Abstract

Long-baseline, high-finesse Fabry-Perot interferometers can be used to make distance measurements that are precise enough to detect gravity waves. This level of sensitivity is achieved in part when the interferometer mirrors are isolated dynamically, with pendulum mounts and high-bandwidth cavity length control servos to reduce the effects of seismic noise. We present dynamical models of the cavity fields and signals of Fabry-Perot interferometers for use in the design and evaluation of length control systems for gravity-wave detectors. Models are described and compared with experimental data.

© 2002 Optical Society of America

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  1. A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
    [CrossRef] [PubMed]
  2. B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
    [CrossRef]
  3. R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1987).
  5. A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  6. G. Franklin, J. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems (Addison-Wesley, Reading, Mass., 1994).
  7. J. Camp, L. Sievers, R. Bork, J. Heefner, “Guided lock acquisition in a suspended Fabry-Perot cavity,” Opt. Lett. 20, 2463–2465 (1995).
    [CrossRef] [PubMed]
  8. M. Regehr, “Signal extraction and control for an interferometric gravitational wave detector,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif.1995).
  9. R. Weiss, Massachusetts Institute of Technology. Cambridge, Mass. 02139 (personal communication, 1992).

1999

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

1995

1992

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

1983

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Abramovici, A.

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Allen, B.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Althouse, W.

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Blackburn, J. K.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Bork, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1987).

Brady, P. R.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Camp, J.

Creighton, J. D. E.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Creighton, T.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Drever, R.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Drever, R. W. P.

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Droz, S.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Emami-Naeini, A.

G. Franklin, J. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems (Addison-Wesley, Reading, Mass., 1994).

Ford, G.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Franklin, G.

G. Franklin, J. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems (Addison-Wesley, Reading, Mass., 1994).

Gillespie, A. D.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Gursel, Y.

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Hall, J.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Heefner, J.

Hough, J.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Hughes, S. A.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Kawamura, S.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Kowalski, F.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Lyons, T. T.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Mason, J. E.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Munley, A.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Owen, B. J.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Powell, J.

G. Franklin, J. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems (Addison-Wesley, Reading, Mass., 1994).

Raab, F. J.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Regehr, M.

M. Regehr, “Signal extraction and control for an interferometric gravitational wave detector,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif.1995).

Regehr, M. W.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Sathyaprakash, B. S.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Savage, R. L.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Shoemaker, D.

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Siegman, A.

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Sievers, L.

J. Camp, L. Sievers, R. Bork, J. Heefner, “Guided lock acquisition in a suspended Fabry-Perot cavity,” Opt. Lett. 20, 2463–2465 (1995).
[CrossRef] [PubMed]

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Spero, R. E.

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Thorne, K. S.

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Vogt, R. E.

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Ward, H.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Weiss, R.

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

R. Weiss, Massachusetts Institute of Technology. Cambridge, Mass. 02139 (personal communication, 1992).

Whitcomb, S.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Whitcomb, S. E.

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Wiseman, A. G.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1987).

Zucker, M. E.

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Appl. Phys. B

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

B. Allen, J. K. Blackburn, P. R. Brady, J. D. E. Creighton, T. Creighton, S. Droz, A. D. Gillespie, S. A. Hughes, S. Kawamura, T. T. Lyons, J. E. Mason, B. J. Owen, F. J. Raab, M. W. Regehr, B. S. Sathyaprakash, R. L. Savage, S. Whitcomb, A. G. Wiseman, “Observational limit on gravitational waves from binary neutron stars in the galaxy,” Phys. Rev. Lett. 83, 1498–1501 (1999).
[CrossRef]

Science

A. Abramovici, W. Althouse, R. W. P. Drever, Y. Gursel, 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 Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Other

M. Regehr, “Signal extraction and control for an interferometric gravitational wave detector,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif.1995).

R. Weiss, Massachusetts Institute of Technology. Cambridge, Mass. 02139 (personal communication, 1992).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1987).

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

G. Franklin, J. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems (Addison-Wesley, Reading, Mass., 1994).

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Figures (13)

Fig. 1
Fig. 1

Fabry-Perot cavity and error signal generation.

