Abstract

We develop an analytical approach in order to understand the causes of sidebands imbalance. The results have been tested by two different and more sophisticated numerical tools. The main static perturbations that can generate sidebands imbalance are described and fully analyzed, with a special attention to the design of the Laser Interferometer Gravitational Wave Observatory I, whose typical parameters have been used for numerical estimations of this phenomenon.

© 2004 Optical Society of America

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References

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  1. J. B. Camp, H. Yamamoto, and S. E. Whitcomb, “Analysis of light noise in a recycled Michelson interferometer with Fabry–Perot arms,” J. Opt. Soc. Am. A 17, 120–128 (2000).
    [CrossRef]
  2. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  3. A. Kostenbauder, Y. Sun, and A. E. Siegman, “Eigenmode expansions using biorthogonal functions: complex-valued Hermite–Gaussians,” J. Opt. Soc. Am. A 14, 1780–1790 (1997).
    [CrossRef]
  4. P. R. Saulson, Interferometric Gravitational Wave Detectors (World Scientific, Singapore, 1994).
  5. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  6. R. Beausoleil, E. Gustafson, M. Fejer, E. D’Ambrosio, B. Kells, and J. Camp, “Model of thermal wave-front distortion in interferometric gravitational-wave detectors. I. Thermal focusing,” J. Opt. Soc. Am. B 20, 1247–1268 (2003).
    [CrossRef]
  7. B. Bochner, “Simulating a dual-recycled gravitational wave interferometer with realistically imperfect optics,” Gen. Relativ. Grav. 35, 1029–1057 (2003).
    [CrossRef]
  8. P. Fritschel, N. Mavalvala, D. Shoemaker, D. Sigg, M. Zucker, and G. Gonzalez, “Alignment of an interferometric gravitational wave detector,” Appl. Opt. 37, 6734–6747 (1998).
    [CrossRef]

2003 (2)

2000 (1)

1998 (1)

1997 (1)

1966 (1)

Beausoleil, R.

Bochner, B.

B. Bochner, “Simulating a dual-recycled gravitational wave interferometer with realistically imperfect optics,” Gen. Relativ. Grav. 35, 1029–1057 (2003).
[CrossRef]

Camp, J.

Camp, J. B.

D’Ambrosio, E.

Fejer, M.

Fritschel, P.

Gonzalez, G.

Gustafson, E.

Kells, B.

Kogelnik, H.

Kostenbauder, A.

Li, T.

Mavalvala, N.

Shoemaker, D.

Siegman, A. E.

Sigg, D.

Sun, Y.

Whitcomb, S. E.

Yamamoto, H.

Zucker, M.

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Figures (11)

Fig. 1
Fig. 1

Laser light enters the cavity through the recycling mirror. There are specific conditions that make the amplitude of the electromagnetic fields identical at the sideband frequencies fCR±fmod in the steady state.

Fig. 2
Fig. 2

Power gain at the beam-splitter symmetric port GRT is simply defined as a function of two longitudinal quantities: fCRδl-±fmodl-, which corresponds to a differential phase, and fCRδl+±fmodl+, which corresponds to a common phase for the sidebands at f=fCR±fmod.

Fig. 3
Fig. 3

Scattering among the cavity modes is represented by the operators m(a) and m(b). In the frequency-independent case, they are unitary matrices that depend on some adimensional parameter, proportional to the distortion, as for tilt or curvature mismatch. In the 2×2 model we have constructed, we generated the unitary matrices from the Pauli matrices, overriding the usual modal-model rules.

Fig. 4
Fig. 4

Peak of the field amplitude after one round trip corresponds to a different position of the beam splitter than in the unperturbed case. The same microscopic tuning that maximizes such amplitude for one eigenmode makes dark the output port for that specific eigenmode. Here a and b are proportional to a change in the radius of curvature. Using the LIGO typical values a=0.01 corresponds to a variation of the thank you order of ΔR400 m over R=14571 m.

