Abstract

We develop a steady-state analytical and numerical model of the optical response of power-recycled Fabry–Perot Michelson laser gravitational-wave detectors to nonlinear thermal focusing in optical substrates. We assume that the thermal distortions are small enough that we can represent all intracavity fields as linear combinations of basis functions derived from the eigenmodes of a Fabry–Perot arm cavity. We have included the effects of power absorption in optical substrates and coatings, mismatches between laser wave-front and mirror surface curvatures, and aperture diffraction. We demonstrate a detailed numerical example of this model using the matlab program Melody for the initial Laser Interferometer Gravitational Wave Observatory detector.

© 2003 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  21. Y. Hefetz, N. Mavalvala, and D. Sigg, “Principles of calculating alignment signals in complex resonant optical interferometers,” J. Opt. Soc. Am. B 107, 1597–1605 (1997).
    [CrossRef]
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  28. See http://www.ligo.caltech.edu/gari/COCAsBuilt.htm.

2001

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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

1994

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. Rüdiger, R. Schilling, and W. Winkler, “Thermal lensing in recycling interferometric gravitational wave detectors,” Phys. Lett. A 194, 124–132 (1994).
[CrossRef]

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

1992

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

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

1991

B. J. Meers and K. A. Strain, “Wave-front distortion in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 43, 3117–3130 (1991).
[CrossRef]

W. Winkler, K. Danzmann, A. Rüdiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[CrossRef] [PubMed]

1990

P. Hello and J.-Y. Vinet, “Analytical models of thermal ab-errations in massive mirrors heated by high power laser beams,” J. Phys. (Paris) 51, 1267–1282 (1990).
[CrossRef]

1984

1983

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

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

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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

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

Anderson, D. Z.

Beausoleil, R. G.

Bork, R.

Brillet, A.

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

Camp, J.

D. McClelland, J. Camp, J. Mason, W. Kells, and S. Whitcomb, “Arm cavity resonant sideband control for laser interferometric gravitational wave detectors,” Opt. Lett. 24, 1014–1016 (1999).
[CrossRef]

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

Danzmann, K.

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. Rüdiger, R. Schilling, and W. Winkler, “Thermal lensing in recycling interferometric gravitational wave detectors,” Phys. Lett. A 194, 124–132 (1994).
[CrossRef]

W. Winkler, K. Danzmann, A. Rüdiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[CrossRef] [PubMed]

Drever, R.

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

Drever, R. W. P.

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

Ford, G. M.

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

Fritschel, P.

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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Gursel, Y.

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

Hall, J. L.

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

Hefetz, Y.

Y. Hefetz, N. Mavalvala, and D. Sigg, “Principles of calculating alignment signals in complex resonant optical interferometers,” J. Opt. Soc. Am. B 107, 1597–1605 (1997).
[CrossRef]

Hello, P.

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

P. Hello and J.-Y. Vinet, “Analytical models of thermal ab-errations in massive mirrors heated by high power laser beams,” J. Phys. (Paris) 51, 1267–1282 (1990).
[CrossRef]

Hough, J.

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

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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

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

Kells, W.

Kostenbauder, A.

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Lez, G.

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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and 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, and A. Brillet, “A high-accuracy method for the simulation of non-ideal optical cavities,” J. Phys. I 2, 1287–1303 (1992).

Mason, J.

Mavalvala, N.

P. Fritschel, R. Bork, G. Lez, N. Mavalvala, D. Ouimette, H. Rong, D. Sigg, and M. Zucker, “Readout and control of a power-recycled interferometric gravitational-wave antenna,” Appl. Opt. 40, 4988–4998 (2001).
[CrossRef]

Y. Hefetz, N. Mavalvala, and D. Sigg, “Principles of calculating alignment signals in complex resonant optical interferometers,” J. Opt. Soc. Am. B 107, 1597–1605 (1997).
[CrossRef]

McClelland, D.

Meers, B. J.

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

B. J. Meers and K. A. Strain, “Wave-front distortion in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 43, 3117–3130 (1991).
[CrossRef]

Mizuno, J.

