Abstract

We develop a detailed spatiotemporal model in the adiabatic approximation of the optical response of the Laser Interferometer Gravitational-Wave Observatory (LIGO) optical system to environmental perturbations. We begin by deriving a first-order linear time-dependent evolution equation that models the electromagnetic field as a function of position within a Fabry–Perot interferometer. This model allows both the length of the resonator and the misalignment angles of the end mirrors to vary in time and describes both resonant and nonresonant phenomena. After defining a biorthogonality relation that must be satisfied by general unperturbed spatial eigenfunctions of the Fabry–Perot interferometer, we expand the intracavity field as a linear combination of these functions and convert the spatiotemporal evolution equation into a linear system of coupled time-dependent iteration and/or differential equations. We then calculate the adiabatic connection equations that link the two LIGO Fabry–Perot interferometers through the power recycling cavity, which comprises two mirrors (the power recycling input mirror and the beam splitter) that have the same mechanical degrees of freedom as those in the Fabry–Perot arm cavities. We develop a detailed instance of this model for the evolution of the intracavity field within a resonator with sufficiently small misalignment angles that a Hermite–Gauss basis set can be used. We develop a detailed general approach to signal demodulation for simulation of servo-control systems and describe its implementation in the Hermite–Gauss approximation for Cartesian split-plane detectors. Finally, we demonstrate the use of this small-angle model to simulate the effect of angular misalignment on the longitudinal response function.

© 1999 Optical Society of America

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  1. A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
    [CrossRef]
  2. A. Giazotto, “The VIRGO experiment: status of the art,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), p. 86.
  3. K. Danzmann, “GEO 600—A 600-m Laser Interferometric Gravitational Wave Antenna,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), p. 100.
  4. K. Tsubono, “300-m Laser Interferometric Gravitational Wave Detector (TAMA300) in Japan,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), p. 112.
  5. J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitational-wave detection,” Phys. Rev. D 38, 433–447 (1988).
    [CrossRef]
  6. M. W. Regehr, F. J. Raab, S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Opt. Lett. 20, 1507–1509 (1995).
    [CrossRef] [PubMed]
  7. J.-Y. Vinet, P. Hello, C. N. Man, A. Brillet, “A highaccuracy method for the simulation of non-ideal optical cavities,” J. Phys. (Paris) I 2, 1287–1303 (1992).
  8. P. Saha, “Fast estimation of transverse fields in high-finesse optical cavities,” J. Opt. Soc. Am. A 14, 2195–2202 (1997).
    [CrossRef]
  9. B. Bochner, “Modelling the performance of interferometric gravitational-wave detectors with realistically imperfect optics,” Ph.D. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1998).
  10. D. Z. Anderson, “Alignment of resonant optical cavities,” Appl. Opt. 23, 2944–2949 (1984).
    [CrossRef] [PubMed]
  11. E. Morrison, B. J. Meers, D. I. Robertson, H. Ward, “Automatic alignment of optical interferometers,” Appl. Opt. 33, 5041–5049 (1994).
    [CrossRef] [PubMed]
  12. Y. Hefetz, N. Mavalvala, D. Sigg, “Principles of calculating alignment signals in complex resonant optical interferometers,” J. Opt. Soc. Am. B 14, 1597–1605 (1997).
    [CrossRef]
  13. J. Camp, L. Sievers, R. Bork, J. Heefner, “Guided lock acquisition in a suspended Fabry–Perot cavity,” Opt. Lett. 20, 2463–2465 (1995).
    [CrossRef]
  14. K. E. Oughstun, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 165–387.
  15. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  16. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).
  17. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 904.
  18. K. An, C. Yang, R. R. Dasari, M. S. Feld, “Cavity ring-down technique and its application to the measurement of ultraslow velocities,” Opt. Lett. 20, 1068–1070 (1995).
    [CrossRef] [PubMed]
  19. M. J. Lawrence, B. Willke, M. E. Husman, E. K. Gustafson, R. L. Byer, “The dynamic response of a Fabry–Perot interferometer,” J. Opt. Soc. Am. B 16, 523–532 (1999).
    [CrossRef]
  20. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).
  21. L. Schnupp, Max Planck Institut für Quantenoptik, D-85748 Garching, Germany (personal communication, 1986).
  22. R. G. Beausoleil, D. Sigg, M. R. Zwikel, “Initial alignment tolerance of the LIGO interferometer,” available from D. Sigg, LIGO Hanford Observatory, P.O. Box 1970 S9-02, Richland, Wash. 99352.