Fig. 5
Fig. 5

When the optical axes of the two branches are misaligned, the position of the beam splitter must be tuned in order to make all the light that is associated with the resonating mode recycled. The same tuning is the one that ensures there is no resonating mode at the output port and restores a symmetric interaction of the sidebands with the interferometer, provided the driving beam is matched with the resonating eigenvector.

Fig. 6
Fig. 6

Power stored in the recycling cavity when one of the two branches is perturbed because of a curvature mismatch. The design value is 14571 m, and the relative sidebands imbalance is ∼20% in the range 13000–13500 m.

Fig. 7
Fig. 7

Example of a frequency-dependent perturbation whose result is a different bright-port power for the two sidebands. The physical mechanism is purely longitudinal: The Fabry–Perot cavities are identical and affected by the same offset, plus the condition L=(2n+1)λmod/4 is not satisfied.

Fig. 8
Fig. 8

If the coupling between the recycling and the Fabry–Perot cavities is not exactly matched, the power stored by the two sidebands may be different if the macroscopic condition L=(2n+1)λmod/4 is not fulfilled. There is no geometrical asymmetry between the two arms, but the frequency dependance of the arm reflectivities gives rise to a different interaction of the sidebands with the perturbed interferometer.

Fig. 9
Fig. 9

If the absolute value of the arm reflectivity is one, we can limit our attention to its imaginary part, which is a function of the phase gained after a full round trip inside the Fabry–Perot cavity. The two sidebands are located symmetrically around the carrier whose round-trip phase is zero by definition. This is not true for the higher-order modes excited by a geometrical mismatch.

Fig. 10
Fig. 10

If the only perturbation is due to misalignment, the mode sustained by the recycled Michelson interferometer is given by a Gaussian beam, whose optic axis is misaligned with respect to the unperturbed cavity axis. The overlap of the electromagnetic fields impinging on the beam splitter, coming from the two branches, is the stable eigenmode, while the remaining part is scattered out of the antisymmetric port and consists of higher-order modes excited by the tilt.

Fig. 11
Fig. 11

For small perturbations l1-l2R1-R2 with li=z(Ri); as predicted by the paraxial approximation for the propagation of Gaussian beams.

Equations (68)

