LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
ON THE ANALYTIC STRUCTUREOF A FAMILY OF HYPERBOLOIDAL BEAMSOF POTENTIAL INTEREST FOR ADV-LIGO
Vincenzo Galdi*Giuseppe CastaldiVincenzo PierroInnocenzo M. PintoJuri AgrestiErika D’AmbrosioRiccardo De Salvo
TWG/University of Sannio at BeneventoLIGO Lab/Caltech
OWG Session
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
Agenda
• Background: Non-spherical mirrors and flat-top beams
• From nearly-flat (FM) and nearly-concentric mesa (CM) beams to Bondarescu-Thorne (BT) hyperboloidal beams
• Analytic structure of BT hyperboidal beams– Rapidly-converging Gauss-Laguerre (GL) expansion– Some results: Beam shapes and mirror corrections– Generalized duality relations (lowest-order mode)
• Complex-order Fourier transform
• Conclusions and perspectives
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
Background
• Use of flat-top (“mesa”) beams suggested for mitigating thermal noise effects in future LIGO interferometers [D’Ambrosio et al., LIGO-G000223-00-D]
– Better averaging of thermally-induced mirror surface fluctuations
– Potential reduction by a factor 3 in thermoelastic noise power [D’Ambrosio et al., gr-qc/0409075]
Gaussian BeamMesa Beam thermally-induced
fluctuations
mirror
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
Background (cont’d) • Nearly-flat mesa (FM) beams
– Synthesized via coherent superposition of minimum-spreading Gaussian beams (GB)
– Supported by nearly-flat mirrors with “Mexican-hat” profile• Prototype cavity developed; experimental tests under way [Agresti
et al., LIGO-G0400412-00-D]• Concerns about tilt-instability [Savov & Vyatchanin, gr-qc/0409084]
• Nearly-concentric mesa (CM) beams– Same intensity distribution at the mirror– Much weaker tilt-instability [Savov & Vyatchanin, gr-qc/0409084]
• FM and CM configurations connected by duality relations– One-to-one mapping between eigenstates [Agresti et al.,
gr-qc/0511062]
• More general (“hyperboloidal”) beams [Bondarescu & Thorne, gr-qc/0409083] may be of interest
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
FM/CM vs. BT hyperboloidal beams
• FM beams– Optical axes are the generators of a cylinder
• CM beams– Optical axes are the generators of a cone
• BT hyperboloidal beams– Optical axes are the generators of a hyperboloid– Parameterized via “twist angle”– Contain FM (= 0) and CM (=) as special
cases– “Optimal” configuration expected in a
neighborood of =
0
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
L
0r r
waist plane
nearly-spheroidal mirrors
z S r h r
z0
BT hyperboloidal beams• Supported by nearly-spheroidal mirrors
• Field distribution on fiducial spheroid [Bondarescu & Thorne, gr-qc/0409083]
0-integral generally needs to be computed numerically– Closed-form (Gaussian) solution for =/2
0
2 22 0 0 00
0 0 0 02 20 00 0
2 cos, exp sin sin 1 cos
2
R r r rrrrU r S dr d r i i
w w
• Fiducial spheroid:
• Waist-plane aperture radius:
• GB spot size: (minimum spreading)
2 2sin 2
2
rLS r
L
0R
00
Lw
k
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
BT hyperboloidal beams (cont’d)
• Mirror profile correction
• Symmetry/duality relations– Field distribution
– Mirror profile correction
0
arg , arg 0,U r S U Sh r
k
arg , constantU r S h
*
*,U U
U U
h r h r
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
GL expansions
• FM and CM beams– Field distributions at the waist plane related via Fourier
transform (FT)– Coincide with “flattened” beams in [Sheppard & Saghafi,
Opt. Comm. 132, 144, 1996]• Gauss-Laguerre (GL) expansions available
• FT
2
22 exp2m mL
GL (orthonormal) basis functions
0 1m
m mA A
00
0 00 0
2 2,0 , ,0m m m m
m m
r rU r A U r A
w w
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
GL expansions (cont’d)• Expansion coefficients [Sheppard & Saghafi, Opt. Comm.
132, 144, 1996]
0 10 20 30 40 50 60 70 80
1E-5
1E-4
1E-3
0.01
0.1
A(
)m
m
0
0
0.5w
R
0
0
0.25w
R
0
0
0.1w
R • Easily computable
• Abrupt fall-off for
P: incomplete Gamma function 2 20 0
2 20 0
21,2m
w RA P m
R w
20202
Rm
w
Rapid convergence
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
GL expansions (cont’d)
• Extension to generic BT hyperboloidal beams [Galdi et al., gr-qc/0602074]
• Allows field computation at any point in space– Use standard GL (paraxial) propagators– Field distribution on the fiducial spheroids
• Symmetry/duality relations verified
2
00 0
cos, exp
2m
m mm
r rU r S ik i A
L w
0 0
2,0 , cos
m
m m m mm
rU r A A A
w
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
Results: Beam shapes
0.0 1.0 2.0 3.0 4.0 5.0 6.01E-15
1E-12
1E-9
1E-6
1E-3
1
Rel
ati
ve e
rror
r/ w0
0.000
0.002
0.004
0.006
0.008
|U(
r.S)
|2
• Parameters
• Truncation criterion
• Reference solution from [Bondarescu & Thorne, gr-qc/0409083]
0 0
0
2.603 , 10.4 ,
1064
w cm R cm
nm FM, CM
Gaussian
targeted accuracy
3
0
, 10MAm MA
0, 18M 0.1 , 0.9 17M 0.2 , 0.8 14M
0.5 0M
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
Results: Mirror corrections
• Same parameters
• For
• Typical errors: ~0.1nm– Well within fabrication
tolerances0.0 1.0 2.0 3.0 4.0 5.0 6.0
1E-18
1E-15
1E-12
1E-9
1E-6
1E-3
1
h/
r/ w0
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
h(
r)/ 18M
0.9 17M 0.8 14M 0.5 0M
Mexican hat
h r h r
0 2
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
Im Re
20 0
2
1
1
100
0
00
0 0 0
2
2 2
Generalized duality relations
• Two arbitrary (1,2)-indexed BT hyperboloidal beams related by fractional FT operators of complex order
• Generalizes (for lowest-order mode) the FM-CM duality relations in [Agresti et al., gr-qc/0511062]
2
1
cos1 log
cos
i
1 2 0
1 2 1
(identity operator)
(standard FT)
Special cases
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
Conclusions and perspectives• Summary
– Focus on the analytic structure of a class of hyperboloidal beams of interest for future LIGO interferometers
– Rapidly-converging, physically-insightful GL expansions for generic BT hyperboloidal beams
• Field computation at any point in space• Validation/calibration against reference solution
– Generalized duality relations• Complex-order FT
• Current/future research– Thorough parametric analysis
• Implications for Adv-LIGO– Extension to higher-order modes (HOM)
• Techniques to depress HOM (parametric instabilities)
LHO, Mar. 19-22, 2006 LIGO-G060087-00-Z
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