcm2 (25 psi)
Maximum 352 kgcm2 (50 psi)
50
TABLB 24
OUTPUT POWBR (TEMOO) SPECIFICATIONS
Wavelength Output Power
(nro) (roW)
All lines 5000
5287 350
5145 2000
501 7 400
4965 600
4880 1500
4765 600
4727 200
4658 150
4579 350
4545 120
351 1 - 3638 200
51
Feedlhroughs Ceramic sleeve Gas fill port
ilI~ l t-1 tl VA
1middotdeg Mirror mount
Gaha ~ae lacke
Tube support s
Inv3r resonator rods o ring seal
Ul tJ
Figure 210 A typical Ar ion laser resonator
As can be seen from figure 21 the optical elements
are resting on two tables The near end table which
holds all the optics except the endcap containing the
return mirror is made of granite weighs 2 tons and its
dimensions are 96X48X14 It sits on an iron frame
which in turn sits on three jacks directly on the tunnel
floor The table at the far end is also a marble plate
with an iron frame
The optics box on the table is connected to the vacuum
and contains most of the optical elements It is made
of aluminum and weighs approximately 200 kg This large
mass helps to keep the optics at uniform and stable
temperature which would otherwise affect the index of
refraction A 20 litis ion pump backed by a turbo pump
maintains the vacuum at 10-4 Torr The turbo pump is
attached directly to one side of the box but currently
there is no bellows between it and the box This made
it necessary to turn the pump off while data were taken
Another turbo pump is used for the insulating vacuum
of the magnets it is occasionally used to pump down the
optics box as well The nominal speed is 120 litis but
because of long lines and small diameter constraints the
effective speed is significantly less
The principal vacuum tube where the magnetic field
is and where the light travels back and forth several
hundred times is pumped by three 20 litis ion pumps and
53
three titanium sublimators We monitor the vacuum with
three ion gauges (figure 24) operating in the range from
10-3 to 10-12 Torr
Inside the optics box there are several mounts and
over 25 remote control motors and cables which outgas
While taking data all the mechanical pumps are off and
the vacuum inside the box is around 10-4 Torr In the
tube the vacuum depends on whether the system was
previously baked and if the sublimators were fired In
that case and when the valve between the box and the tube
is closed the vacuum attained is in the 10-9 Torr range
and presumably 10-10 Torr could be reached for a clean
system With the valve open and the pressure in the box
around 10-5 Torr we had 33 x 10-6 Torr and 89 x 10-7
Torr at the two ends of the beam tube
23 optics
Immediately following the laser head there is a
polarizer followed by a power stabilizer with EO crystal
(see section 33) Following that we have installed a
54
telescope to match the cavity a V2 Plate (HWP) a
periscope and a 45deg mirror All these components are
outside the vacuum (see fig 21)
Polarizers
The light enters through a window in the vacuum box
and is incident on the first polarizer as shown in figure
21 The polarizers are air spaced for high laser power
vacuum compatible and the exit surfaces are tilted by
3deg to eliminate etalon effects They were manufactured
by Karl Lambrecht corporation in chicago6 and the nominal
extinction is better than 10-6 over two thirds of the
surface This is a very conservative specification because
10-7 is routinely obtained We have achieved extinctions
down to 10-8 several times Extinction is the ratio of
the light coming through two crossed polarizers to the
light just before the second polarizer Following that
there is a mirror with a hole through which the light is
allowed to pass and a shunt mirror which is used for
diagnostics After the ray returns from the cavity it
bounces off the mirror with the hole and then off another
spherical mirror with radius of curvature 4 meters It
then passes through a Quarter Wave Plate (QWP) a Faraday
cell and the second polarizer (analyzer)
55
QWP
The QWP mount sits on two translators with one inch
travel each and is used only in ellipticity measurements
A two inch total travel is needed in order to make
successive ellipticityrotation measurements The QWP is
a device that has two axes a fast axis and a slow axis
In birefringent materials where the ordinary and
extraordinary rays have different velocities (and
therefore the index of refraction is different) the phase
difference between the two components is
-2n64gt=64gt -64gt =-(n -n )d (22)
0 A 0
where - is the light wavelength ne and no are the
corresponding indices of refraction for the extraordinary
and ordinary rays and d is the thickness of the crystal
that the light crosses When 164gtI=n2 then jn -n o d=A4
hence the name of such a device The polarization of the
light that is parallel to the fast axis is gaining a 90 0
phase with respect to the polarization component parallel
to the slow axis From figure 211a we see how a QWP
converts an ellipticity to a rotation Let the fast axis
be along x and the light be REP (right elliptically
polarized ie Ex lags behind Ey by n(2) then after
the QWP the Ex component will gain a phase n2 with respect
to the Ey and therefore they will have the same phase
Now we can recombine the vectors and we see that the net
56
x
y
a)
x
y
b)
Figure 211 a) Fast axis of QWP along x axis b) Fast axis along y
57
result is linearly polarized light but with the
polarization vector rotated by an angle ~ with respect to Ey
the x axis tan 11 = E It is clear that the fast axis must x
be oriented along one of the principal axes of the ellipse
if the fast axis is along y (fig 211b) then the Ex and
-Ey components have the same phase yielding again linearly
polarized light but with the polarization vector rotated
in the opposite direction Table 25 shows the sense of
rotation for LEP and REP light going through a QWP
TABLE 25
position LEP REP
Fast axis along X Ey
tan ~ =shyEx
Eytan ~ =-shy
Ex
Fast axis along Y Ey
tan~=--Ex
Eytan ~ =shy
Ex
Half wave plate
This plate simply rotates the polarization vector
The component that is parallel to the fast axis gets ahead
of the components parallel to the slow axis by 180deg The
phase difference is
58
2nllcent=n=--(n -n )d (23)A e 0
We find the new polarization vector by replacing Ex ~
-Ex where Ex is the component parallel to the fast axis
Faraday cell
Our Faraday cell (fig 212) consists of a plate of
BK7 glass inside a coil (solenoid) A tube holds the
glass in place and there are through cuts in both mounts
which serve to reduce any heating effects due to eddy
currents The coil actually is wound in two coils one
on top of the other The length of the coil is 344-inch
and the diameter I-inch The wire used is gauge (AWG)
No 28 (32 mils in diameter) The first (inner) coil
consists of ten layers 1500 turns and the resistance
is Ri = 28D The second (outer) coil consists of eight
layers 1100 turns and the resistance is Ro = 29D The
inductance of the inner coil is Li = 12mH and that of the
outer Lo = 15mH The magnetic field produced is (taking
into account the fact that the length of the coil is not
infinite)
B i = 4nNI =206 x J[Amps] gauss (24 )
for the inner coil and
B =4nNl=1506 X I[Amps] gauss (25)o
The impedances of the two coils are
59
AlumInum tube Part I side view 22middot~ OOSmiddot --
193~
l
bull 06 toOOSmiddot OSSmiddot OOOS 04Smiddot0005
bull
42tO05middot OSOOS
0 5
2tO05StO05middot AlumInum Coil 2 Side vIew
L~===~3~44~~0~0~05~-==~J7750005
0063 0005- 750005middot
Sharo edges ---------- shy 04005 ~ 8l2taps ~
(aligned)
Figure 212 Support tubes for the Faraday cell
60
(26)
and
where we used w = 2nv and v = 260 Hz as an example with
the two coils connected in parallel the magnetic field
is Bt = Bi + Bo with il-VIZ (V is the voltage acrossI
the two terminals) and i 0 = V I Z 0 206 1506)
B E = -+ [= 10XV[Volts] gauss (28)( Zi Zo
When polarized light goes through a crystal with a
Verdet constant V and there is a magnetic field along
the axis of light propagation then the polarization plane
rotates by an angle
e == V Bl (29)
where B is the magnetic field component along the axis
of propagation and I the total length of the crystal Our
crystal ( W - 13P 1-inch length furnished by optics for
Research Co) has a Verdet constant V = 40 radTm at A
= 633nm Since V is proportional to the wavelength A we
can find the constant at 514 5nm (Green) V = 325 radTm
Thus
(210)
The advantage of using the Faraday effect is that a properly
aligned crystal does not distort the light polarization
61
(extinctions are excellent) The main disadvantage of the
Faraday cell is that the coil acts as a low pass filter
as is apparent from equations 26 and 27 This makes
it very difficult to drive it at very high frequencies
Mounts
Most of the optics inside the vacuum box reside on
ORIEL Mounts For most of the elements the x y (which
define the plane normal to the beam direction) and ecent
motions are standard In some of the elements (where the
orientation of the birefringence or polarization axis
matters) the 3600 mounts are used allowing the adjustment
of the rotational degree of freedom as well
The mounts are motor controlled and some have position
readout with resolution Ol~ (~ 361 x 10-6 rad) We
calibrated the readout microns to rotation in rad by
rotating one mount360deg and dividing by the motor total
travel in microns The motor controllers are connected
with the motors inside the vacuum box through vacuum sealed
feedthroughs
62
24 cavity
The cavity consists of two high reflectivity (998)
interferential mirrors ground and coated by the optics
shop at the University of Rochester The mirrors have a
diameter of 112Scm and a radius of curvature of 1903rn
One mirror has a hole of S7mm diameter at its center
From equations 112 and 1 28 it is apparent that to maximize
the effect to be measured the light should go through
the magnetic field region several times since the effect
is proportional to the number of bounces This is achieved
by reflecting the light on the mirrors several hundred
times The necessary expressions for calculating the spot
position on the mirrors is given in reference 7 Usually
the light passes through a hole in the first mirror and
with the system al igned the successive spots form an
ellipse the number of bounces depends on the distance D
of the mirrors and the radius of curvature The number
of reflections achieved in this manner is limited by the
spot diameter and the diameter of the mirrors (in reality
the limitation is more stringent because of the reentrance
condition)7 For that reason we deform one mirror (so
that its focal length is different in one direction than
in the other) and when the ellipse is closed for the
first time the spot misses the hole and traces out more
ellipses Thus a Lissajous pattern8 is formed this is
63
shown in figure 213 where there are about 400 reflections
(a total of 800 for both mirrors) We use two methods
to measure the number of reflections
A) When the mirror reflectivity is known we can measure
the attenuation the light suffers inside the cavity The
intensity of the light emerging from the cavity is
- I I
o e Ao (211)
where nO is the number of reflections needed to attenuate
the light intensity to its lIe value n is the number of
reflections and IO is the initial intensity no is related
