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P. Ewart
The science of light
Oxford Physics: Second Year, Optics
• Geometrical optics: image formation
• Physical optics: interference → diffraction
• Phasor methods: physics of interference
• Fourier methods: Fraunhofer diffraction
= Fourier Transform
Convolution Theorem
• Diffraction theory of imaging
• Fringe localization: interferometers
The story so far
Oxford Physics: Second Year, Optics
t
2t=x
sourceimages
2t=x
path difference cos x
circular fringe
constant
Parallel reflecting surfaces Extended source
Fringes localized at infinity
∞
Circular fringe constant
Fringes of equal inclination
Oxford Physics: Second Year, Optics
Wedged Parallel
Point
Source
Non-localised
Equal thickness
Non-localised
Equal inclination
Extended
Source
Localised in plane
of Wedge
Equal thickness
Localised at infinity
Equal inclination
Summary: fringe type and localisation
Oxford Physics: Second Year, Optics
• Dispersion: dq/dl, angular separation of wavelengths
• Resolving Power: lDl, dimensionless figure of merit
• Free Spectral Range: extent of spectrum covered by interference pattern before
overlap with fringes of same l and different order
• Instrument width: width of pattern formed by instrument with monochromatic light.
• Etendu: or throughput –
a measure of how efficiently the instrument uses available light.
Optical Instruments for SpectroscopySome definitions:
Oxford Physics: Second Year, Optics
• Fringe formation – phasors
• Effects of groove size – Fourier methods
• Angular dispersion
• Resolving power
• Free Spectral Range
• Practical matters, blazing and slit widths
The Diffraction Grating Spectrometer
Oxford Physics: Second Year, Optics
0
I
N = 2
4
N-slit grating
N-1 minima
I ~ N2
Peaks at = n2
Oxford Physics: Second Year, Optics
0
I
N = 3
4
9
N-slit grating
Peaks at n2, N-1 minima, I ~ N2 1st minimum at 2/N
Oxford Physics: Second Year, Optics
N
N m N
I
0
4
N2
N-slit grating
Oxford Physics: Second Year, Optics
• Dispersion: dq/dl, angular separation of wavelengths
• Resolving Power: lDl, dimensionless figure of merit
• Free Spectral Range: extent of spectrum covered by interference pattern before
overlap with fringes of same l and different order
• Instrument width: width of pattern formed by instrument with monochromatic light.
• Etendu: or throughput –
a measure of how efficiently the instrument uses available light.
Optical Instruments for SpectroscopySome definitions:
N-slit diffraction grating
Order
Order
0 1 2 3 4
0
2
(a)
(b)
(c)
I(q) = I(0) sin2{N/2} sin2{/2}
sin2{/2} {/2}2
= (2l) d sinq (l) a sinq
. slit separation slit width
x 2 ,
N-slit diffraction grating
Order
Order
0 1 2 3 4
0
2
(a)
(b)
(c)
. slit separation slit width
x 2 ,
Oxford Physics: Second Year, Optics
l ldl
D min2N
2n
(a)
l ldl
q
Dq min Dql
qp
(b) = (2l)dsinq
Oxford Physics: Second Year, Optics
Diffractedlight
Reflectedlight
Diffractedlight
Reflectedlight
q
q
(a)
(b)
Diffracted lightReflected
light
Oxford Physics: Second Year, Optics
Order
Order
0 1 2 3 4
0
2
(a)
(b)
(c)
Unblazed
Blazed
Oxford Physics: Second Year, Optics
(a)
Dxs
(b)
Dxs q
grating
f1
f1
f2
f2
Oxford Physics: Second Year, Optics
Instrument width =
Resolving Power = lDlInst
= nDnInst
= 2Wl
Maximum path difference in units of wavelength
Maximum path difference
1 .