Fig. 2
Fig. 2

Fields at a mirror.

Fig. 3
Fig. 3

Fields in a Fabry-Perot interferometer.

Fig. 4
Fig. 4

Dependence of cavity fields on past cavity history.

Fig. 5
Fig. 5

Cavity simulation algorithm for time scales less than nbeamsτ.

Fig. 6
Fig. 6

Cavity simulation algorithm for time scales greater than nbeamsτ.

Fig. 7
Fig. 7

Field intensities as the cavity goes through resonance. The static field was computed with Eq. (20) and the dynamic field with Eqs. (14) and (24).

Fig. 8
Fig. 8

Constant velocity sweep showing signals at the input mirror detector. The static field was computed with Eq. (20) and the dynamic field with Eqs. (14) and (24).

Fig. 9
Fig. 9

Measured and predicted control system error signal (in-phase demodulated signal) as the cavity goes through resonance at a constant velocity.

Fig. 10
Fig. 10

Roots of the cavity mirror transfer functions (rational approximation). From Eqs. (67) and (68).

Fig. 11
Fig. 11

Source phase transfer function example, computed with Eq. (61).

Fig. 12
Fig. 12

Mirror A transfer function example, computed with Eq. (62) or (67).

Fig. 13
Fig. 13

Mirror B transfer function example, computed with Eq. (63) or (68).

Tables (2)

Tables Icon

Table 1 Example Parameters for a 40-m Interferometer

Tables Icon

Table 2 Example Parameters for a 4-km Interferometer

Equations (74)

Equations on this page are rendered with MathJax. Learn more.