to the reflectivity
1 n =-- (212)
o 1 - R
After n reflections
and for (l-R)laquol we can approximate
and therefore no=ll-R)
The ratio IIIo is the ratio of the photodiode signal
when the light is traveling inside the cavity to that
when the shunt mirror is in place In such a measurement
the two polarizers must not be exactly crossed since the
ratio IIIo is usually around 05 and would be obscured
by extinction fluctuations if a good extinction was used
64
bull
Figure 213 Lissajous pattern on end mirror
65
If the reflectivity of the mirrors is not known then
nO can be found from equation 211 by counting a small
number of reflections This is however difficult for
reflectivities better than 998 because the attenuation
of the light intensity (for say 50 reflections) is very
small
B) The other method is based on measuring the time
delay for the traversal through the cavity We are using
a beamspl i tter at point A (fig 2 1) to feed one photodiode
and a chopper to chop the light (at -2 KHz) before point
A We feed this output to the first channel of an
oscilloscope (which we use at the alternate configuration)
and we trigger with this channel What we see on the
oscilloscope is a square wave NOw we feed to the second
channel of the oscilloscope the photodiode signal (fig
21) The oscilloscope screen shows two square waves
shifted relative to one another shifted (fig 214 a
b) by the time the light spends inside the cavity Figure
214b shows the null measurements that is the signal
photodiode sees the light before the cavity and therefore
the two square waves coincide The photodiode outputs
should be fed to the oscilloscope input directly (with a
500 termination for fast rise time) and one should beware
of introducing false phase delays (amplifiers) The slow
rise time in the figure is due to the chopper which limits
the accuracy of the method
66
a)
b)
---- Time (5x10-6 sDiv)
Figure 214 a) Time delay between the two paths b) Null measurement
67
Interferential mirrors introduce ellipticity because
they have a slow and a fast axis Therefore it is
important to align the mirrors so that the fast (or slow)
axis coincides with the polarization pl~ne of the laser
light In order to find this axis we setup a small test
system as shown in figure 215 The first polarizer
polarizes the light which is then reflected back and
forth between the two mirrors several times forming a
circular pattern Typical values are for the distance
between the two mirrors D = 50 cm and for the number of
reflections approximately 50 on each mirror (a total of
100 reflections) We need about this many reflections
because the total ellipticity which is cumulative is
100 times more than in a single bounce and the effect
becomes measurable If the axis of one of the mirrors
is known the measurement is straightforward If this is
not the case the procedure is much more time consuming
and the following measurements must be performed
a) Form a circle (fig 216) with several bounces on
each mirror and catch the exiting beam with the mirror
with the hole After this exiting beam goes through the
analyzer we rotate the latter to extinguish it We record
the intensity (as seen on the FFT at the chopper frequency) bull
b) We rotate one mirror by 5deg and try to extinguish
the beam recording at the end the minimum value We
repeat this until we complete a rotation through 360deg
68
a w
~
(I) 0 0IV shy
-NOshygt0 as 0
laquo 0 c s
c I 0
laquo 0
8bull 0
Q) ~~ 0 () Tmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoty
(I)
8 c
(I) ~
(I) 0
~ (I)
0
(I) N ~
as 15 0
15 0
en C IV E IV (I) as (I)
E 25c (I) enc ~
Q5 zs E c (I)
C)
Iii N e en
u
~
~
69
bullbullbullbullbullbull bull bull bull bull bull bull bull bull bull bull bull bull bullbullbullbullbullbullbull
bull bull t bull
bull
Figure 216 Circle for birefringence axis
70
The total readout looks like figure 217
c) We find the minimum of the above minima and rotate
the mirror so that its axis (it is not the real axis yet)
coincides with the polarization plane of the light
d) We take the same measurements but now we rotate
the other mirror with the same increment of 5deg We find
again the minimum of the minima
e) Now we align the two axes of the mirrors found
from above and take exactly the same measurements by
rotating both mirrors together 5deg for each measurement
and find again the minimum of the minima The real axes
of both mirrors are along the new minima For data taking
in the experiment these axes should be along the direction
of the light polarization In addition when the light
is reflected off a mirror it acquires an ellipticity910
which is proportional to the square of the angle of
incidence and independent of the orientation of the fast
or slow axis of the mirror Therefore special care should
be taken to readjust the circles on the mirrors so that
they are always the same circles Another detail to be
concerned about is that different points on the mirrors
might have different reflectivity In order to exclude
this case we used a QWP in front of the analyzer and
tried to extinguish the beam In this way we can check
whether the non-extinction is due to ellipticity or due
to a local absorption
71
600t-2
500r2
400-2
300(gt-2
200-2
100-2
I r r
iA
~ j
0 to 20 30
Figure 217 Extinction (arbitrary units) vs rotation
Position 33 corresponds to 360 degrees rotation
72
40
The above procedure is tedious because the minimum in
the first stage (a) is not so obvious and because
readjustments must be made while rotating both mirrors
Fortunately this procedure needs to be performed only
once the final result shown in figure 218ab
25 Ray Transfer
A paraxial ray moving through an optical system will
change position and slope according to the following
equation (fig 219)11
(213)
where XIX I are the position and slope of the original
ray and X2X~ that of the transformed one Table 26
shows the ray transfer matrices for different optical
systems they have unit determinant (AD - Be = 1) assuming
no absorption The focal length of the system and the
location of the principal planes are related to the matrix
element by11
f = -1 IC
D-l h l =-shy (214)
C
A-I h =-shy
2 C
73
Figure 218 Extinction (arbitrary units) vs rotation
In the polar graph R is extinction in -dBV The two axes (fast amp slow) appear clearly
74
_----=_bull- - ~ - - shy ~ y p I I-----~~I ---shy TH 1 1 - ________
I I I I I I I I I I
X~
INPUT OUTPUT PLANE PLANE
Figure 219 Parameters of a paraxial ray
75
where the notation for h1 and h2 is given in figure 219
When the light bounces back and forth between two spherical
mirrors it undergoes a periodic focusing which is
equivalent (if we imagine the system unfolded) to the
light passing through a series of focusing lenses The
ray transfer matrix of this system is evaluated by means
of Sylvesters theorem 11
I~ BII1=_1_Asin8-Sin(n-l)8 Bsin n8 (2 IS) D sin8 Csinn8 Dsinn8-sin(n-l)8
where cos 8 = I 2 (A + D)
In order for periodic sequences to be stable
(periodically focused) they mus t obey - 1 lt 2 1
( A + D) lt 1 i
otherwise the sines and cosines become hyperbolic
functions and the spot size becomes bigger and bigger as
it passes through the elements In the case of a cavity
with spherical mirrors its dual a sequence of lenses
have focal lengths equal to those of the mirrors and the
distance between them is equal to the distance between
the two mirrors In that case the above inequality is
given by
(216)
Figure 220 shows the stable and unstable regions of the
cavity Our cavity (d = 1256m R = 1903m) corresponds
to point A far away from any unstable region
76
CONfOCAL (lIItalllt z d)
I
N I
~~~
Figure 220 Stable (white) regions and unstable (shaded) regions of mirror resonators Point A is for the cavity used in this experiment
n
26 Telescope
When the light exits the laser head it has a diameter
of 2w = 15mm at the lie point of the electric vector
and divergence 05 mrad The beam waist is 2wo = 12 mm
and is located 140 cm behind the output coupler of the
laser head at the output the beam has a Gaussian profile
A complete treatment of the propagation of a laser bear
can be found in references 11 and 12 the main results
being
(217)w(z) = W[ I + (n~n and
[ (nw2)2]R (z) =z 1 + l A ZO bull (218)
Here 2wO (beam waist) is the minimum diameter of the
Gaussian beam and at this point the phase front is plane
A is the light wavelength 2w is the beam diameter at the
lie points after a distance of travel z measured from
the point where w = wO- R(z) is the radius of curvature
of the wavefront at z If we SUbstitute in equation 217
A = 5145 nm Wo = O6mm for a distance z = 18m we would
have a beam diameter 2w = 10mm This actually would be
the spot size on the far end mirror To avoid this we
are using a telescope to match the beam to the cavityl011
78
We only need one middotlens in order to transform the beam
characteristics Then
and
(220)
where fo=nwIW2f f must be larger than fO and K=plusmnl
d1 (d2) is the distance between the position of the waist
of the original (transformed) beam and the lens similarly
w1 (w2)is the waist of the original (transformed) beam
ButWO(=w2) is given by
W 4 = (~)2 D(R 1-D)(R2 - D)(R 1+ R2 - D) (221)
o n (R 1 +R 2-2D)2
where R1 R2 are the radii of curvature of the two mirrors
and Wo the cavity waist For R1 = R2 the waist Wo is
located at the center of the cavity due to the symmetry
otherwise the distances t1 and t2 from the first and
second mirror respectively are given by
(222)
In our case R1 = R2 = R = 1903 m and D = 1256 m Therefore
W 4 = (~)2 D(R - D)2(2R - D) = (~)2 D(2R - D) =9
o n 4(R-D)2 n 4
Wo= 121mm (223)
79
ie the beam waist in the cavity diameter 2wO = 24mm
The beam diameters on the mirrors are
and
(224)
In our case (AR)2 D w 4 =w 4 =w 4 =i- =w=148mm (225)
1m 2m n 2R - D
ie beam diameter 2w = 3 mm
From the above when R = D (confocal cavity) we have
2 AR 2 AR w =- and w =shy (226) o 2nn
That is if we had a confocal cavity (R = D = 1256 m)
then the beam waist diameter and the beam diameter at the
mirrors would be
2wo =203mm and 2w=287mm (227)
As is apparent now our telescope must match the beam
waist w1 = 06mm which is located 140cm behind the output
coupler (from laser specifications) to the beam waist
at the center of the cavity w2 = 12mm (from equation
223) We see from equations 219 220 that once d11
d2 are fixed then f is fixed too since fO = 44m In
our case d1 = 2m d2 = 9m and
80
--e
Je
Ie N N Q) I 2l
Ji lL o
Ibull
81
Table 26
Ray Transfer Matrices of six Elementary Optical
structure
No OPTICAL SYSTEM RAY TRANSFER MATRIX
1
-d - I I 1 I
I 1 I I
[~ ~J
2
f
4shy I I
r shy1 0shy
1 1-7
3
--d-shy ~f
I 2
r -1 d d1-shy I shy -
f f
4
d_-df-~
~f of I
bull I 2
[ d did1-shy d l +d 2-shy11 I
l-~- d2_~+ d l d 21 1 d 2-Tt- 12 + 112 11 12 12 1112
5
- ---- ~~~~ n t j
bull~
n n~-inrl I I
- If~ 1 n 2cosd - Sind -
no ~nOn2 no
- ~non2sin d~ ~cosd -no no
6
~-7~ ---~ gt)
Y (
~1 ~~ middoth [~ dn]
1
82
=9f=641m (228)
We can use the above technique or use a more practical
approach which employs a combination of two lenses one
divergent and the other convergent put in series (fig
221) In the above figure H HI are the positions of
the principal planes of the telescope11 with
(229)
where f1 is the focal length of the convergent lens f2