Instrument function
DnInst =1
2W
nInstrument width (Grating spectrometer):
• Interference by division of wavefront:
The Diffraction Grating spectrograph
• Interference by division of amplitude:
2-beams -
The Michelson Interferometer
Oxford Physics: Second Year, Optics
Optical Instruments for Spectroscopy
Oxford Physics: Second Year, Optics
Michelson interferometer
• Fringe properties – interferogram
• Resolving power
• Instrument width
Albert Abraham Michelson
1852 –1931
Oxford Physics: Second Year, Optics
t
Light source
Detector
M2
M/
2 M1
CP BS
Michelson
Interferometer
Oxford Physics: Second Year, Optics
t
2t=x
sourceimages
2t=x
path difference cos x
circular fringe
constant
Fringes of equal inclinationLocalized at infinity
Oxford Physics: Second Year, Optics
xn
Input spectrum Detector signal Interferogram
x = 2t
I(x) = ½ I0 [ 1 + cos 2nx ]
½ I0
I(x)
Oxford Physics: Second Year, Optics
x
x
x
xmax
I( )n
I( )n
I( )n
1
2
(a)
(b)
(c)
Oxford Physics: Second Year, Optics
DvInst = 1/xmax
Instrument width = 1 .
Maximum path difference
Resolving Power = Maximum path difference
in units of wavelength
-
Michelson Interferometer
Size of instrument
Oxford Physics: Second Year, Optics
LIGO, Laser Interferometric Gravitational-Wave Observatory
4 Km
LIGO, Laser Interferometric Gravitational-Wave Observatory
Vacuum ~ 10-12 atmosphere
Precision ~ 10-18 m
Oxford Physics: Second Year, Optics
Michelson interferometer
• Path difference: x = pl
measure l by reference to known lcalibration
• Instrument width: DvInst = 1/xmax
WHY?• Fourier transform interferometer
• Fringe visibility and relative intensities
• Fringe visibility and coherence
Lecture 10
Albert Abraham Michelson
1852 –1931
Oxford Physics: Second Year, Optics
x
x
x
xmax
I( )n
I( )n
I( )n
1
2
(a)
(b)
(c)
Oxford Physics: Second Year, Optics
Coherence
Longitudinal coherence
Coherence length: lc ~ 1/Dn_
Transverse coherence
Coherence area: size of source or wavefront
with fixed relative phase
Oxford Physics: Second Year, Optics
Transverse coherence
r
wsd
r >> d
d sin < l
Interference
Fringes < ld
d defines coherence area
Michelson Stellar Interferometer
Measures angular size of stars
Oxford Physics: Second Year, Optics
Division of wavefront
fringes
Young’s slits (2-beam) cos2(/2)
Diffraction grating (N-beam) sin2(N/2)
sin2(/2)
Division of amplitude
fringes
Michelson (2-beam) cos2(2)
Fabry-Perot (N-beam) ?
Sharper fringes
Oxford Physics: Second Year, Optics
d
Eo
t r1 2
t1
Eot t1 2
t r r1 2 12
t r r1 2 13 2
t r r1 2 1
Eot t r r1 2 1 2e-i
t r r1 2 12 2
Eot t r r1 2 1 22 2 -i2
e
Eot t r r1 2 1 23 3 -i3
e
q
t2t1
Oxford Physics: Second Year, Optics
d
Eo
tr
t
Eot2
tr3
tr5
tr2
Eot r2 2 -i
e
tr4
Eot r2 4 -i2
e
Eot r2 6 -i3
e
q
Oxford Physics: Second Year, Optics d
Eo
tr
t
Eot2
tr3
tr5
tr2
Eot r2 2 -i
e
tr4
Eot r2 4 -i2
e
Eot r2 6 -i3
e
q
Airy Function
Oxford Physics: Second Year, Optics
m2 (m+1)2
I( )
The Airy function: Fabry-Perot fringes
(l)d.cosq
Oxford Physics: Second Year, Optics
ExtendedSource
Lens
Lens
Fabry-Perotinterferometer
d
Screen
(l)2d.cosq
Fringes of equal inclinationLocalized at infinity
Oxford Physics: Second Year, Optics
m2 (m+1)2
I( )
The Airy function: Fabry-Perot fringes
DFWHM
Finesse = DFWHM
Finesse = 10 Finesse = 100
Oxford Physics: Second Year, Optics
Multiple beam interference:
Fringe sharpness set by Finesse
F = √R
(1-R)
• Instrument width DvInst
• Free Spectral Range FSR
• Resolving power
• Designing a Fabry-Perot
Fabry-Perot Interferometer
Oxford Physics: Second Year, Optics
m2 (m+1)2
I( )
The Airy function: Fabry-Perot fringes
DFHWM
Finesse, F = DFHWM DInst = F
Oxford Physics: Second Year, Optics
DvInst = 1 .