1999

1997

1996

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

1995

1994

1992

J.-Y. Vinet, P. Hello, C. N. Man, A. Brillet, “A highaccuracy method for the simulation of non-ideal optical cavities,” J. Phys. (Paris) I 2, 1287–1303 (1992).

1988

J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitational-wave detection,” Phys. Rev. D 38, 433–447 (1988).
[CrossRef]

1984

Abramovici, A.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Althouse, W.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

An, K.

Anderson, D. Z.

Beausoleil, R. G.

R. G. Beausoleil, D. Sigg, M. R. Zwikel, “Initial alignment tolerance of the LIGO interferometer,” available from D. Sigg, LIGO Hanford Observatory, P.O. Box 1970 S9-02, Richland, Wash. 99352.

Bochner, B.

B. Bochner, “Modelling the performance of interferometric gravitational-wave detectors with realistically imperfect optics,” Ph.D. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1998).

Bork, R.

Brillet, A.

J.-Y. Vinet, P. Hello, C. N. Man, A. Brillet, “A highaccuracy method for the simulation of non-ideal optical cavities,” J. Phys. (Paris) I 2, 1287–1303 (1992).

J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitational-wave detection,” Phys. Rev. D 38, 433–447 (1988).
[CrossRef]

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Byer, R. L.

Camp, J.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

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

Danzmann, K.

K. Danzmann, “GEO 600—A 600-m Laser Interferometric Gravitational Wave Antenna,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), p. 100.

Dasari, R. R.

Feld, M. S.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 904.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Giaime, J. A.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Giazotto, A.

A. Giazotto, “The VIRGO experiment: status of the art,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), p. 86.

Gillespie, A.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Gustafson, E. K.

Heefner, J.

Hefetz, Y.

Hello, P.

J.-Y. Vinet, P. Hello, C. N. Man, A. Brillet, “A highaccuracy method for the simulation of non-ideal optical cavities,” J. Phys. (Paris) I 2, 1287–1303 (1992).

Husman, M. E.

Kawamura, S.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Kuhnert, A.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Lawrence, M. J.

Lyons, T.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Man, C. N.

J.-Y. Vinet, P. Hello, C. N. Man, A. Brillet, “A highaccuracy method for the simulation of non-ideal optical cavities,” J. Phys. (Paris) I 2, 1287–1303 (1992).

J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitational-wave detection,” Phys. Rev. D 38, 433–447 (1988).
[CrossRef]

Mavalvala, N.

Meers, B. J.

E. Morrison, B. J. Meers, D. I. Robertson, H. Ward, “Automatic alignment of optical interferometers,” Appl. Opt. 33, 5041–5049 (1994).
[CrossRef] [PubMed]

J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitational-wave detection,” Phys. Rev. D 38, 433–447 (1988).
[CrossRef]

Morrison, E.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 904.

Oughstun, K. E.

K. E. Oughstun, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 165–387.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

Raab, F. J.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

M. W. Regehr, F. J. Raab, S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Opt. Lett. 20, 1507–1509 (1995).
[CrossRef] [PubMed]

Regehr, M. W.

Robertson, D. I.

Saha, P.

Savage, R. L.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Schnupp, L.

L. Schnupp, Max Planck Institut für Quantenoptik, D-85748 Garching, Germany (personal communication, 1986).

Shoemaker, D.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Siegman, A. E.