that of the divergent (f2 is negative) and
where d is the distance between the two lenses As is
shown in figure 221 d1 d2 of equations 219 and 220
are with respect to the principal planes It should be
noted that equations 229 and 230 can be found from table
26 and equations 214
Therefore our telescope must satisfy the following
equations
(231 )
83
middot ddl=l+[ +fshyfl
d =D -(l+(+d-f~)2 lot f 2
with 1 = 140cm d ~ 5cm I ~ 40cm and Dtot ~ 11m By
solving these equations for fl f2 we can obtain the
parameters of the telescope In reality we used a computer
program in which we chose fl with a value that is available
commercially and varied d (within reasonable limits) so
that f2 is also available
27 Jones vectors and Matrices
The treatment of monochromatic light going through
polarizers QWP Faraday cell etc is greatly simplified
with the use of the Jones vectors and matrices Table
27 shows the Jones vectors for different kinds of
polarizations13 and table 28 shows the Jones matrices
for different optical elements 13 When the laser light
goes through an optical element the final Jones vector
can be found by multiplying the original Jones vector by
the Jones matrix of that element As an example we are
going to use here this formalism to treat our system in
two configurations
84
Table 27
Jones vectors
Light NormalizedVector
Vector
Linearly Polarized light along the x axis [Ax~middotmiddotJ [~J Linearly Polarized light along the y axis [~J[A~middot] Linearly Polarized
Axe [COSSJ[ ]light in an angle 9 SineA 14gt~with respect to x axis ye
Linearly Polarized [ COse ]axis in an angle -9 [ Axmiddot] - sin eA Ifgtxwith respect to x axis - ye
Left Circularly Polarized light (LCP) [ Amiddotmiddot ] ~[~Ji~-n2)
Axe
Right Circularly Polarized light (RCP) [ Amiddot ] ~[~iJi~ n2)
Axe
Elliptically Polarized light [A ] [ cose -]
lltIgtyAye sin Se
85
2
Table 28
Jones Matrices
optical Element 8= deg qeneral 8
Ideal Polarizer at an angle 8 with
respect to the axis [~ ~J [ cose cosesin eJ cos8sin 8 sin 2 8
14 plate (QWP) with
fast axis at an angle 8 [~ ~J [ cos
2 e-isin
2 e cosesin9(1 +i)J
cosesin 9( 1 + i) -icos 2 e+sin 2 9
AJ2 plate (HWP) with
fast axis at an
angle e [~ _deg1 ] [ cos28 sin 28 ]
sin28 - cos29
Plate introducing phase delay ltP with fast axis at e [~ e~~ ]
[ cos 2e sin 2ee-middotmiddot cosesineCl-e-)] cos9sineCI -e-middotmiddot) cos 2 ee-middotmiddot sln 2e
0
Note e is the angle between the polarization direction
or the fast axis and the x axis The Jones matrices for
e degare obtained from those for 8 =degby using the rotation
matrices ie rotate first the x axis along the fast
axis (polarization axis) of the element perform the
multiplication with the Jones matrix of the element at
0 and then rotate the x axis back to the original
direction
86
a) Ellipticity Polarizer at 0middot phase shift ~ from the
cavity due to the magnetic field with fast axis at e (the
fast axis is along the direction orthogonal to the magnetic
field see equation 110) a QWP with fast axis at 0middot
Faraday cell (rotation) and an analyzer at i+a where a
is the misalignment from the perfect crossing between the
polarizer and the analyzer The final Jones vector is
(from tables 27 28)
0 0J[ cosa sinaJ[ cosTj sin TjJ[ 1 [ deg 1 -sina cosa_ - sin Tj cos Tj deg
cos2S+sin2Se-i4gt (232)[ cos8sin8( l-e- i
4raquo
The misalignment angle a is of the order of 10-6 rad
Tj bull Tj (t F) is the Faraday rotation with a frequency of 260
Hz and is of the order of 10-3 rad1 ~ contains the time
dependence of the magnetic field since it is the phase
shift introduced to the light inside the magnetic field
This is much smaller than any other phase shift introduced
into the system and we take it to be between 10-8 - 10-12
rad Therefore the above matrices can be rewritten
(approximately) as
87
cos 2e+Sin 2e(l-icent) [ tcentcosesine
(233)
Therefore the current detected by a photodiode after the
analyzer is
then
(234 )
where we have used T)=T)OCOSWFt 1p=cent2=ljJocosw M t cent is
the phase shift and 1p the ellipticity and 10= IAxl2 the
light intensity after the polarizer It should be noted
that the signal of interest is maximum when e= plusmnnl4
b) Rotation We use the same configuration but without
the QWP
(235)
where E = Eo COSW Mt is the rotation introduced by the magnetic
field Expanding equation 235 we have
88
(236)
and as before the photocurrent is
(237)
In the above case we have used E as an angle of optical
rotation In reality there is an attenuation E (axion
case see figure 15) of the photon component parallel
to the magnetic field For an angle -8 between the first
polarizer and the magnetic field the Jones matrices become
sin8J[1-e[0 0J[ 1 aJ[ 1 11J[ cosedeg 1 - a 1 - T] 1 - si n e cose deg
[cose -sin8][1 sin e cose deg
neglecting the small terms in the above we obtain
(238)
89
comparing equations 236 and 238 we conclude that
for 6=plusmnn4 or
E=(E2)sin26
in general where E is the rotation of the polarization
plane of the beam and E is the attenuation of the component
parallel to the magnetic field Therefore
(239)
90
References
1) This experiment was first proposed by E Iacopini and
E Zavattini Phys Lett ass 151 (1979) see also E
Iacopini B smith G Stefanini and E Zavattini II
Nuovo Cimento 61S 21 (1981)
2) Magnet Measurements Analysis Analysis amp Measuring
Group BNL Report numbers TMG-259 (1982) and TMG-270
(1983) E J Bleser et al Nucl Instrum Meth A23S
435 (1985)
3) E Iacopini G Stefanini and E Zavattini Effects
of a magnetic field on the optical properties of dielectric
mirrors CERN-EP83-55 1983
4) Orazio Svelto Principles of Lasers translated and
edited by David C Hanna second edition Plenum Press
(QC688 S913) 1982
5) Rudi Wiedemann Lasers and Optronics October 1987
p 55
6) The code name of our polarizers is MGLQD-12
manufactured by Karl Lambrecht corporation 4204 N
Lincoln Avenue Chicago Illinois 60618 USA
91
7) D Herriott H Kogelnik and R Kompfner Appl opt
3523 (1964)
8) D Herriott and Harry J Schulte Appl opt bull 883
(1965)
9) S Carusoto et al The Ellipticity Introduced by
Interferential Mirrors on a Linearly Polarized Light Beam
orthogonally Reflected1 CERN-EP88-114 1 1988
10) M A Bouchiat and L Pottier Appl Phys B29 1 43
(1982)
11) H Kogelnik and T Li Proc IEEE 5 1312 (1966)
12) Miles V Klein and Thomas E Furtak OPTICS second
edition John Wiley amp Sons Inc 1986
13) I Spyridelis et al Askiseis Optikis Tefhos ~
ARISTOTELEIO PANEPISTIMIO THESSALONIKIS 1980
92
Chapter 3 Data Acquisition
31 Electronics
In the previous chapters we described a system designed
to search for ellipticity and rotation introduced by a
magnetic field The sources of ellipticity are both the
QED vacuum polarization and axions (depending on the axion
mass and coupling constant) while the source of rotation
is only the axions As we saw in the last chapter (equations
232 235) the two effects require a different setup
To search for ellipticity we must have a QWP in the optical
path in which case any induced rotation gives no effect
to first order To search for a rotation we remove the
QWPi in the latter case it is the ellipticity that gives
no effect to first order The reason for this is that
ellipticity does not mix (interfere) with the Faraday
rotation and therefore the useful signal rather than being
linear in the field amplitude remains proportional to the
square of the effect Of course the signal of interest
is always proportional to the sought after effect times
the Faraday amplitude (see equations 234 and 239) If
we Fourier analyze IT we obtain an unwanted signal at the
Faraday cell frequency w F and two satellites containing
93
the signal of interest at frequencies W F plusmn W M where W M
is the magnet modulation frequency The magnet frequency
however is in the tens of millihertz region and it becomes
apparent that the width of the Faraday signal must be
small and noise free Also all the clocks must be locked
together to avoid any phase jitter
Figure 31 shows the electronics setup of our system
The Fast-Fourier Transform or FFT (HP 35660A Dynamic
Signal Analyzer) has an internal synthesizer with very
stable frequency output We use this synthesizer at 260
Hz and 35V zero-peak to feed a TECHRON 5515 voltage
amplifier which in turn drives the Faraday cell The
width of the Faraday frequency line is much less than 1
mHz as measured with the same FFT
The transmitted light is fed to a silicon photodiode
(table 31) The area of this photodiode is kept small
so as to minimize its capacitance which restrains the high
frequency response Because of this we use a focusing
lens before the photodiode with a focal length between
Bmm to 10mm We make sure the focal point is not exactly
on the photodiode surface to avoid surface current density
limitations In order to avoid Etalon effects we removed
the glass cover of the photodiode otherwise the incoming
ray would interfere with the multiple reflections in the
glass and the signal would become very sensitive to the
beam pointing stability
94
II W () () if t-
ATCOMPAllBLE
COMPUTER
HPI8
HP35660A
FFT
INTERNAL
INPUT
RS232C ORIEL
18011
ENCODER
MOTOR CONTROLLER
BAND PASS FILTER
MOTORS
E SYNTHESIZER CURRENT PREAMPLIFIER
260Hz 3SV
260 Hz 10V
FARADAY
CELL
OPTOCOUPLER
SCALAR
t3328
OPTOCOUPLER MAGNET TRIGGER
Figure 31 Electronics setup
95
Table 31
Silicon Photodiode
Type No S1336 - lSBQ 1 - TO - lS
11 x 11Size (rom)
Range (nm) 190 - 1100
Radiant Sensitivity (AW) 027 5145nm
Short Circuit Current Ishl 100 lux Min (jJA) 1 Typ (jJA) 12
Dark Current Id VR = 10mV Typ (pA) 2 Max (pA) 20
Temperature Dependence of Dark Current Typ 115 (Timesoc)
Shunt Resistance Rsh VR = 10mV Min (GD) 05 Typ (GD) 5
Junction Capacitance CjVR = OV Typ (pF) 20
Rise Time tr VR = OV RL == 1kD Typ (Ils) 01
4 x 10-15NEP Typ (WHz12)
3 x 1013D Typ (cm Hz12W)
5
Temperature Range Operating (OC)
Reverse Voltage VRmax (V)
-20-+60 Storage COC) -55-+S0
96
The photodiode output goes to a EGampG model 181 Current
sensitive preamplifier This amplifier has six different
feedback resistors selectable with a front panel switch
(10-4 VIA to 10-9 VIA in increments of factors of 10)
Figure 32 shows the noise gain and input impedance vs
frequency for the different settings A battery-powered
Hamamatsu C1837 amplifier with two feedback resistor
settings (R = 107 0 and R = 109 0) was used for the early
measurements (until October 10 1989) The amplifier is
mounted far away from any metal surface to avoid
capacitively coupled ground loops which could pickup
transient noise The output cable was two conductor
shielded (one is used as the signal carrier and the other
as the return) the outer shield was grounded on the
amplifier end only Similar cable is used between the
TECHRON 5515 voltage amplifier and the Faraday cell The
above configuration seems to have eliminated interference
from transient noise
It should be noted that the optics (vacuum box with
the table laser head and one of the endcaps) are in a
temperature controlled enclosure The only electronics
that are inside the enclosure are the preamplifier the
motor controller and the laser power stabilizer This
minimizes the need for accessing the enclosure the heat
load and electrical noise The output of the preamplifier
if fed to a bandpass filter (Frequency Devices 9002 used