2d.F
= 1/xmax
Instrument width = 1 .
Maximum path difference
-
Fabry-Perot Interferometer: Instrument width
effective
Oxford Physics: Second Year, Optics
m2 (m+1)2
I( )
The Airy function: Fabry-Perot fringes
DInst
Finesse = D
Free Spectral Range
Oxford Physics: Second Year, Optics
FSR
(m+1) d
n nDn
mth th
I( )d
Oxford Physics: Second Year, Optics
I( )d
d
n nDnR
DnInst
Resolution criterion:
DnR DnInst
Oxford Physics: Second Year, Optics
qm-1
mth fringe (on axis)
rm-1
Aperture size to admit only mth fringe
Typically aperture ~10
rm-1
Centre spot scanning
Oxford Physics: Second Year, Optics
Maxwell’s equations
2
22
t
EE roro
2
22
t
HH roro
).( rkti
oeEE
Hn
HEo
o
ro
ro
1
EnH constant
E.M. Wave equations
indexrefractiven
Oxford Physics: Second Year, Optics
n2
E1
E2
E1
n1.n2
Boundary conditions:
Tangential components of E and H are continuous
Oxford Physics: Second Year, Optics
n2
Eo
ET
E1
Eo E1
.no.n1
..nT
ko.k1
.k1ko
.kT
(a) (b)
l
Layer
TT
OO Ekn
nikEE
1
1
1 sincos
TTOOO EknkiEEn 11 cossin)(
(9.7)
(9.8)
;cos 1kA ;sin1
1
1
kn
iB
;sin 11 kinC 1cos kD
TTOO
TTOO
O
O
DnCnBnAn
DnCnBnAnr
E
E
TTOO
O
O
T
DnCnBnAn
nt
E
E
2
(9.10)
(9.11)
l l
l l
l l l l
Quarter Wave Layer:
A=0, B=-i/n1, C=-in1, D=0
,4l
TTOO
TTOO
O
O
DnCnBnAn
DnCnBnAnr
E
E
R =
2
2
1
2
12
nnn
nnnr
TO
TO
Anti-reflection coating, n12 ~ nO nT , R → 0
e.g. blooming on lenses
l
Oxford Physics: Second Year, Optics
Ref
lect
ance
0.04
0.01
Uncoated glass
400 500 600 700 wavelength nm
3 layer
2 layer
1 layer
Oxford Physics: Second Year, Optics
Eo
Eo
.no .nT
ko
ko
n2.nH nLn2.nH nL
Multi-layer stack
TT
OO Ekn
nikEE
1
1
1 sincos
TTOOO EknkiEEn 11 cossin)(
(9.7)
(9.8)
;cos 1kA ;sin1
1
1
kn
iB
;sin 11 kinC 1cos kD
1 + r = ( A + BnT )t
nO( 1 – r ) = (C + DnT )t
1 1 A B 1
+ r = t
nO -nO C D nT
l l
l l
l l l l
Oxford Physics: Second Year, Optics
.nT.nH.nHnL .nH
nL nL.nH
.nH.nHnL nL
l
Interference filter; composed of multi-layer stacks
Oxford Physics: Second Year, Optics
z
yx
EozE
Eoy
Oxford Physics: Second Year, Optics
z
y
x
Source
Circularly polarized light: Ez = Eosin(kxo – t) ; Ey = Eocos(kxo – t)
Oxford Physics: Second Year, Optics
z
y
x
E = z E sin( ) o okx Source Source
xo E = y E cos( ) o okx
E
z
y
x
E = 0z
xo E = y E o
E
(a) 0 x = x , t = o(b) x = x , t = kx /o o
Oxford Physics: Second Year, Optics
z
y
x
Source
Looking towards source:
Clockwise rotation of E
Right circular polarization
Oxford Physics: Second Year, Optics
EL
ER
EP
Superposition of equal
R & L Circular polarization
= Plane polarized light
EL
ER
Superposition of unequal