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

Sievers, L.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

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

Sigg, D.

Y. Hefetz, N. Mavalvala, D. Sigg, “Principles of calculating alignment signals in complex resonant optical interferometers,” J. Opt. Soc. Am. B 14, 1597–1605 (1997).
[CrossRef]

R. G. Beausoleil, D. Sigg, M. R. Zwikel, “Initial alignment tolerance of the LIGO interferometer,” available from D. Sigg, LIGO Hanford Observatory, P.O. Box 1970 S9-02, Richland, Wash. 99352.

Spero, R.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

Tsubono, K.

K. Tsubono, “300-m Laser Interferometric Gravitational Wave Detector (TAMA300) in Japan,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), p. 112.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

Vinet, J.-Y.

J.-Y. Vinet, P. Hello, C. N. Man, A. Brillet, “A highaccuracy method for the simulation of non-ideal optical cavities,” J. Phys. (Paris) I 2, 1287–1303 (1992).

J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitational-wave detection,” Phys. Rev. D 38, 433–447 (1988).
[CrossRef]

Vogt, R.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Ward, H.

Weiss, R.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Whitcomb, S.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Whitcomb, S. E.

Willke, B.

Yang, C.

Zucker, M.

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Zwikel, M. R.

R. G. Beausoleil, D. Sigg, M. R. Zwikel, “Initial alignment tolerance of the LIGO interferometer,” available from D. Sigg, LIGO Hanford Observatory, P.O. Box 1970 S9-02, Richland, Wash. 99352.

Appl. Opt.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. (Paris) I

J.-Y. Vinet, P. Hello, C. N. Man, A. Brillet, “A highaccuracy method for the simulation of non-ideal optical cavities,” J. Phys. (Paris) I 2, 1287–1303 (1992).

Opt. Lett.

Phys. Lett. A

A. Abramovici, W. Althouse, J. Camp, J. A. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Phys. Rev. D

J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitational-wave detection,” Phys. Rev. D 38, 433–447 (1988).
[CrossRef]

Other

B. Bochner, “Modelling the performance of interferometric gravitational-wave detectors with realistically imperfect optics,” Ph.D. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1998).

A. Giazotto, “The VIRGO experiment: status of the art,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), p. 86.

K. Danzmann, “GEO 600—A 600-m Laser Interferometric Gravitational Wave Antenna,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), p. 100.

K. Tsubono, “300-m Laser Interferometric Gravitational Wave Detector (TAMA300) in Japan,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), p. 112.

K. E. Oughstun, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 165–387.

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

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 904.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

L. Schnupp, Max Planck Institut für Quantenoptik, D-85748 Garching, Germany (personal communication, 1986).

R. G. Beausoleil, D. Sigg, M. R. Zwikel, “Initial alignment tolerance of the LIGO interferometer,” available from D. Sigg, LIGO Hanford Observatory, P.O. Box 1970 S9-02, Richland, Wash. 99352.

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Figures (7)

Fig. 1
Fig. 1

Schematic of a generalized version of the LIGO two-aperture standing-wave resonator. The incident electric field amplitude function F(r, t) is transmitted through the (possibly misaligned) ITM and continues to propagate in the positive z direction. For LIGO we have A2=D1, B2=B1, C2=C1, and D2=A1, with the specific values A1=D1=1, B1=L04 km, and C1=0.

Fig. 2
Fig. 2

Positive and negative propagation models corresponding to the resonator schematic shown in Fig. 1. Aj, Rj, and Mj represent, respectively, the aperture, the radius of curvature, and the anomalous phase distortion (caused by misalignment, for example) of either the ITM (j=1) or the ETM (j=2).

Fig. 3
Fig. 3

Cavity decay curve computed by integrating Eq. (30) for the case where τc=1.14 μs and Tm=0.12 μs.