97
FREOUENCY Hz
Figure 32 EGampG low noise preamplifier characteristics
Typicil noIbullbull CUFfnl It bull function 01 frqulncy nd nl shy11 lty
I)
t 0
~
-
Hl ~
u U Z
0 laquo
~ ~
i ~ i
FREQUENCY Hz
Tplcal inrpu1lmpednc bull bullbullbull function of tvlty nd I CIUM1C
FREQUENCY Hz
Tplcal fquency ponn a tunction Olnliivlly
shy~ Cl II 0 shy0 J
Z r c
FREQUENCY Hz
98
in differential input frequency band 220 Hz to 300 Hz
gain 1) used to eliminate the DC component and to attenuate
the signal at twice the Faraday frequency This allows
the FFT to detect lower level signals otherwise obscured
due to dynamic range limitations The output of the
bandpass is fed to the input of the FFT A typical FFT
setting is the following
Center 260 Hz (same as the Faraday frequency)
Full bandwidth (400 Channels) 15625 Hz
Window Hanning
Trigger External
Input AC Coupling Autorange
We use vector average since the noise (in volts) H z)
decreases as fN due to the randomness of the noise phase
where N is the number of averages whereas the signal
phase is of course fixed and therefore the signal vector
remains unaffected We take one measurement (one record)
at a time and calculate the vector average OFF line A
preliminary vector average is performed ON line for
diagnostic purposes
The output display of the FFT is read through an HPIB
(Hewlett-Packard Interface Bus) and a 37203A HPIB
extender by an AT compatible personal computer After
one measurement the computer commands the FFT to change
bandwidth (usually to 0 - 800 Hz) and take one average
from which it obtains the DC amplitude and the signal
99
at twice the Faraday frequency It stores the data on
the hard disk and updates the vector average on the screen
A note is need here about the HP 3660 FFT and its units
The setup for the units is dBVrms ie if the readout
is A dBVrms this corresponds to B Vrms
11
10 20B = V rms M A = 2010g B dBV rms
When we read a spectrum on the display through HPIB the
FFT always sends the values in linear units and in zero
to peak amplitudes independently of the displayed units
In case we read the marker as we do for the DC and the
signal at twice the Faraday frequency then the data
transferred through the HPIB is the same as the ones
displayed (ie dBVrms )
A more important detail concerns the DC readout from
the HP35660A FFT When the displayed values are zero
to peak amplitudes the DC readout is twice as much as the
actual value that is there is an offset of 6dB Now
if we change the displayed values from zero to peak to
rIDS (root mean square) the FFT just subtracts -3dB from
every channel (ie divides every displayed number by f2 ) even the channel with the DC Therefore if the
displayed values are in rIDS in order to obtain the correct
DC value we have to subtract only -3dB and not 6dB In
our case it is most convenient to use zero to peak
amplitudes and just subtract 6dB at the DC readouts
100
32 Misalignment Correction
From equations 234 and 239 we realize that our signals
are very close to the Faraday frequency where is present
the unwanted peak proportional to a 110 Because the
satellites are so close together there is no filter that
can remove the center peak If a is very large then a
problem arises because of the dynamic range limitations
(see section 41) Therefore a should be kept as low as
possible (below 10-5 rad) The task of keeping the
misalignment low is handled by the AT computer with the
help of the ORIEL 18011 motor controller (MC) The two
instruments communicate via an RS232 serial interface
The motor controller has three inputs and is connected
with the motor that rotates the analyzer another motor
that rotates the polarizer and the third one moves the
theta motion of the polarizer (theta is defined in the
polarization plane of the laser beam) Before the AT
commands the MC to move any motor it checks with the FFT
the amplitude at the Faraday frequency W F and at 2w F bull
Since at w F the signal is proportional to 2a110 and at 2WF
proportional to n~2 then the ratio is equal to 4ano
The Faraday amplitude no is kept between 10-4 and a few
times 10-3 rad Therefore our goal is to keep the above
ratio at about 100 (40 dB) which means that a is kept in
101
the 10-5 to 10-6 rad range If the misalignment is already
in this range the AT does not try to move any motor and
sets up the FFT for the next measurement Otherwise it
moves either the analyzer rotation or the polarizer theta
depending on the misalignment level In principle since
we know ~o very precisely we could rotate the analyzer
by a fixed amount (in linear dimensions 1~ of micrometer
motion corresponds to 361 x 10-5 rad) Furthermore if
we know the phase of the misalignment component we would
also know in which direction to move The position readout
of the motor controller has a resolution of Ol~m but the
motors move reliable only for distances in excess of O 5~m
(not to mention the backlash problem which is of the order
of a few microns) Thus in order to reach rotation
resolution of 10-6 rad we would have to have O 04~ readout
resolution on the micrometers Therefore the program
moves the analyzer rotation only if the misalignment is
within the range of the micrometers otherwise it moves
the polarizer theta By moving the theta motion of the
polarizer a few microns the extinction remains unchanged
whereas the projection of that motion on the rotation
plane gives us the necessary resolution to lower the
misalignment to the 10-6 rad range
There are two alternatives to the above scheme One
is to use a Faraday cell and apply a DC current through
the coil to rotate the polarization plane and to match
102
the extinction plane of the analyzer This could be done
with a HP 3325A synthesizer or a computer controlled DC
current source The other scheme is to use negative
feedback on the Faraday glass and keep the misalignment
continuously low
When the misalignment is at an acceptable level the
AT commands the FFT to change center frequency and
bandwidth After the FFT settles its digital filters
etc the AT enables the ARM command of the FFT which can
now accept a trigger to start the measurement The settl ing
time is a considerable part of our overhead because it
takes about 50 of the actual data taking time If the
time needed for one average is 512 sec then the settling
time is 256 sec however if the input of the FFT is
overloaded by a transient signal while it is settling a
new 256 sec period is initiated
Trigger
We have chosen to vector average so that we can have
both amplitude and phase information from our signals
To do that we must trigger the FFT and the magnets with
the same signal The output of the Faraday frequency
signal from the FFT I S internal synthesizer is split with
a tee connector One output is fed to the TECHRON 5515
Power Supply amplifier to amplify the voltage and drive
the Faraday coil and the other is fed to the a scaler
103
bull
which divides the original frequency by 3328 to obtain f
= 78125 mHz This is the magnet modulation frequency
(T = 128 sec) and care has been taken so that this
frequency falls in the center of an FFT channel
To eliminate any ground loops between the FFT the
scaler and the magnet electronics we use an optocoupler
to feed the scaler from the FFT and another one to feed
the magnet trigger from the scaler The optocoupler
consists of an LED (Light Emitting Diode) whose light is
picked up by a photodiode which in turn generates a current
with the same frequency This signal passes through a
currentvoltage amplifier with a peak value of about 15
volts which in turn is fed to a TTL converter
33 Laser Power Stabilizer
The laser amplitude noise at the lasers output is
much higher than the Shot Noise Limit (SNL) even though
the laser has a feedback system Figure 33 shows the
light spectral density versus power A major goal of the
experiment is to have very good extinction so that the
SNL dominates the amplitude noise
Nevertheless we can further reduce the light intensity
104
------
20 Laser sre_~ClI_density at 1400_H_z_____
r---1
N shy-
1 5 -shy1--
N I - L-J
-
(f) 1 0 ~ C - (J)
U ---shy~
0 ~ ~- L 0 +- O I) -ltshy
UI shyU (]) 0_
(f)
00 ---~--+__--+---_t_---+-___I
o 400 8 12 1600 2000 wer [mW]
Figure 33 Laser light spectral density (in units of 1E-S) VS laser output power
bull
fluctuations and any other thermal noise introduced by
the optics by using a Laser Power Controller The laser
light passes through many optics elements and bounces off
many mirrors All these elements introduce llf noise due
to thermal fluctuations (eg index of refraction etalon
effects position fluctuations etc) Furthermore the
cavity itself with many hundreds of bounces could move
the beam so that there are amplitude fluctuations caused
by different absorptions at the different positions of
the optics elements
A Laser Power Controller (LPC) consists of an
Electrooptic Crystal (EO) a polarizer and a photodiode
Polarized light goes through the Electrooptic Crystal
then through a polarizer and finally through a beamsplitter
which redirects 2 of the light to a temperature controlled
(33 0 C) photodiode with a Smm x Smm sensitive area The
output of the photodiode is fed to an electronics module
which monitors this readout and tries to keep it stable
by modulating the EO crystal The polarizer before the
beamsplitter allows only one component of the light to
pass through namely the one that is parallel to the
polarization axis By applying voltage to the EO crystal
the original vector is rotated (so that the output of the
polarizer is attenuated or amplified) by an amount equal
(negative in sign) to the change in amplitude of the laser
light that the photodiode observes Actually applying
106
voltage to an EO crystal introduces ellipticity to the
original polarized beam but effectively it is the same
as if the beam was rotated by an angle equal to the
ellipticity because the polarizer that follows rejects
one component no matter what its phase relative to the
other component
Of course we can use the photodiode at a different
position on the light path and stabilize the power there
by modulating the EO crystal In our experiment the light
returning from the cavity is split to two parts by the
analyzer one part is the transmitted 11 (or extraordinary)
ray which has all the signal information and the other
is the rejected (or ordinary) ray which has no signal
information but has the same amplitude noise as the
transmitted light Therefore we placed our photodiode
in the path of the rejected light Figure 34 shows
the nominal noise reduction versus frequency and figures
35 and 36 show our data with LPC OFF and ON respectively
Figure 37 shows the 11f and amplitude noise reduction
using the LPC The shot noise limit corresponds to -105
dBV (R = 1070 DC = 9 dBV) the first ten channels are
not a faithful representation of the intensity because
the AC coupling attenuates these very low frequencies
At the beginning of each run we complete a form (table
32) so that we keep track of the runs and files we write
on the hard disk
107
o~____________________________________________~LPC Noise Reduction IX)
0 to
C 0 0 0Jshyc Q)
-+oJ laquo
0 ~
0
0 0 000
0000 000
0 0
000
000 ltXgt
0 0 deg0 o~