EAb=tAEAa+rAEAc,
EAd=tAEAc-rAEAa.
RA+TA+AA=1,
EAbt=tAEAat-rA exp2jkδAtEAct,
EAdt=tAEAct+rA exp-2jkδAtEAat.
EBbt=tBEBat,
EBdt=-rB exp-2jkδBtEBat.
EBat=exp-jkLEAbt-τ,
EAct=exp-jkLEBdt-τ.
EAat=tBSESt.
EDt=rBSEAdt.
EAbt=tAtBSESt+gtEAbt-2τ.
gt=rArB exp-2jkL+δBt-τ-δAt.
EDt=rArBS exp-2jkδAtESt-tArBSrB exp-2jkL+δB×t-τEAbt-2τ.
EAbt=EAbt-2τEAb,
ESt=ESt-2τES,
δAt=δAt-2τδA,
δBt-τ=δBt-3τδB.
EAb=tAtBSES1-rArB exp-2jkL+δB-δA.
ED=rArBS exp-2jkδA-tA2tBSrBSrB exp-2jkL+δB1-rArB exp-2jkL+δB-δAES.
EAb=tAtBSES1-rArB=tAtBSESi=1nrArBi+rArBn+11-rArB.
=tAtBSESrArBn+11-rArBEAb=rArBn+1.
nbeams=log log rArB-1.
EAbt=tAtBSESt+gttAtBSESt-2τ+gt-2τtAtBSESt-4τ+gt-4τtAtBSESt-2n-1τ+gt-2nτtAtBSESt-2nτ.
EAbt=tAtBSESt+n=1nbeamsi=0n gt-2iτESt-2nτ.
Ft=EAbt-2τ.
EAbt=tAtBSESt+gtFt.
Fs=exp-2τsEAbs.
exp-2τs=1-2τs+2τs2-.
exp-2τs11+2τs.
Fs=11+2τs EAbs.
Ft=EAbt-2τ ddt Ft.
ddt Ft=12τ1-gtFt-tAtBSESt,
EAbt=gtFt+tAtBSESt.
ddtxt=A·xt+B·EABt,
Ft=C·xt+D·EABt.
ddtxt=A+BCgt1-Dgt·xt+BtAtBS1-Dgt·Est,
EABt=11-DgtgtC·xt+tAtBS·ESt.
ESt=ElasertJ0Γ+J1Γexpjωmodt-J1Γexp-jωmodtexpjω0t.
ESt=ES0texpjω0t+ES1texpjω1t-ES-1texp-jω-1t.
ES0t=ElasertJ0Γ,
ES1t=ElasertJ1Γ,
ES-1t=-ElasertJ1Γ.
λ±1=2πc/ω0ωmod=2πc/ω±1.
Lnom=nλ0/2, where n=integer.
fmod=c/λmodnmodc/4L.
EDt=ED0texpjω0t+ED1texpjω1t-ED-1texp-jω-1t.
IIn=2ImagED0tED-1*t-ED1*t.
Iquad=2RealED0tED-1*t+ED1*t.
g0t=rArB exp-2jkδBt-τ-δAt,
g1t=g-1t=-rArB exp-2jkδBt-τ-δAt,
ES0t=Es0 exp-jkδs.
Δnt=2 i=1n δBt-2i-1τ-2 i=1n δAt-2iτ+δst-2nτ.
Rg=rArB,
Rr=tA2tBSrBSrA.
ED0t=rA exp-jk2δAt+δst+Rrn=1nbeamsRgn exp-jkΔntEs0.
ED-1t-ED1t=2rA exp-jk2δAt+δst+Rrn=1nbeams-Rgn×exp-jkΔntEs1.
IInt=4Es0Es1ImagrA2-rA2Rr2n=1nbeamsRgn exp-jkΔnt-2δAt-δst×rA2Rr2n=1nbeams-Rgn exp-jkΔnt-2δAt-δst+Rr4n=1nbeams Rgn exp-jkΔnt×n=1nbeams-Rgn expjkΔnt.
IInt=4Es0Es1ImagrA2-rA2Rr2n=1nbeamsRgn1-jkΔnt-2δAt-δst×-rA2Rr2n=1nbeams-Rgn1-jkΔnt-2δAt-δst+Rr4×-n=1nbeams Rgn1-jkΔnt×-n=1-nbeams-Rgn1-jkΔnt.
IInt=4Es0Es1-Rr2Rg1-Rg2δAt+δst+n=1nbeamsRgnΔnt-Rr2Rg1+Rg2δAt+δst-n=1nbeams-RgnΔnt+Rr4Rg1-Rgn=1nbeams-RgnΔnt+Rg1+Rgn=1nbeams RgnΔnt.
IInδss=4kEs0Es1-2Rg1-Rg2+1-tA2rA2Rg1-RgRg exp-2τs1+Rg exp-2τs+1+tA2rA2Rg1+RgRg exp-2τs1-Rg exp-2τs.
IInsδA=8kEs0Es1-2Rg1-Rg2+-11+Rg+tA2rA2×c Rg1-Rg2Rg exp-2τs1+Rg exp-2τs+11-Rg+tA2rA2Rg1-Rg2Rg exp-2τs1-Rg exp-2τs.
IInδBs=8kEs0Es111+Rg-tA2rA2Rg1-Rg2×Rg exp-τs1+Rg exp-2τs-11-Rg+tA2rA2Rg1-Rg2Rg exp-τs1-Rg exp-2τs.
exp-2τs=-s-σs-σ±jωs-σ±2jωs+σs+σ±jωs+σ±2jω.
σ=π2τ,
ω=πτ.
IInδAs=s+σn=1s+σ+jnωn=1s+σ-jnωs+αn=1s+α+j2ωn=1s+α-j2ω,
IInδBs=IInδAss-2σn=1s-2σ+j22n-1ωn=1s+2σ-j22n-1ωs+2σn=1s+2σ+j22n-1ωn=1s+2σ-j22n-1ω,
δBt=X cos ωt=RealX expiωBt.
EBdt=-rB1-jkX expjωBt+jkX exp-jωBtEBat.
EBa=u=-11ν=-11 EBauν exp-iuωmod+νωBt.
Bd=-rBu=-11EBau0 exp-iuωmodt+ν=-1,1EBauν+ikXEBau0exp-iuωmod+νωBt.
IIn=2Realu=-10ν=-10EBauν*EBau+1ν+1+EBauν+1EBau+1ν*.
Iquad=2Realu=-10ν=-10EBauν*EBau+1ν+1-EBauν+1EBau+1ν*.

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