R & L Circular polarization
= Elliptically polarized light
Oxford Physics: Second Year, Optics
E
qy
z
EEoz
Eoyy
z
(a) (b)
Oxford Physics: Second Year, Optics
Polarization states defined by:
Ey = Eoy cos(kx - t) Ez = Eoz cos(kx - t - )
• = 0 Linearly Polarized
• ≠ 0 Elliptically Polarized
Oxford Physics: Second Year, Optics
Polarization states defined by:
Ey = Eoy cos(kx - t) Ez = Eoz cos(kx - t - )
• = 0 Linear
• = + /2 Eoy = Eoz Left/Right Circular
• /2 Eoy ≠ Eoz Left/Right Elliptical
axes along y,z
• ≠ /2 Eoy ≠ Eoz Left/Right Elliptical
axes at angle to y,z
Oxford Physics: Second Year, Optics
Unpolarized light defined by:
Ey = Eoy cos(kx - t) Ez = Eoz cos(kx - t – (t)
• = (t) Phase varies randomly in time
unpolarized light
Oxford Physics: Second Year, Optics
= 0, 5
Note: if Eoy = Eoz
these ellipses become circles
Polarized Light
• Production
• Analysis
• Manipulation
Oxford Physics: Second Year, Optics
Oxford Physics: Second Year, Optics
z z
x,y x,y
nex
nexno
no
(a) (b)Positive Negative
Uniaxial Crystals
Optic Axis
Oxford Physics: Second Year, Optics
z z
x,y x,y
nex
nexno
no
(a) (b)Positive Negative
Uniaxial Crystals
o-ray
e-ray
Epropagation
Phase shift:
= (2l)[no-ne]l
Oxford Physics: Second Year, Optics
z z
x,y x,y
nex
nexno
no
(a) (b)Positive Negative
Uniaxial Crystals
o-ray
e-ray
E
o-ray
Oxford Physics: Second Year, Optics
Linear Input change Output
q = 45O, Eoy=Eoz , l Left/Right Circular
q ≠ 45O, Eoy≠Eoz , l L/R Elliptical
q ≠ 45O, Eoy≠Eoz ≠ ,l Elliptical tilted axis
q ≠ 45O, Eoy≠Eoz , l Linear at q
to input
Oxford Physics: Second Year, Optics
z
yx
EzEP
Ey
q
z
yx
Ez
EP
Ey
q
Half-wave plate: introduces -phase shift
Oxford Physics: Second Year, Optics
e-ray e-raye-ray
o-ray
o-ray o-rayOptic axis Optic axis
Prism Polarizers
Oxford Physics: Second Year, Optics
d1
d2
d1
d2
(a) (b)
A
B
Babinet-SoleilBabinet
Oxford Physics: Second Year, Optics
Linearly polarized lightafter passage through the
/4 platel
Elliptically polarized light
E
y
z
yz
q
E
y
z
yz
(a) (b)
Analysis of polarized light
Oxford Physics: Second Year, Optics
Linearly polarized lightafter passage through the
/4 platel
Elliptically polarized light
E
y
z
yz
q
E
y
z
yz
(a) (b)
Analysis of polarized light
Find ratio of
major:minor axes by
rotating linear polarizer
Find angle of
Linear polarization using
l/4 plate and linear polarizer
Eoz/Eoy = tan (q)→q, tanq→cos
Find Eoz:Eoy and
Oxford Physics: Second Year, Optics
AB
(a)
AB
C
(b)
AB
C D
(c).
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