101 2 102 2 103 2
Hz
Figure 34 Nominal noise reduction in dB vs frequency
108
-50shy
70
-90 gt m u
110 I bull I It fill II JIlfll ampfIn lflLI II
I- I0 I 11ft Vlni~
_ 1 3 0 l 1 r T~ ~~ ~ Irq I0
150+---~--~--~--
200 -120 l-----imiddot ---1--__1_---1
1 0 40 120 200
CHANNfl Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 35 Typical data with LPC OFF
bull
50
70
-middot90 ~shygt CD u
1 10 shy
~ ~L 1 lh a ~I~ ~J-~A ~r~ ~l ~V~ shyo
-130
-150 -tshy200 -120 L1 0
bull
--------------------------shy
~
40 120 200
CHANNEL
Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -80 dBV
Figure 36 Typical data with LPC ON The noise floor around 78 mHz is 10 times smaller than in previous figure
AVERACE COMiCETE s-c
A Mo~ka X 0 Hz VI -B6961 dBv~mc____---
-45 --1-- ------r---1 l-- -- f Ibull
dBVmc
LogMog ~~I+I~~I~I~--~----~----~----~----~------~-----------------10 dB
d i v 1uV It I All I J I I I I
~
~ ~
i
~ I yJ
I bull p amp - f I ~ I If IdWI I I I J I f Ii ~ fI- I- ----- I NYi ~ WChjJ ~~~~JcJ--1 I-
-05 i
I~ I 1 - I 1---------- shy
-1251 I I~ Stot C Hz Stop 1 e kHz Spctum ChClM 1 RMS 100
Figure 37 1f and amplitude noise reduction using the LPC The two
spectra shown are taken with the LPC OFFON
Table 32
DATE ITIME ILOGFILE
Laser Power FFT Trigger
LPC ON Constant Power Constant Transmission LPC OFF Transmission
EGampG Preamplifier Resistance n
MAGNET ON OFF Cavity Shunt mirror Shape
TM = s MIN Current of reflections MAX Current Polarization RAMP RATE UP As QWP IN OUT
DOWN As
Filter OUT DC (Source off) DC (Source on) 12wF
Filter IN DC
12wF
DC Offset 2W F Offset 12wFllwF
Voltage at the Faraday Cell 11 0 BW 0 2 1V
Cavity with Gas Vacuum TYPE PRESSURE Pressure ENCLOSURE END TEMPERATURE DOOR END 1 2 3 BOX
Notes
112
Chapter 4 Analysis of Results
41 Noise Sources
In equations 232 and 235 we used the matrices for
the ideal crossed polarizersanalyzers
[~ ~J [0deg 0J 1 and
In this case the light going through the pair of crossed
polarizers is zero which is the ideal case In reality
there is always light after the second polarizer equal
to the extinction (ie ITIO = 0 2 see chapter 2 under
optics) times the light intensity before the second
polarizer To accommodate for this the above matrices
can be rewritten
[~ ~J [~ ~J and
so that the light intensity after the analyzer is I =
1002 Taking this into account we can rewrite equation
234 to lowest order
(41 )
113
Equations 237 and 239 change accordingly We will call
our signal Wo and we will know that this corresponds to
Wosin26 for ellipticity Eo for rotation and ~sin26 for
attenuation
Shot Noise Limit
The DC current IDC = Io (0 2 + ~ + a 2 ) flowing through a
resistor fluctuates1 resulting to a rrns current noise
given by
JSNL(rms) = ~2eJ dcBW ( 42)
where e = 1 6 x 10-19 C and BW is the measurement bandwidth
This noise is white (same in amplitude at every frequency)
It is also the limit for most precision experiments today
even though new techniques that can beat this noise are
becoming available 2
In the case where the shot noise is the limit the
signal to noise ratio can be deduced from equation 41
( 43)
where q = 065 is the quantum efficiency of the photodiode
(027 AW radiant sensitivity at 5145 nmmiddot 241 eV = ftw
see table 31) and N is the total number of current
electrons collected In equation 43 we have used the
114
zero to peak shot noise and neglected the term a 2 bull
Therefore
SNR= ( 44)
where P is the laser power before the analyzer and T the
measurement time In practice we use 02=n~2 (see the
discussion of the amplitude noise below) Then for a
signal to noise ratio 1 laser power P = 05 watts before
the analyzer and Wo = 10-12 rad the integration time
needed is
2(SN R)]2rw)T= -=SSdays ( 4S)[ Wo pq
It should be noted that for Wo = 4 x 10-12 rad the integration
time becomes less than 35 days From equation 44 it
is apparent that SNR is maximum when n~raquo 0 2 but the
function n~(02+n~2) is slowly varying and it only changes
from 1 (for n~2 02) to 2 (for n~raquo02) It is however
difficult to reach the Shot Noise Limit for intense light
Therefore we keep the Faraday rotation angle at such a
2level that n~2 0 bull
Amplitude Noise
We can now include in the above calculation the laser
amplitude noise which is proportional to the light power
115
(46 )
where A is the amplitude noise or spectral density at
the signal frequency In the case where the amplitude
noise is dominant then
(ie the SNR is independent of the laser power) The
SNR given in equation 47 has a maximum when n~2=a2 and
becomes
( 48)
from which we conclude that the SNR improves for better
(smaller) extinction
From equation 46 we obtain the limit of the noise
spectral density if we are to be shot noise dominated
154 x 10-9 ~BW[Hz] ( 49)
~T~ a +shy2
that is
Alaquo 154 x 10-6~Hz=-1162dB~Hz for a2+n~2= 10- 6
Alaquo487x 10-6~Hz=-1062dB~Hz for a2+n~2= 10- 7
and (410)
Alaquo 2flw
116
for P = 05 watts q = 065 and hw = 241 eVe
Preamplifier noise dynamic range
Any resistor at finite temperature generates a white
noise voltage across its terminals known as Johnson noise
The rms voltage noise is given by
V rms ( R ) = J4 k T R B W (411)
where R is the resistance in n BW the bandwidth k the
Boltzmann constant and T the absolute temperature 4kT
= 162 x 10-20 V2HZD at room temperature (200 C = 68 0 F
= 293 0 K) The current noise is obtained from Vrms divided
by R
I R ) = ~ 4 k T B 11 (412)rms( R
Figure 32 shows the current noise for the different
settings of the EGampG amplifier versus frequency Even
though Johnson noise is white Irms depends on frequency
because R is the feedback impedance of the amplifier
Usually there is a capacitor in parallel with the feedback
resistor to prevent the amplifier from oscillating at high
frequency and therefore the impedance becomes smaller at
higher frequencies Also the photodiode capacitance (and
hence the sensitive area) should be kept as small as
possible for the same reason Figure 41 shows 1G2 where
G is the normalized gain versus frequency and R = 107n
117
(f)
gt
o --
N I
x Ql
Ie ~
r
118
----- N
-shy(f) (l
E laquo
L-J
(]) (f)
0 I- L_I- 0
1E-10
1 E - 1 1
1E-12
1 E 13
1E 14
1 E 15
E - 1 6 1 E
r------shy
Amplitude Noise
Initial laser power 05 watts
Shot Noise
[GampG Arnplifier Noise
J bull _l--LL l~tlL J~~-----LL J-LLd -LlJshybull
10 1E-9 1 E --8 1 [ 7 1E-6 1E-5
DCF
Figure 42 Noise sources In the experiment The amplitude noise drawn here corresponds to spectral density of 1E-5 per root Hz
using the Hamamatsu amplifier (C1837) Using this curve
we found the 3dB (ie 112 ) gain point to be at 1420
Hz At zero frequency G = 1 Figure 4 2 shows the
contribution from different noise sources in Amps~Hz as
a function of the DC factor (DCF) ie a2+~~2
When an analog signal is converted to digital with a
certain sampling rate an error is intr~d~ced because of
the finite spacing of the digital readout Assume that
Vs is the resolution (spacing) of the digital readout
Then the rms noise associated with one measurement is
1 JV12 V 2ltV2gt=_ V2dV =_s~
o V 5 -V2 5 5 12
I 2 V 5J ltV gt=-- (413)
o 23
and for Ns measurements the error becomes
Vs a --== (414)
q 2J3N s
If further the measurement consists of N
photoelectrons the measurement error is a SN =IN (shot
noise) Supposing that the 2N photoelectrons correspond
2n 2nto the full scale of the digital converter - 1 ~
where n is the number of bits then the resolution is
Vs = 2N2n and the quantization error becomes
N a =--== (415)
q 2 n J3N s
120
If we want the quantization error to be equal to the shot
noise the sampling rate must be
(416 )
Let us take as an example 05 watts of laser light
which corresponds to 85 x 1017 photoelectronss and
assume a typical extinction of 10-7 (ie N = 85 x 1010
photoelectronssec) and n = 12 bits The sampling rate
must be at least Ns = 1700 Hz If we require the
quantization error to be 10 times less than the shot noise
the sampling rate must be 170 KHz
Our FFT (HP35660A) has 12 bits 125 KHz sampling rate
and a dynamic range of 72 dB This means that when the
highest level signal is A volts the FFT is insensitive
to any signal that is more than 72 dB (- 3981 times) weaker
than A volts From equation 41 we see that in our spectrum
we have a DC component with relative strength of 10-6 to
10-8 (in units of IO) and a signal at 2w F also with
relative strength of 10-6 to 10-8 Our signal ~o~o is
supposed to have a relative strength of 10-14 to 10-16
Therefore we need to use a bandpass filter to eliminate
or attenuate the DC and the 2w F signal The problem
becomes more difficult when we consider the misalignment
peak at W F and our signals at W Ft W M where W M is in the
tens of mHz The ratio of the two peaks is 2aljJo with a
in the 10-5 to 10-6 rad range and Wo in the 10-12 range
121
then we would need a dynamic range of 120 dB it is obvious
that the dynamic range of the FFT is not adequate to
accommodate this range and there is no filter with such
narrow bandwidth as to separate the sidebands from the
W F peak
The way to overcome this difficulty is to introduce
random noise3 4 the level of which is within the dynaDic
range of the instrument Using vector averaging the
noise reduces as 1N a where Na is the number of averages
and the peaks can appear after an adequate number of
averages Actually we don I t have to introduce any random
noise because we have the laser shot noise (equation
42)
As an example we can look at laser power P = 05 watts
(which corresponds to 10 = 85 x 1017 photoelectronss) 2
2 noBW = 4 mHz and 0 =2 Then the ratio
2anoo a n 10 -= 0 ~ 72dB (417)2 el DCBW ~ el 0 ll~BW
determines that a must be less than 27 x 10-7 rad In
the above we have used the zero to peak shot noise
lf or flicker noise
Equation 234 shows that the signal is given by Is =
10 1l0Wo and the misalignment peak by I w =21 0a1l0 Because
the two signals are very close together in frequency 10
122
has to remain stable throughout the measurement In
reality however 10 is subject to lf noise the source
of which has been discussed earlier (see section 33)
42 Data Analysis
Using equation 128 we can set limits on the axion
coupling constant 5
(418)
The external magnetic field is Bxt = (BDC+BO)2 = B~c+Bt
+ 2B DC B obull where Boc is the DC magnetic field and Bo is
its amplitude modulation In our case the dominant term
is 2BocBO and we therefore use B~xl=2BDCBo I is the
magnetic field length and we use 1 = 2 x 439 m (see table
21) E is the rotation we measure or a limit on the
rotation angle In these units (natural Heaviside -
Lorentz) a magnetic field of 1T can be expressed as 195
eV2 and 1 cm as 5 x 104 eV-1 therefore for e = 45deg
I
M = 2 1 4 G eV x ~ 2 B DC B 0 [ T 2] 2 (419)
123
Rotation
A) Data with magnet off Figure 43 presents a single
record of data with the magnets off and no QWPi the
bandwidth is 390625 mHz (single channel bandwidth 0976
mHz) The number of reflections is 790 plusmn 35 (the time
delay is found to be 33 plusmn 1 5 ~s in a cavity 1256 m long)
FigurEs 44 and 45 show the rms (taking into account
only the amplitude of the data points) and vector (taking
into account both the amplitude and phase of the data
points) average of 26 records (files) The small peaks
that appear are due to the FFT external trigger (most
probably from ground loops) and are absent when the trigger