Fig. 4
Fig. 4

Cavity decay curve computed by integrating Eq. (30) for the case where τc=1.14 μs and Tm=0.26 μs.

Fig. 5
Fig. 5

Time evolution of the TEM10 component of the stored intracavity field at the optical carrier frequency as M2 is misaligned sinusoidally in the xz plane. The misalignment amplitude is Θm=0.01, and the oscillation period is 100τ0. The input field is purely TEM00 and has a power of 1 W.

Fig. 6
Fig. 6

Schematic of primary optical components, electric fields, and conventions used in the LIGO IFO simulation. The coordinate systems for all fields propagating to the right or downward are right-handed, while those for propagation to the left or upward are left-handed. (The unit vector zˆ is always parallel with the direction of propagation, and yˆ is always directed toward the reader.) All angular rotation axes are defined so that a positive mirror misalignment causes the propagation vector—reflecting from the vacuum–coating interface of the mirror—to deflect away from the z axis in the positive direction.

Fig. 7
Fig. 7

Constant velocity scan of the in-line arm cavity rear mirror. Shown is the measured power of the demodulated quad-phase signal at the antisymmetric port with the use of a monolithic photodetector. The solid curve shows the perfectly aligned case, whereas the dashed curve represents a misalignment of the scanning mirror of 2 μrad. The scanning velocity is 1 μm/s over an interval of 0.025λ.

Tables (3)

Tables Icon

Table 1 Operating Characteristics of the LIGO FPI

Tables Icon

Table 2 Optical Parameters of the Components of the IFO Power Recycling Cavitya

Tables Icon

Table 3 Matrix Operators Required by the IFO Power Recycler Connection Equations in the Adiabatic and Hermite–Gauss Approximations, Based on the Conventions Shown in Fig. 6

Equations (88)

