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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