is removed The voltage at the Faraday coil is VFC = 79
Vo-p (from equation 210 we find TJomiddotOo- p = 826 x 10-5
(radV) x 79 V = 653 x 10-4 rad) and the peak at 2WF
is 12wF = -10 dBVO-p The Faraday frequency is WF = 260
Hz and the signal peaks are supposed to appear at (260 plusmn
001953) Hz (ie plusmn 20 channels away from the center peak)
The amplitude of these peaks is -106 dBVO-p and using
the following equations we can deduce the rotation angle
that this corresponds to The amplitude at 2WF is (using
equation 237)
( 420)
whereas at the signal frequency
124
---------------------~~----------~ 0
~
o
f ~ 1~
0 ~
- CO L -CO - 0
r --
0 c
CO I I ~
-0 0+--
I--z5~ i
I ~
8I
III IIII
6lI c
(j)
j
I I III+~ c
l
1 en u
0 0 N
0 0 0 0 0 0t1) f- v) - I) Lr)
I - - -shyI I I
8 0
125
50
-70 ~
--90 gtm D
-110
tIJ 0
-130
JIrIVH LJj~~~~
150+---+---+---~---~----~--
200 -120 t10 40 120 200
CHANNEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953mHz Amplitude of first harmonic -105 dBV
Figure 44 RMS average of 26 files
--
50
-70
-90 -
CD -0
-110
fJ
-130
_---------------------
~ 150+---~~------
200 -120 -40 40 120 200
CHAN~JEL
Center frequency 260 Hz Full BW 390 mHz Magnet frequency 1953 mHz Amplitude of first harmonic ~107 dBV
Figure 45 Vector average of 26 files (the same as in previous figure)
Is=oTJoEo (421)
From equations 420 and 421 we have
TJo Is E =--- (422)
o 2 12~
and substituting the values of I zw = -10 dBVO-PI Is =
-106 dBVO-PI we obtain
6 53 x 10-4 d 10-106120V ra o-p -9 Eo= 2 -)020 =517XIO rad (423)
10 V O-P
The above rotation limit corresponds to the false peak
(induced by the FFT trigger via ground loop) at 1953
mHz (the magnet period at the time was 512s but this
is data with the magnets off) Assuming the false peak
is removed Is = -113 dBVO-p and the rotation becomes 23
x 10-9 rad The noise around 80 mHz from the center peak
is -125 dBVo-p This corresponds to Eo = 58 x 10-10 rad
and comparing this with equation 423 the need to drive
the magnets as fast as possible becomes apparent
To acquire one record takes 1024 s (for a total
bandwidth of 390625 mHz) With the 26 measurements we
can place a limit of Eo = 58 x 10-10 rad therefore the
sensitivity of the experiment is
Es = EoX [T = 58 x 10- 10 rad x ~ 1024 26 =
=86X 10-sradlJHz (424 )
128
It should be noted that in this run the Laser Power
Controller (LPC) which can give us another factor of 10
in sensitivity was not in place During a later run
when the LPC was in operation we achieved a limit of Eo
= 54 x 10-10 rad corresponding to a sensitivity of Es =
18 x 10-8 radJHz
B) Data with magnet on Figure 46 shows the data
with the magnets on (log file name FNM30811ASC which
means it is the third run on 081189) The vacuum inside
the cavity is 65 x 10-6 Torr and 5 x 10-7 Torr at the
two ends and 10-5 Torr in the box The Faraday voltage
is VFC = 17 VO-P (ie no = 14 x 10-3 rad) 12wf = 004
dBVO-p total BW = 780 mHz and the magnet frequency is
3906 mHz (magnet period is 256s) The peaks are at Is
= -85 dBVO_p and from equation 422 we have
(425)
The number of reflections is the same as before 790 plusmn
35 whereas the magnetic field is modulated from 1100 to
1600 Amps with a ramp rate 100 Ampss and from 1600 Amps
to 1100 with a rate of 35 Ampss At these currents the
magnet transfer function is 1563 gaussAmp (see fig 26
and 27) that is Bl = 172 Tesla and Bh = 25 Tesla +8 8-8 1 8 2_8 28 11
Using B DC =-2- B O=-2- =+ 2BDCBOT we conclude
from equations 419 and 425 that
129
50
70 ~-
--g gt rn TJ hN~ vJ~Wyen
w o
130
-150+------4------~----middot+---
200 - 120 - 40 40 120 200
CHANNE Center frequency Full BW 780 mHz
260 Hz
Magnet frequency 39 mHz Amplitude of first harmonic -85 dBV
Figure 46 Typical rotation data with the magnets ON
790 M=214GeVx 8 392x 10shy
( 426)
assuming this limit to be correct within 10 due to
uncertainties on the number of reflections the Faraday
rotation (ie Faraday cell calibration) and the
polarization direction (angle e) bull
In order to find the source of these peaks we took
data with the shunt mirror (fig 21) blocking the light
from entering the cavity This way any electronic pickup
would appear again at the right frequency with the same
amplitude To account for the excess of light seen by
the signal photodiode (the cavity attenuates the light
which is now bypassed) we used neutral density filters
Figure 47 shows that under these conditions the peaks
appear at -105 dBVO_p This level is the same as the
peaks which appeared during the first run without the
magnet We have I zw = -85 dBVo-p VFC = 17 VO-p (~o = 14 x 10-3 rad) and therefore Eo = 105 x 10-8 rad Thus
the modulation of the light intensity increases by a factor
of 3 when the light traverses the cavity and we must
search for the source of this effect
The amount of gas inside the cavity cannot produce
such a large rotation and the endcaps were shielded against
any stray magnetic field below 1 mgauss without the
131
-50
7 () -shy
-90middotmiddotmiddot gt m TJ
-110 w I)
-130
I I ~~-150 -200 120 -- 4 0 40 120 200
C11AN r-j r-L Center frequency 260 Hz Full BW 780 mHz Magnet frequency 39 mHz Amplitude of first harmonic -1 05 dBV
Figure 47 Shunt mirror data with the magnets ON
magnetic shielding we measured a stray DC magnetic field
of 24 mgauss and a modulation of 3 mgauss at the far end
endcap where the lead pot is located (the above field
strengths are measured along the light propagation
direction only this component produces Faraday rotation
on the mirrors) Any Faraday effect on the mirrors is
linearly proportional to the magnetic field and the 3
mgauss could only cause a modulated rotation of less than
10-9 rad In any case further shielding the endcaps
with Il-metal reduced the stray field (even the DC component)
below the range of our magnet probe (1 mgauss) This had
no effect at the level of the peaks (actually the data
shown in figure 46 are with the Il-metal covering both
endcaps) In the next step we established that the effect
was proportional to both the BDC and Bo which implies a
quadratic dependence on the magnetic field
The ultimate test is rotating the polarization to 0deg
(ie parallel to the external magnetic field) Trying
this we found that the amplitude of the peaks was around
the same value (within 50) After that we positioned a
Mitutoyo displacement gauge on the back of the far end
end-cap We observed a signal at the magnet frequency
shown in figure 48 which corresponds to a 6 nm periodic
motion We also established the source of this signal
to be the magnet motion of about 04 JlID Table 41
133
bull
R~AL-TJM~ Ave COMPLETE
A Mark X 390625
-54 dBVm
LogMag 10 dB
div
w
-134
stct t 0 Hz Spct um Chctn I
---I-----middot~
____L-____
StOpl 15625 Hz OVLD RMSS
mHz
s c Tlk
YI -BlBB dBVrma
Figure 48 The cavity mirrors move due to magnetic motion
summarizes some of our rotation data with the magnetic
field on and off and for light polarization at 45middot or
omiddot with respect to the external magnetic field
Ellipticity
To search for ellipticity we include a QWP in front
of the Faraday cell to convert it to a rotation Data
taking is the same as previously except for making sure
that the QWP has reached thermal equilibrium Table 42
presents the results of the ellipticity data Several
trials with different elliptical orientations established
the fact that there was a strong correlation between
results and orientation The data shown in table 42 are
with different orientations of the cavity ellipse
Figure 49 shows the limits obtained from the rotation
data with N = 790 reflections Eo == 2x IO-Brad and B~xt
= 117 T2 assuming the signal to be due to axion
production Superimposed are the ellipticity limits for
induced ellipticity tlJo = 29 x IO-8 rad N = 38 reflections
and B~Xl = 1 36 T2 If both the rotation and ellipticity
peaks were due to axions then the axion mass and its
coupling constant could be found from the intersection
points of the two curves in figure 49 Assuming this
is the case we obtain an axion mass of about 10-3 eV and
a coupling constant somewhere between
2x I04GeVltMlt6x l04GeV due to many intersection points
135
Our noise floor during data taking was dominated by
amplitude noise which at best was a factor of 5 from the
shot noise We did not take particular care to lower
this noise because the spurious signals where at much
higher level
In order to find the absolute rotation of our signals
we have used the calibration given in equation 210 for
the Faraday cell An alternative approach is to use the
extinction to find the Faraday rotation It is easy to
show following equation 41 that
n J2W110 = 20 - shy
J DC (426)
where J~c is the DC light measured while the Faraday cell
is off
We saw in the various sets of data gathered with the
two methods a rate of agreement from 5 - 20 We decided
therefore to use the calibration of the Faraday cell for
consistency Applying the Faraday cell calibration to
the measurement of the Cotton-Mouton constant of N2 and
comparing our results with previous measurements (which
are very accurate) we found a 5 agreement
136
Table 41
Rotation data
Log file name FNM
00725 02728 02810 01811 03811
Magnetic field
(2BDC B O[T2]) 00 117 1 65 00 165
Polarizashytion
(degrees) 45 45 45 45 45
Faraday rotation
[rad] 65X10-4 12x10-3 21x10-3 21x10-3 14X10-3
Extinction 14X10-7 22x10-7
790
3x10-7 3X10-6
Number of Reflections 790 790 790
0 Shunt Mirror
Rotation Eo[rad]
52X10-9 2X10-8 43x10-8 45X10-9 10X10-8
Phase of the two
harmonics (-+) [ 0 ]
70 -70 No FFT trigger
79 -79 20 -20 86 -86
Frequency [mHz]
Visible peaks
1953
YES
1953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] NA 39 37 NA NA
Vacuum in the cavity ends [Torr]
65x10-6 5X10-7
2x10-6 2X10-6 NA
Number of records
26 4 5 36 49
Table continued on next page
137
Rotation data (Continued)
Log file name FNM
30S12 00S13 10S13 00S14
Magnetic field
( 2 B DC B0 [ T 2 ])
074 0S2 123 1 65
Polarization (degrees
45 45 45 45
Faraday rotation
[rad]
1 4X10-3 14X10-3 14X10-3 14X10-3
Extinction 3X10-6 3x10-6 3x10-6 3x10-6
Number of Reflections 790 790 790 790
Rotation Eo[radJ 17X10-S 92X10-9 16x10-S 35x10-S
Phase of the two
harmonics (-+) [ ]
No FFT trigger
No FFT trigger
No FFT trigger
No FFT trigger
Frequency [mHz]
Visible peaks
3953
YES
3953
YES
3953
YES
3953
YES
Coupling constant
M [105 GeV] 39 57 53 41
Vacuum in the cavity ends [Torr]
2x10-6 2x10-6 2x10-6 2x10-6
Number of records
45 35 6 3S
Table continued on next page
13S
Rotation data (continued)
Log file name FNM
30816 40816 10818 21005 31005
Magnetic field
(2B DC B 0[T2]) 078 082 165 1 36 1 36
Polarizashytion
(degrees) 45 45 0 0 45
Faraday rotation
[rad] 1 4x10-3 14X10-3 14X10-3 95X10-4 12X10-3
Extinction 3X10-6 3X10-6 1 5X10-5 3X10-7 14XIO-6
Number of Reflections 790 790 790 33 33
Rotation Eo[rad]
23X10-8 27x10-8 2X10-7 14X10-8 16x10-8
Phase of the two
harmonics (-+) [ 0 ]
No FFT trigger
No FFT trigger
No FFT trigger
2 -2 23 -23
Frequency [mHz]
Visible peaks
39
YES
39
YES
39 78
YES YES
78
YES
Coupling constant
M [105 GeV] 35 33 17 12 11
Vacuum in the cavityends [Torr]
2x10-6 2x10-6 1x10-5 1x10-4 1x10-4
Number of records
10 9 5 10 15
Table continued on next page
laquo
139
Rotation data (continued)
Log file name FNM 11006 21006