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F(r, t)Re{eF(r, t)exp[i(k0z-ω0t)]}.
F(r, t)qFq(r, t)exp[i(kqz-ωqt)],
E(r, t)Re{eE(r, t)exp[i(k0z-ω0t)]},
2E(r, t)-i2k0zE(r, t)=0,
E(x, y, z2)=A1dx dy ρ1(x, y)K21(x, y; x, y)×E(x, y, z1)K^21[E(x, y, z1)],
K21(x, y; x, y)=1iλBexpi πλB[A(x2+y2)-2(xx+yy)+D(x2+y2)].
E(x, y, z1+)=ρ1(x, y)E(x, y, z1-)
ABCD=A1B1C1D1 10-2/R11=A1-2B1/R1B1C1-2D1/R1D1.
E(x, y, z1)=A2dx dy ρ2(x, y)K12(x, y; x, y)×E(x, y, L0)K^12[E(x, y, z2)],
ABCD=A2B2C2D210-2/R21=D1-2B1/R2B1C1-2A1/R2A1,
E(x, y, 0)=A1dx dy ρ1(x, y)×K(x, y; x, y)E(x, y, z1)Kˆ[E(x, y, 0)],
K(x, y; x, y)=A2dx dy ρ2(x, y)×K12(x, y; x, y)K21(x, y; x, y),
γmnumn(x, y, 0)=A1dx dy ρ1(x, y)×K(x, y; x, y)umn(x, y, 0),
Adx dy umn*(x, y, z)umn(x, y, z)=δmmδnn
γmnvmn(x, y, 0)=ρ1(x, y)A1dx dy KT(x, y; x, y)×vmn(x, y, 0),
A1dx dy vmn(x, y, 0)umn(x, y, 0)
=-dx dy vmn(x, y, 0)umn(x, y, 0)=δmmδnn
Eq(r, t)=mnEmnq(z, t)umn(r),
Emnq(z2, t+τ)=mnKmn;mn(t)Emnq(z1, t),
Kmn;mn(t)=-dx dyA1dx dy ρ(x, y)×vmn(x, y, z2)K(x, y; x, y; t)×umn(x, y, z1).
z˙<12πF λτ0,θ˙<12F λπwτ0,
E(x, y, L0, t+L0/c)=P^21(t)[F(x, y, 0, t)]+exp[-i2k0z1(t)]×K^21(t)[E(x, y, 0, t)],
E(x, y, 0, t+τ0)=exp[i2k0z2(t+τ0/2)]K^12(t+τ0/2)×[E(x, y, L0, t+τ0/2)]=exp[i2k0z2(t+τ0/2)]×Pˆ(t)[F(x, y, 0, t)]+exp{i2k0[z2(t+τ0/2)-z1(t)]}×Kˆ(t)[E(x, y, 0, t)],
E(x, y, 0, t)=n=1 exp[i2k0z2(tn+τ0/2)]×m=1n-1 exp{i2k0[z2(tm+τ0/2)-z1(tm)]}×Kˆ(tm)[Pˆ(tn)[F(x, y, 0, tn)]],
Emnq(t+τ0)=exp{i2kq[z2(t+τ0/2)-z1(t)]}×mnKmn;mn(t)Emnq(t)+exp[i2kqz2(t+τ0/2)]×mnPmn;mn(t)Fmnq(t),
Kmn;mn(t)=γmnδmmδnn|γmn|exp(-iφmn)δmmδnn.
exp{i[2kqL(t)-φ00]}
=exp[i(2k0L0-φ00)]exp[i2k0ΔL(t)]×exp(i2ΔωqL0/c)exp[i2ΔkqΔL(t)]exp{i[2k0ΔL(t)+Δωqτ0]},
Emnq(t+τ0)=exp{i[2k0ΔL(t)+Δωqτ0]}×mnKmn;mn(t)Emnq(t)+exp[i2k0z1(t)]×mnPmn;mn(t)Fmnq(t),
Eq(t+τ0)=exp{i[2k0ΔL(t)+Δωqτ0]}{K(t)Eq(t)+exp[i2k0z1(t)]P(t)Fq(t)},
Eq(t+nτ0)SDnS-1Eq(t)+β 1-Dn1-DS-1PFq,
τ0E˙q(t)=-{1-exp{i[2k0ΔL(t)+Δωqτ0]}×K(t)}Eq(t)+exp{i[2k0Δz2(t)+Δωqτ0]}×P(t)Fq(t),
τ0E˙q(t)=-τ02τc+i[2k0ΔL(t)+Δωqτ0]+ln K(t)Eq(t)+exp{i[2k0Δz2(t)+Δωqτ0]}P(t)Fq(t).
E˙00(t)=-12τc+itTm2E00(t)+1,
E00(t)=1-i2 Tm exp[ζ2(t)]×π[erf ζ(t)-erf ζ(t0)]-exp[-ζ2(t0)]ζ(t0),
