Abstract

The influence of angular mirror-orientation errors on the length of a Fabry–Perot resonator is analyzed geometrically. Under conditions in which dominant errors are static or vary slowly over time, the analysis permits a simple prediction of the spectrum of short-term cavity length fluctuations resulting from mirror-orientation noise. The resulting model is applicable to the design of mirror control systems for the Laser Interferometer Gravitational-Wave Observatory, which will monitor separations between mirrored surfaces of suspended inertial test bodies as a way to measure astrophysical gravitational radiation. The analysis is verified by measuring the response of the Laser Interferometer Gravitational-Wave Observatory’s 40-m interferometer test-bed to the rotation of its mirrors.

© 1994 Optical Society of America

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References

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  1. A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
    [CrossRef] [PubMed]
  2. R. E. Vogt, “The U.S. LIGO project,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1991), p. 244.
  3. S. L. Smith, “A search for gravitational waves from coalescing binary stars using the Caltech 40 meter gravity wave detector,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1988), p. 37.
    [PubMed]
  4. Equation (1) and the rest of the discussion can be readily applied to any cavity geometry by the suitable redefinition of α, β, and γ. In general, β and γ are not linearly dependent.
  5. R. G. Brown, Introduction to Random Signal Analysis and Kalman Filtering (Wiley, New York, 1983), Chap. 2, p. 85.
  6. S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), pp. 133–294.
  7. M. E. Zucker, “The LIGO 40 m prototype laser interferometer gravitational wave detector,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1991), p. 224.
  8. S. Kawamura, “Test mass orientation noise in the LIGO 40 m prototype,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1991), p. 1486.
  9. Remote LIGO sites exhibit approximately one tenth the seismic amplitude found in the campus laboratory at frequencies that determine the rms mirror angle fluctuation. In addition, current seismic isolation stack and suspension designs show one fifth the resonant amplification effect found to enhance rms mirror motion in the 40-m interferometer. Although the hundredfold increase in length would proportionately increase the dνrms arising from a given rms angle fluctuation, these factors should reduce that angle fluctuation by a factor of the order of 50, leading us to expect only a doubling in net dνrms in full-scale interferometers.

1992 (1)

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Abramovici, A.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Althouse, W. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Brown, R. G.

R. G. Brown, Introduction to Random Signal Analysis and Kalman Filtering (Wiley, New York, 1983), Chap. 2, p. 85.

Drever, R. W. P.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Gürsel, Y.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Kawamura, S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

S. Kawamura, “Test mass orientation noise in the LIGO 40 m prototype,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1991), p. 1486.

Raab, F. J.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), pp. 133–294.

Shoemaker, D.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Sievers, L.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Smith, S. L.

S. L. Smith, “A search for gravitational waves from coalescing binary stars using the Caltech 40 meter gravity wave detector,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1988), p. 37.
[PubMed]

Spero, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Thorne, K. S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Vogt, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

R. E. Vogt, “The U.S. LIGO project,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1991), p. 244.

Weiss, R.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Whitcomb, S. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Zucker, M. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

M. E. Zucker, “The LIGO 40 m prototype laser interferometer gravitational wave detector,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1991), p. 224.

Science (1)

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the laser interferometer gravitational wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Other (8)

R. E. Vogt, “The U.S. LIGO project,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1991), p. 244.

S. L. Smith, “A search for gravitational waves from coalescing binary stars using the Caltech 40 meter gravity wave detector,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1988), p. 37.
[PubMed]

Equation (1) and the rest of the discussion can be readily applied to any cavity geometry by the suitable redefinition of α, β, and γ. In general, β and γ are not linearly dependent.

R. G. Brown, Introduction to Random Signal Analysis and Kalman Filtering (Wiley, New York, 1983), Chap. 2, p. 85.

S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), pp. 133–294.

M. E. Zucker, “The LIGO 40 m prototype laser interferometer gravitational wave detector,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1991), p. 224.

S. Kawamura, “Test mass orientation noise in the LIGO 40 m prototype,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1991), p. 1486.

Remote LIGO sites exhibit approximately one tenth the seismic amplitude found in the campus laboratory at frequencies that determine the rms mirror angle fluctuation. In addition, current seismic isolation stack and suspension designs show one fifth the resonant amplification effect found to enhance rms mirror motion in the 40-m interferometer. Although the hundredfold increase in length would proportionately increase the dνrms arising from a given rms angle fluctuation, these factors should reduce that angle fluctuation by a factor of the order of 50, leading us to expect only a doubling in net dνrms in full-scale interferometers.

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Figures (6)

Fig. 1
Fig. 1

Geometry of a half-symmetric Fabry–Perot cavity consisting of flat (M1) and concave (M2) test-body mirrors.

Fig. 2
Fig. 2

Suspended test-body and orientation control system used in the LIGO 40-m interferometer. The quadrant photodetector and electronic processor determine angular error signals from the position of an auxiliary laser beam reflected from the mirrored surface of the test body. These signals are filtered and amplified and applied to pairs of electromagnetic coils near the suspension point. The coils interact with permanent magnets (poled oppositely to induce torque) on an intermediate platform, which is suspended by a single wire so that it is free to rotate. The torque developed on this platform is transmitted to the test body below by the suspension wires, inducing rotation of the mirror and closing the feedback loop. Probe torques are introduced by the addition of currents to the feedback signals driving the coils.