Magnetic field ( 2 B DC B 0 [ T 2 ]) 1361 36
Polarizashytion degrees 4545
Faraday rotation [rad] 11X10-362x10-4
3x10-7 3x10-7Extinction
Number of Reflections 38 38
47X10-8 59x10-8Rotation Eo[radJ
-31 31 harmonics (-+)
Phase of the two -36 36 [ 0 ]
78 Visible peaks Frequency [mHz) 78
YES
Coupling constant M (105 GeV]
YES
2022
5x10- 4
ends [Torr]
Number of records
5x10-4Vacuum in the cavity
59
140
Table 42
Ellipticity data
Log file name
FNM 01012 21015 31016 21016
Magnetic
field
( 2 B DC B 0 [ T 2 ])
00 1 36 1 36 1 36
Polarization
(degrees) 45 45 45 45
Faraday
rotation
[rad]
2X10-3 2X10- 3 2X10-3 2X10- 3
Extinction 1 5x10-7 15X10-7 15x10-7 15X10-7
Number of
Reflections 38 38 38 38
Ellipticity
V o[rad] 1lX10-8 29X10-8 11X10-7 22X10-7
Phase of the
two harmonics
(-+) [ 0 ]
-81 81 76 -76 -137 137 -128 128
Frequency
[mHz]
Visible
peaks
78
NO
78
YES
78
YES
78
YES
Vacuum in the
cavity ends
[Torr]
10-4 10-4 10-4 10-4
ltI
141
CJ
I
f
t I
-l-
I w
T)
I 0 w ----- ~ W E
gtQ
0(f) (f) shy0 -fIJ
sectE 2c 0
c 9shy
~ w I c -
W 0
- ctl 0a iri ori Q) ) 0)
u
L()
w ltD w shy
L() w shy
~ w shy
[18~ ] 1 +UO+SUO~ 6uldno)
w0shy0 0 shy
142
References
1) The art of electronics Paul Horowitz Winfield Hill Cambridge Univ Press TK7815H67 1980
2) Talk given at Brookhaven National Lab by Richard E Slusher ATampT Bell Labs NJ
3) J Butterworth et al J Sci Instrum 44 1029 (1967)
4) Gary Horlick Anal Chem 47 352 (1975)
5) This experiment was first proposed by L Maiani R Petronzio and E Zavattini Phys Lett 175B 359 (1986)
bull
143
Chapter 5 Cotton-Mouton Coefficients of the Noble Gases
Phenomenologically the origin of the cotton-Mouton
effect is similar to the QED vacuum polarization where the
role of the e+e- pairs is played by the electrons in gas
atoms In the latter case the electric field of the light
induces a polarization of the atoms
We can measure the Cotton-Mouton constant of different
gases by filling up the cavity with the particular gas while
including the QWP in our system When a polarized light is
traveling in a direction orthogonal to the external magnetic
field it is gaining an ellipticity1 equal to
lJ = nCM sin (28) f dx(i3 x k)2 (51 )
where CM is the Cotton-Mouton constant of the gas at given
pressure and temperature B is the external magnetic field
vector k is the unit vector along the light propagation
and e is the angle between the light polarization (ie the
electric field of the photon) and Bxk For our apparatus
we can safely assume the ellipticity as
lJ = nCM sin (28)B 2 IN (52)
where N is equal to the number of bounces and 1 is the
magnetic field length The eM constant is defined as
(53)
144
where np and no are the parallel and orthogonal indices of
refraction respectively B is the magnetic field and A the
light wavelength
We have measured the Cotton-Mouton constant of N21 He
Ar and Ne The CM constant of N2 and Ar had been previously
measured the former rather accurately23 and the latter
with less precision Therefore we checked our system
against any systematics by measuring the CM constant of N2
improved the measurement of Ar and measured the constants
for He and Ne for the first time
The gas pressure was monitored by a WALLACE amp TIERNAN
manometer mounted at the center of the cavity The manometer
had an offset of 12 mmHg and the precision of the readout
was 05 Torr Prior to every measurement we pumped down
to 4 x 10-4 Torr filled up with gas and pumped down a
second time again to 4 x 10-4 Torr (cavity pressure) he
purpose of this was to prevent any gas that was previously
trapped on the box walls to be flushed out by the incoming
gas and thus contaminating our system This is important
because the Cotton-Mouton constant of 02 is 10 times larger
than that of N2 whose constant is in turn more than 103
greater than that of He The cavity was opened at the
beginning of the run so that the sublimators were saturated
not absorbing any more gas All the pumps (including the
ion pumps) were off during the measurements and constant
pressure was maintained throughout
145
The temperature was monitored at the two cavity ends and
at the cavity center There was consistently a temperature
gradient the cavity end close to the enclosure being warmer
than the end near the tunnel exit
The number of reflections was kept small (between 30 and
40 the light spots formed one ellipse on each mirro) so
that the light dispersion due to the gas was slight The
Laser Power Controller was always in constant transmission
mode (ie effectively off) because the signals were well
above the noise a Hanning wiridow (the data is multiplied
in the time domain by a sine wave with a period equal to
the time record this way the frequency width of the
misalignment peak at W F is kept small) was used in the FFT
The magnetic field period was 128 sec with modulation from
1250 ~ 1600 Amps and ramp rate of 115 As both ways The
magnet current modulation is shown in figure 51 The
corresponding magnetic field (using figures 26 and 27
where we find the transfer function to be TF = 1563 gaussAmp
with an accuracy of 1) is BI = 1250 Amp x 1563 gaussAmp
~ BI = 1954 Tesla and Bh = 25 Tesla Therefore BDC = 22
Tesla and B~ = 0274 Tesla In order to find BO of the
first harmonic we used a diagnostics voltage from the
electronics driving the magnets This voltage is
proportional to the magnet current with calibration of 2
mVAmp Fourier analyzing (fig 52) the voltage we find
BO = 199 Amps x 1563 gaussAmp = 031 T that is
146
--
------
-----------------
---~-----
------ ---- shy-~------
en -CD N en - lcr Q)
C 0gt Q) c ta-Q)
E 1=
bull
=I--
Q) - c-c Q)0) ta l ~ o
147
N J E CD -ta-en c E laquo 0 0 ltD -0 If)
c E laquo 0 IJ) CI--2 ni 5 C 0 E E c 0) ta ~
Lri Q) l u 0)
-----J--shy
bull i----middotf --1----1
middoti It R-I--II--middotH illmiddot ----+---1------1
-B3 Lshy__--shy__-
REAL-TIME AVG COMPLETE Sr-c TIl-lt
A Mar-ke XI 7B 125 mHz V -11 026 dBVr-ma I -~-- -- ------------------------shy
-3 dBVr-m_
-----I------r-----~-----~---~LcgMag 10 dB
dlv -----+1----+----shy
~ 0)
Start 0 Hz Stopa 15625 Hz Spcactrum Chan 1 RMS5
Figure 52 Fourier analysis of the magnet current
o -------------------shy
-20
40
gt -60-shym u
-80 b It)
100
-120 200 -120- 110 40 120 200
CHI~JN EI Center frequency 260 Hz Full BW 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic ~22 dBV
Figure 53 Typical N2 data at 147 Torr
bull
- 40 - -----shy
60
-80 -shy
gtCD 0 -100
VI o -120
-140 I
-200 -120 -40 40 120 200
CHANNEL Center frequency 260 Hz Full 8W 156 Hz Magnet frequency 78 mHz Amplitude of first harmonic -85 d8V
Figure 54 Vacuum ellipticity drih
BO = 1135 XB~ Therefore 2BOBDC = 136 T2 Figure 53
shows typical N2 data whereas figure 54 shows data without
gas The peaks which appear without gas are attributed to
the motion of the magnet which moves the light ray to a
different spot on the QWP Figures 55 56 57 58 show
the ellipticity versus N2 Ar Ne and He pressure
respectively From the above figures it can be concluded
that the induced ellipticity is proportional to the gas
pressure We used this assumption to derive the Cotton-Mouton
(CM) constants because the slope of the lines (ellipticity
vs pressure) is proportional to the CM constants per Torr
with a proportionality factor given from equation 52
nSin(2B)S2lN = (135007) x 1013 gauss2-cm with N = 36
reflections B2 = 136 T2 and B = 45deg10deg Using this
method the CM constant becomes independent of the absolute
measured ellipticity (in case the Faraday cell is not
calibrated correctly) The two methods one using the slope
of the above graphs and the other using the absolute
calibration of the Faraday cell give the same values for
the CM constants within 5 for all the gases We therefore
conclude the Faraday cell calibration to be correct within
5 Table 51 shows the CM constants for the various gases
at 760 Torr and temperature of 255degC
In the case of Ar Ne He (monoatomic gases) a two-level
atom model is utilized4 in which the Cotton-Mouton constant
151
L()
G
1
agt~-1 l
q- () ()
e0 c c Q)r-- 0)
IshyIshy 0 0 -Z
L-l ui
gt
~
amp n 0
t
0 (f) Qen (J) Qi
Ishy 0 Q) ()
L l~~ ~J 0
- - E
Lri Lri
C-J Q) 0 0)= u
bull
shy
f-shyI
lJ1 qshy
f-shyI
W n
f-shyi
W C-J
f--O I
W
[pOJ] 4 I J14 dl ll-lbullbull bull J
l52
I I
j
1
shy
shy
o
Ci Al)qd3L
153
0 0 ~
0 f-shy i
shy
as I
0 cent -shy
--- I-lshy
0 i-
VJ VJ Q) e laquo
L-J ~ (lj ~ 0
-shyIshy
J U
u ~ (f) Q CD
0 rmiddot L
ID 0
0 Ishy
lt I -g
OJ cO ampri Q) I enu
0 11)
o N
I r
-
~ l-- -- 0
1 N I
f--o---
0
fshy
0 ~
--7-4
0
- j I 0
1 CO
0 L()
0 N
fshy fshy fshy ro CO I I I I
w w w w w 0 ill N 0 0 N -shy -shy ro ~
r ~
LpOJ J j(~J~d113
---- L L r-
f-L-J
C
~ fJ
(I)
Q) shy
r-L
Q) 7
e Ugt Ugt (l) ~
0 (l)
Z ui
2 -gt gtshy
poundshy(l)
0 (l) () 0 E 0 Q) 2gt u
154
16E-7
j r- U j(1 L
L-J L12E 7 gt
J
()
-1-
O
-W J
I- 80E 8 ~
111 111
40E - 8 I L __~_-L--L- ~_-L-_----l_--------
120 160 200 240 280 370 360 400 440
11 0 Fres-ure [T (lrr]
Figure 58 Induced ellipticity vs He pressurP
1
~
bull
is related to the linear dimensions of the excited states
of the atom
40nmc 2 5CAQ----+-- (54 )a(n-l) 2n
where Wo is the energy difference betvleen the ground state
10 14and the first exciter stat euro I W= 2nv whEre v is 583 x
Hz for the green light (see equation 21) me is the electron
mass Ae (-hmec) is 2n times the electron Compton wavelength
rl is the radius of the excited state c the speed of light
in vacuum a the fine structure constant and n is the index
of refraction of the gas Table 52 delineates the parameters
for the three gases and table 53 presents their
Cotton-Mouton constants the corresponding ~ltrTgt and the
atomic sizes of the closest atoms
bull
156
Table 51
Cotton-Mouton constants
Gas SlopeX1010 [radTorr] CMX1020 at 255degC and 760 Torr [1gauss2-cm]
N2 7550plusmn100 -4300plusmn100
Ar lllplusmn2 62plusmn2
Ne 97==004 54 01
He 34plusmn01 19plusmn01
CM = Slope(nlB 2 N) = slope x 737Xlo-14gauss2-cm at T
= 255degC assuming e = 45deg The assigned error above is
the statistical one the CM values could be as much as 5
higher due to the uncertainty in the polarization angle 9
which introduces a biased (systematic) error The signs of
the CM-constants are derived from the phase of the vector
signal and are relative to that of N2 whose sign is assumed
from references 2 and 3
bull
157
Table 52
Gas Wo
[cIs]
W (green)
[cIs]
n - 1
Ar 1 46x10 16 367x1015 296x10-5
Ne 221x1016 367X1015 685X10- 5
He 21xI0 16 367xI015 357xlO- 5
Table 53
Gas
Cotton-Mouton
Constant
[1gauss2-cm]
Jlt rf gt
[A]
Alkali
Atom
Ref 5
ro[A]
Ar 62x10-19 240 K 238
Ne 54x10-2O 187 Na 190
He 19x10-2O 185 Li 1 55
The reported values are at 255degC for the CM-constants and
Jltr~gt and 17degC for the rest
158
References
1) F Scuri et al J Chern Phys 85 1789 (1986)
2) A D Buckingham Tras Faraday Soc 63 1057 (1967)
3) S Carusotto et al Jour opt Soc Am Bl 635 (1984)
4) S Carusotto et al CERN-EP83-181 (1983)
5) American Institute of Physics Handbook McGraw Hill NY 3rd ed (1972) bull
bull
159
Chapter 6 Conclusions
bull
bull
In the above pages I have described an experiment
performed at Brookhaven Laboratory where we have searched
for the effect of a magnetic field on the propagation of
light in vacuum In view of the success of QED in other
physical processes in particular the Lamb shift and g-2 of
the electron and muon one expects that the vacuum will be