ζ(t)1+i2Tmt+iTm22τc.
limB0K(x, y; x, y)=δ(x-x/A)δ(y-y/A)|A|×expπλAC(x2+y2).
M(x, y; x, y; Δθx, Δθy)=δ(x-x)δ(y-y)×expik2(xΔθx+yΔθy)-x2+y2R.
G21=G12g0G0,
G0mn;mng0m+nδmmδnn,
g0exp(-iφ00/2)=exp(-i cos-1 g1g2)=g1g2-i1-g1g2.
Mmn;mn(t)-dxdy -dxdy ρ(x, y)×vmn(x, y, z+)umn(x, y, z-)×M(x, y; x, y; Δθx, Δθy)-rM˜mm[Θx(t)]M˜nn[Θy(t)],
M˜mm(Θx)12m+mm!m! 2πw21/2×-dxHm2xwHm2xw×exp-2xw2+i4xwΘx,
M(t)=exp{-2[Θx2(t)+Θy2(t)]}×1i2Θx(t)i2Θy(t)i2Θx(t)1-4Θx2(t)-4Θx(t)Θy(t)i2Θy(t)-4Θx(t)Θy(t)1-4Θy2(t).
K(t)=r1r2G0M2(t+τ0/2)G0M1(t),
P(t)=-it1r2G0M2(t+τ0/2)G0,
Θ2x=Θm sin(ω2xt),Θ1x=Θ1y=Θ2y=0.
Δk^k^-kˆ=2Δθxx^+2Δθyy^,
Δk^=2Δθxx^+2Δθyy^.
K^15=-ir5t6 exp{i[ωq(τC+τD)/2-4Δπz5]}G^16G^65M^5.
E5=(P^5115+P^5225)F5+P^51E1+P^52E21-(K^5115+K^5335),
K^5115K^51K^15,
K^5225K^52K^25,
P^5115K^51P^15,
P^5225K^52P^25.
F1=K^15E5+P^15F5,
F2=K^25E5+P^25F5.
E7=P^71E1+P^72E2+K^71F1+K^72F2.
E(φ00; t)=Re{G(φ00)[E0(t)+E+q(t)exp(-iΔωqt)+E-q(t)exp(+iΔωqt)]exp(-iω0t)},
Gmn;mn(φ00)exp[-i(m+n)φ00]δmmδnn
S(φ00; t)=ET(φ00; t)DE(φ00; t).
S¯(φ00; t)=Re{E0(t)D˜(φ00)[E+q(t)+E-q(t)]cos(Δωqt)}-Re{E0(t)D˜(φ00)×[E+q(t)-E-q(t)]i sin(Δωqt)},
S¯(φ00; t)SI(φ00; t)cos(Δω0t)+SQ(φ00; t)sin(Δω0t).
exp[-iβ cos(Δω0t)]=n=-i-nJn(β)exp(-inΔω0t)J0(β)-iJ1(β)exp(-iΔω0t)-iJ1(β)exp(+iΔω0t),
SI(φ00; t)=Im[V+q(φ00; t)+V-q(φ00; t)],
SQ(φ00; t)=Re[V+q(φ00; t)-V-q(φ00; t)],
E˜0(t)E0(t)/J0(β),
E˜±q(t)iE±q/J1(β)
V±q(φ00; t)J0(β)J1(β)E˜0(t)D˜(φ00)E˜±q(t).
S¯(φ00; t)=Re[S˜(φ00; t)exp(-iΔω0t)],
S˜(φ00; t)SI(φ00; t)+iSQ(φ00; t)=i[V-q*(φ00; t)-V+q(φ00; t)],
SI(φ00; t)=Im[V+q(φ00; t)+V-q(φ00; t)],
SQ(φ00; t)=-Re[V+q(φ00; t)-V-q(φ00; t)].
Dmn; mn-dxdyvmn(x, y)umn(x, y)Dˆ(x, y).
Dmn; mn-dy0dx--0dx×vmn(x, y)umn(x, y).
Dmn;mn=Dx;m,mDy;n,n.
D=π201010120120.
exp(iωrτ0)=exp(iωnrτ0)=-1,
exp(iωrτC)=exp(iωnrτC)=±i.
g1A-n1BR1,
g2D-n2BR2,
πw12λ=Bg2g1(1-g1g2)1/2,
πw22λ=Bg1g2(1-g1g2)1/2.
Δθj=λπwj 1+πnjwj2λRj21/2.
umn(x, y, 0)=12m+nm!n! 2πw121/2×Hm2xw1Hn2yw1×exp-ik2R1+1w12(x2+y2).
φmn2(m+n+1)cos-1g1g2
νmnq=c2Lq+(m+n+1)φ002πqΔνL+(m+n+1)ΔνG,
|γ˜jmn|2Ajdxdy|umn(x, y, zj)|2=12m+nm!n!π 0ζjdρ ρ exp(-ρ2)×02πdϕ Hm2(ρ cos ϕ)Hn2(ρ sin ϕ),

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