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Pround-tripexp[i(k±kmod)l+]cos[(k±kmod)l-],
k=2πfCRc=2πλ,kmod=2πfmodc=2πλmod,
kl+=nπ,kl-=mπ,m,nI,
δl+, δl-0Pround-tripexp[i(kδl+±kmodl+)]cos(kδl-±kmodl-)
exp(ikmodl+)=±1,exp(ikmodl-)=±1,
exp[i(k±kmod)l+]r1+r22 cos[(k±kmod)l-]
+ir1-r22 sin[(k±kmod)l-]
Pround-tripexpi(kδl+±kmodl+)±i arctan (r1-r2)sin kmodl-(r1+r2)cos kmodl-,
exp[i(k±kmod)l+]pm1+m22 cos[(k±kmod)l-]
+im1-m22 sin[(k±kmod)l-]p,
exp[i(k±kmod)l+]pi m1+m22 sin[(k±kmod)l-]
+m1-m22 cos[(k±kmod)l-]p,
p=exp(iθG)00exp(2iθG),
m1=1-2ii1-2,
m2=1-2-i-i1-2,
v1=1-i sin(kδl-)2 cos(kδl-)sin θG,
v2=i sin(kδl-)2 cos(kδl-)sin θG1,
λ1=cos2(kδl-)-2×exp2iθG+i 2 tan2 kδl-2 tan θG×exp[i(kδl+±kmodl+)],
λ2=cos2(kδl-)-2×exp4iθG-i 2 tan2 kδl-2 tan θG×exp[i(kδl+±kmodl+)],
v1±=1-i sin(kδl-±kmodl-)2 cos(kδl-±kmodl-)sin θG,
v2±=i sin(kδl-±kmodl-)2 cos(kδl-±kmodl-)sin θG1,
λ1±=cos2(kδl-±kmodl-)-2×exp2iθG+i 2 tan2(kδl-±kmodl-)2 tan θG×exp[i(kδl+±kmodl+)],
λ2±=cos2(kδl-±kmodl-)-2×exp4iθG-i 2 tan2(kδl-±kmodl-)2 tan θG×exp[i(kδl+±kmodl+)],
vOUT=-i sin(kδl-)exp-iθG-i22 tan θG12 cos(kδl-) sin2 kδl-tan θG+i(1+cos2kδl-)×exp(3iθG+ikl+),
δp=i exp(iθG) dθGdRΔR003i exp(3iθG) dθGdRΔR,
R=radiusofcurvature,
m1=exp(iα)exp(-iβ)cos γi sin γi sin γexp(iβ)cos γ,
exp[i(kδl+±kmodl+)]pm1+m22 cos(kδl-±kmodl-)
+im1-m22 sin(kδl-±kmodl-)p,
m1=m(a)=exp(i2a)exp(ian·σ),
n·σ=σH=12 -1111,
h(x, y)-λ2πΦ(x, y),
h(x, y)-λ2πΦ(x, y)
=λ4πHm2xwHn2yw γ2m+nm!n!,
umn|m|u00=iγ-γ22β,
u00|m|u00=1-γ22,
umn|m|umn=1+i2β+O(γ2),
m=exp2ikh(x, y)-λ2πΦ(x, y)
m=1+2ikh(x, y)-λ2πΦ(x, y)+12 2ikh(x, y)-λ2πΦ(x, y)2+,
umn|2kh(x, y)-λ2πΦ(x, y)|umn=2βγ,
mexp(iβ)×1-γ22exp(-iβ)iγiγ[1-O(γ2)]exp(iβ)
|ΔR|2500m|a|=kw22 1Rpert-1Rdesign0.05,|b|=0,
w=3.6cm,k=2πfCRc=2πλ,
λ=1.064 10-6 m,
r1(f)=-r1+exp(2πif2L1/c)1-r1 exp(2πif2L1/c),|r1(f)|=1,
r2(f)=-r2+exp(2πifL2/c)1-r2 exp(2πifL2/c),|r2(f)|=1,
exp[i(kδl+±kmodl+)]×r1(fCR±fmod)+r2(fCR±fmod)2 cos(kδl-±kmodl-)+i r1(fCR±fmod)-r2(fCR±fmod)2 sin(kδl-±kmodl-),
r1(fCR±fmod)
=expi arctan (1-r12)sin[2(k±kmod)L1](1+r12)cos[2(k±kmod)L1]-2r1,
r2(fCR±fmod)
=expi arctan (1-r22)sin[2(k±kmod)L2](1+r22)cos[2(k±kmod)L2]-2r2,
sin[2(k±kmod)L1]=sin[2(k±kmod)L2]=0,
cos(2kmodL1)=cos(2kmodL2)=-1,
r1(fCR±fmod)-expi 1-r11+r12kδL1,
r2(fCR±fmod)-expi 1-r21+r22kδL2,
exp(ikmodl+)=-1;
expikδl++1-r11+r1δL1+1-r21+r2δL2=1
r1(fCR)expi 1+r11-r12kδL1/c,
r2(fCR)expi 1+r21-r22kδL2/c,
δl-+1+r11-r1δL1-1+r21-r2δL2=0
r(fCR±fmod)
=expi arctan (1-r2)sin(±2kmodL)(1+r2)cos(±2kmodL)-2r×expi (1-r2)2kδL1+r2-2r cos[±2kmodL)
δl++2 1+r1-rδL=0,
r(f)=-r+exp(2πif2L/c)pmp1-r exp(2πif2L1/c)pmp,
Φ(z)=kz-(m+n+1)arctan 2zkw02
umn(x, y, z)=2πw2(z) exp-iΦ(z)-(x2+y2)×1w2(z)+ik2R(z)×12m+nm!n! Hm2xw(z)Hn 2yw(z)
kδl-=2/(4 tan θG),=kwθtilt,
kδz=kR(1-cos θtilt)=2kR sin2 θtilt/2,

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