birefringent due to Delbruck scattering The present level
of sensitivity does not allow us to observe this effect as
yet
On the other hand our experiment has set a limit on the
coupling of light scalar or pseudoscalar particles to two
photons of
1 1 gayylt M = 4x lOsCeV
This can be translated to a quark coupling
1 1 lO-4 C V-Igaqq--- - e fa Nag ayy
with N the number of flavors This limit is an improvement
of two orders of magnitude as compared to laboratory
experiments searching for such particles More stringent
limits exist from astrophysical observations and in
particular the evolution of the sun Light pseudoscalars
160
coupling to two photons would be produced in the solar
interior but because of their weak coupl ing they could
escape thus increasing the cooling rate beyond their
observed values The limit on gavv from the sun is M ~ 108
GeV which will be also reached by the present apparatus in
the near future
We have also measured for the first time the Cotton-Mouton
coefficient for the noble gases Neon and Helium which was
previously unknown These results fit reasonably well within
existing calculations
The technical aspects of this work involved the
integration of two massive superconducting magnets with a
highly sensitive optical system We have demonstrated that
this is feasible and established the present limits of
sensitivity These arise from the coherent motion of the
cavity mirrors which induces a rotation of order
e 0 -6x 10-9 rad
for 33 reflections in the cavity The observed rotation is
proportional to the number of reflections The noise level bull ignoring the induced signal is at
e-6X 10- 10 rad
which corresponds to a sensitivity of
Es-2x 10-8rad~Hz
The analysis of the data is carried out in the frequency
domain the basic modulation frequency of the magnetic field
being -78 mHz and limited by eddy current heating Thus
161
bull
lf and phase noise become the dominant contributions
Nevertheless we were able to reach within a factor of five
of the limit imposed by the shot noise of the laser
By eliminating the spurious noise introduced by the
cavity mirror motion and by increasing the dynamic range of
our analyzer we should be a~le to rea=h a sensitivity of
Es-IO-9rad~Hz
Thus a measurement of 106s (approximately 20 days of data
taking) should allow the observation of the QED Delbruck
scattering with a signal to noise ratio of approximately
three
The present apparatus is l1mited to a full field of 5T
at a modulation rate of OlTsec However new magnets for
accelerator applications are being constructed with peak
fields of lOT The availability of such magnets in the
future will make possible the measurement of Delbruck
scattering with precision Our sensitive ellipsometer and
the further improvements in the apparatus are an essential
contribution to the achievement of this goal which has eluded
physicists for over half a century
It is also natural to ask whether the noise level can
be improved by increasing laser power Our experience is
that this is not necessarily true because increased power
introduces heating of the optical elements with resultant
noise as well as instabilities We believe that 1 W of
162
light exiting the cavity is optimal An alternate approach
to increasing the sensitivity is to increase the optical
path length with our present delay-line method we feel
that 1000 traversals are a good match to the mirror (and
also magnet aperture and stability) Higher reflectivity
mirrors will be available in the future 1 - R lt 10-4 but
in that case one should examine a Fabry-Perot resonator
This does not require a great aperture and can accommodate
a large finesse but for a 10m spacing its stability is a
serious problem necessitating highly sophisticated feedback
techniques
The future direction we will explore is to replace the
Faraday cell with an electrooptic modulator (Pockel s cell)
This removes any limit on the modulation of the beat frequency
but does not solve the magnet modulation problem It does
however eliminate the need of a Aj4 plate for the measurement
of ellipticity and is expected to yield better noise figures
Combined with increased modulation amplitude and larger
dynamic range in the detector (ADC) we should gain at least
a factor of 10 in sensitivity the principal limit being
the laser shot noise
With regards to elementary particle theories our
experiment has set limits on axion-photon-photon coupling
gayy ~ 25X IO-6 GeV -1
bull
163
by a purely laboratory experiment for ma lt 10-3 eV As
discussed this eliminates a certain class of theories We
also set limits for a neutrino magnetic moment by
substituting the electronpositron pair with the
neutrinoantineutrino pair in the figure 12 and assuming
an ellipticity limit of 2 x 10-8 rad
With m v10eV this corresponds to Ilvlt 10-5 1lB which is not
very stringent as compared to the GUT prediction of
Ilv = 10- 19 118lt
Our limit on gayy is weaker than the solar limit but the
improved apparatus should reach and exceed that limit We
are also not as sensitive as the cosmic axion searches which
establish
but over a very narrow mass range 45 x 10-6 eV ltmalt 16 x
10-5 eV Furthermore the cosmic axion search is based on bull
the existence of an axion halo which provides the closure
of the universe clearly our experiment is free of this
condition and is thus a direct test of models of elementary
particles independent of cosmological assumptions
Several proposals have been made for experiments to
search for light scalars and pseudoscalars in view of the
strong belief that these particles must exist However
164
none of these experiments has been mounted as yet and no
results other than ours have been so far obtained One
possible class of experiments is to produce and detect
axions as compared to our approach where we observe only
the production by its effect on the incident beam Such
experiments require higher sensitivity since they involve
g4 as compared to our g2 Van Bibber et ale proposed the
use of a very high power pulsed laser available at Livermore 1
Buckmuller and Hoogeveen2 propose the use of X-rays from a
synchrotron light source and axions are produced in the
Bragg scattering from a crystal In this case a larger mass
range is available for the search In contrast Vorobyev
et al 3 propose the use of microwaves incident on a ferrite
whereby axions are produced by their coupling to the aligned
spins 4 A limit on axion production by microwaves in a
magnetic field has been given by Rogers et ale 5 We mention
these proposals to indicate the current interest in the
subject and possible future directions
Another approach is to search for axions produced in the
sun 6 In that case the axions would have energies in the
KeV range (solar interior) and one would have to coherently
convert the axions to X-rays which would then be detected
Pointing toward the sun would improve the signal to noise
ratio
Finally we note that our experiment is also sensitive
to light spin-2 particles that couple to two photons as it
165
is the case for the graviton In that case of course M =
MPlanck =1019 GeV indicating a very weak coupling as compared
to the sensitivity of our experiment which can reach only
M = 108 GeV The common feature of all these processes is
that light particles propagate with the same speed as photons
and thus the two fields can remain coherent over long
distances This enhances the production rate and indeed we
have a mixing (or oscillations) between the two eigenstates
of the field one state is that of the incident photons
the other that of the sought after particle such processes
are thought to be occurring in stars with strong magnetic
fields 7
In conclusion our experiment is the first attempt to
exploit the coherent transformation of photons into other
weakly coupled light particles Whether these predicted
particles do actually exiampt or not can only be answered by
further experiments bull
bull
166
References
1) K van Bibber et al Phys Rev Lett 59 759 (1987)
2) W Buchmuller and F Hoogeveen Coherent Production of Light Scalar Particles In Bragg Scattering ITP-UH 989 Univ of Hannover FRG (1989)
3) P V Vorobyev H V Kolokolov and B F Fogel JETP Lett 50 58 (1989)
4) R Barbieri et al Phys Lett B226 357 (1989)
5) J Rogers et al Appl Phys Lett 52 2266 (1988)
6) K van Bibber et al Phys Rev D in press (1989)
7) S L Adler Ann of Phys (NY) 87 599 (1971)
bull
167
Index
Accelerator 7 8 34 162 Amplifier 94 97 Analysis 7 113 123 Argon 2 5 45 145
151 157 Astrophysical 8 160 Attenuation 25 64 89 Axial symmetry 18 Axion 8 16 19 23 25 89 123 135 163 165 Axion mass 19 29 135 Bandwidth 99 117 124 Berrys phase 9 Birefringence 13 56 160 BK7 glass 59 Bounces 45 63 144 Branching ratio 19 Brewster 48 Calibration 62 131 51 Capacitance 94 96 cavity 40 55 63 78 87 106 124 CBA 7 34 40 Chiral 32 Coherence 26 Coil 59 102 Compton 156 Cosmic I 21 164 Cotton-Mouton 5 136 144 158 Coupling constant 18
23 29 32 123 135 137 CP symmetry 16 CPT 17 critical fields 11 Cryogenics 34 Curvature 48 55 63 DCF 120 Delbruck 5 16 160 Electric 10 11 14 144 Electric dipole moment 17 Electric displacement 11 Electronics 93 94
Electrooptics 106 163 Electroweak scale 19 Ellipticity 14 87 135 141 151 Euler-Heisenberg 11 Extinction 55 12 136 137 Extraordinary ray 56 107 Fabry-Perot 163 Faraday cell 59 94 Faraday rotation 40 4587115133 Feedback 48 97 103 Feedthroughs 62 Filter 62 97 Fourier 4 93 146 Frequency 4 11 40 87 104 137 141 Gauge 16 54 133 Goldstone boson 19 Green light 14 48 156 Ground loops 97 104 124 Helium 5 7 145 151 157 Higgs 18 21 HWP 55 58 86 Index of refraction 11 53 145 Interferential 63 68 Isotropic 10 Jitter 94 Jones matrices 84 86 89 Jones vectors 84 85 Lagrangian 11 16 23 Lamb shift 160 Laser 45 48 78 104 165 Lifetime 19 21 Magnetic field 11 23 26 34 46 53 59 61 123 137 141 146 Magnetic field length 26 29 123 Maxwell 15 Microwaves 165 Mirror 9 26 45 48 63 68 163
168
Misalignment 87 101 122 Mixing 25 166 Muon 15 160 Neon 5 145 151 157 Neutron 17 Neutron stars 15 Nitrogen 5 145 151 157 Optocoupler 104 Ordinary ray 56 107 oscillation length 29 Parity 16 Peccei-Quinn mechanism 18 Phase 4 14 18 25 58 94 103 137 141 Photodiode 94 106 Photons 1 5 8 21 26 160 165 166 Pion 19 Planck mass 166 Pockels cell 163 Polarizer 45 55 86 Power controller 106 129 Power stabilizer 104 Primakoff 21 32 Pseudoscalar 8 23 160 164 Pump 53 QeD 16 18 QED 5 10 15 160 Quantization 120 Quarks 1 Quench current 34 QWP 56 58 86 135 Ray transfer 73 82 Rayleigh 16 Reflections 26 64 124 137 151 Reflectivity 48 63 64 163 Relics 1 Rotation 8 9 26 56 88 102 124 135 137 161 Satellites 93 101 Scalar 1 821 32 160 Sensitivity 2 9 128 161 Silicon 96
Spectral density 104 116 Spurious 9 45 Standard theory 18 Stray magnetic field 40 131 Sublimator 54 145 Sun 9 160 165 Supersymmetric theo~ies
8 Systematic 145 157 Telescope 78 Tensor 23 Tcffison 16 Topological 16 Transfer function 40 129 Triangle anomaly 21 Units 11 14 23 26 100 Vacuum 4 21 48 54 129 138 141 Vacuum polarization 10 11 Verdet constant 45 61 Virtual 2 5 10 23 Wavelength 14 51 61 X-rays 15 165
Noise 11f 106 107122 161 amplitude 104 107 115 flicker 122 shot 104 107 114 136 bull
169
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