Fig. 3
Fig. 3

Linear mirror angle-cavity length coupling coefficient l ˜ ( f 0 ) / ε ˜ 2 ( f 0 )as a function of beam axis position d ¯ 2. The dashed curve is the curve l ˜ ( f 0 ) / ε ˜ 2 ( f 0 ) = d ¯ 2 predicted by relation (9). The zero for the d ¯ 2 axis has been chosen for best fit.

Fig. 4
Fig. 4

Interferometer displacement caused by band-limited random mirror angle noise as a function of d ¯ 2. The coupling magnitude appears proportional to | d ¯ 2 | down to | d ¯ 2 | 0 . 4 mm mm, below which it is approximately constant. By integrating the spectrum of position fluctuations in the transmitted cavity mode, we found that d2rms ≈ 0.2 mm ± 0.1 mm during this experiment, which agrees with the transition from linear behavior predicted by relation (15).

Fig. 5
Fig. 5

Spectrum of low-frequency beam axis position fluctuations (thick curve, upper frequency scale and right-hand magnitude scale) replicated as upper and lower sidebands of an artificially induced 250-Hz probe angle fluctuation in the interferometer displacement spectrum (thin line, lower and left-hand scales). We chose the relative scales in accord with relation (12) by using the known 1.5 × 10−9 radrms amplitude of the 250-Hz probe. Both spectra are normalized to bandwidth BW = 1 Hz.

Fig. 6
Fig. 6

Spectral density of apparent cavity mirror displacement in the 40-m interferometer with the original orientation control system (upper thick curve) and after installing the new orientation feedback electronics (middle thick curve). The lower, thin curve depicts the estimated noise contributed by rotation of one of the test bodies about its horizontal axis. We formed this estimate by assuming that assumed d ¯ 2 1 mm in relation (15) to project the typical influence of the new control circuit’s residual electronic noise. The peak at 212 Hz is a resonance of the wire suspension system, where the coupling of external torque to the mirror is locally enhanced.

Equations (19)

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l = l 0 + αθ 1 2 + βθ 2 2 + γθ 1 θ 2 ,
α = 1 2 ( R + a 2 L ) , β = γ 2 = 1 2 ( R + a 2 ) .
l ˜ ( f ) = l 0 δ ( f ) + α θ ˜ 1 θ ˜ 1 ( f ) + β θ ˜ 2 θ ˜ 2 ( f ) + γ θ ˜ 2 θ ˜ 2 ( f ) ,
a ˜ b ˜ ( f ) a ˜ ( f ) b ˜ ( f f ) d f .
θ ν ( t ) = θ ¯ ν + ε ν ( t ) ,
θ ¯ ν 1 T T / 2 T / 2 θ ν ( t ) d t
l ( t ) l 0 + 1 2 d ¯ 1 [ θ ¯ 1 + 2 ε 1 ( t ) ] + 1 2 d ¯ 2 [ θ ¯ 2 + 2 ε 2 ( t ) ] ,
d 1 2 αθ 1 2 βθ 2 , d 2 2 β ( θ 2 θ 1 )
l ˜ ( f ) d ¯ 1 ε ˜ 1 ( f ) + d ¯ 2 ε ˜ 2 ( f ) ( f = 0 ) ,
θ ˜ ν ( f ) = θ ˜ ν w ( f < w ) + ε ˜ ν ( f > w ) ,
w w | θ ˜ ν w ( f ) | 2 d f | ε ˜ ν ( f ) | 2 d f ,
l ˜ ( f ) w w d ˜ 1 ( f ) ε ˜ 1 ( f f ) d f + w w d ˜ 2 ( f ) ε ˜ 2 ( f f ) d f ( f > 2 w ) .
( d ν rms ) 2 1 T 0 T [ d ν ( t ) d ¯ ν ] 2 d t = w w | d ˜ ν ( f ) | 2 [ 1 δ ( f ) ] d f
| l ˜ ( f ) | 2 [ d ¯ 1 2 + ( 2 d 1 rms ) 2 ] | ε ˜ 1 ( f ) | 2 + [ d ¯ 2 2 + ( 2 d 2 rms ) 2 ] | ε ˜ 2 ( f ) | 2 ( f > 2 w ) .
S l ( f ) [ d ¯ 1 2 + ( 2 d 1 rms ) 2 ] S ε 1 ( f ) + [ d ¯ 2 2 + ( 2 d 2 rms ) 2 ] S ε 2 ( f ) ( f > 2 w ) ,
| ε ˜ 2 ( f ) | { 0 , f 200 Hz K , 200 Hz f 315 Hz 0 , f 315 Hz ,
θ ˜ 2 ( f ) = θ 2 a δ ( f a ) + ε 2 δ ( f b )
l ˜ ( f ) = 3 . 4 × 10 13 m r m s × [ δ ( f b + f a ) + δ ( f b f a ) ] ,
S θ ν 1 / 2 ( f ) 10 17 rad Hz

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