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. Rüdiger, R. Schilling, and W. Winkler, “Thermal lensing in recycling interferometric gravitational wave detectors,” Phys. Lett. A 194, 124–132 (1994).
[CrossRef]

Morrison, E.

Munley, A. J.

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

Nelson, P. G.

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. Rüdiger, R. Schilling, and W. Winkler, “Thermal lensing in recycling interferometric gravitational wave detectors,” Phys. Lett. A 194, 124–132 (1994).
[CrossRef]

Ouimette, D.

Raab, F.

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

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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Robertson, D. I.

Rong, H.

Rüdiger, A.

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. Rüdiger, R. Schilling, and W. Winkler, “Thermal lensing in recycling interferometric gravitational wave detectors,” Phys. Lett. A 194, 124–132 (1994).
[CrossRef]

W. Winkler, K. Danzmann, A. Rüdiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[CrossRef] [PubMed]

Saha, P.

Savage Jr., 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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

Schilling, R.

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. Rüdiger, R. Schilling, and W. Winkler, “Thermal lensing in recycling interferometric gravitational wave detectors,” Phys. Lett. A 194, 124–132 (1994).
[CrossRef]

W. Winkler, K. Danzmann, A. Rüdiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[CrossRef] [PubMed]

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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

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

Siegman, A. E.

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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

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

Sigg, D.

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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

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

Strain, K. A.

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. Rüdiger, R. Schilling, and W. Winkler, “Thermal lensing in recycling interferometric gravitational wave detectors,” Phys. Lett. A 194, 124–132 (1994).
[CrossRef]

B. J. Meers and K. A. Strain, “Wave-front distortion in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 43, 3117–3130 (1991).
[CrossRef]

Sun, Y.

Thorne, K.

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

Vinet, J.-Y.

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

P. Hello and J.-Y. Vinet, “Analytical models of thermal ab-errations in massive mirrors heated by high power laser beams,” J. Phys. (Paris) 51, 1267–1282 (1990).
[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 Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[CrossRef]

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

Ward, H.

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

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

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

Whitcomb, S.

D. McClelland, J. Camp, J. Mason, W. Kells, and S. Whitcomb, “Arm cavity resonant sideband control for laser interferometric gravitational wave detectors,” Opt. Lett. 24, 1014–1016 (1999).
[CrossRef]

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

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

Winkler, W.

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. Rüdiger, R. Schilling, and W. Winkler, “Thermal lensing in recycling interferometric gravitational wave detectors,” Phys. Lett. A 194, 124–132 (1994).
[CrossRef]

W. Winkler, K. Danzmann, A. Rüdiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[CrossRef] [PubMed]

Zucker, M.

P. Fritschel, R. Bork, G. Lez, N. Mavalvala, D. Ouimette, H. Rong, D. Sigg, and M. Zucker, “Readout and control of a power-recycled interferometric gravitational-wave antenna,” Appl. Opt. 40, 4988–4998 (2001).
[CrossRef]

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

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

Appl. Opt.

Appl. Phys. B

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

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Y. Hefetz, N. Mavalvala, and D. Sigg, “Principles of calculating alignment signals in complex resonant optical interferometers,” J. Opt. Soc. Am. B 107, 1597–1605 (1997).
[CrossRef]

J. Phys. (Paris)

P. Hello and J.-Y. Vinet, “Analytical models of thermal ab-errations in massive mirrors heated by high power laser beams,” J. Phys. (Paris) 51, 1267–1282 (1990).
[CrossRef]

J. Phys. I

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

Opt. Lett.

Phys. Lett. A

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. Rüdiger, R. Schilling, and W. Winkler, “Thermal lensing in recycling interferometric gravitational wave detectors,” Phys. Lett. A 194, 124–132 (1994).
[CrossRef]

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

Phys. Rev. A

W. Winkler, K. Danzmann, A. Rüdiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[CrossRef] [PubMed]

Phys. Rev. D

B. J. Meers and K. A. Strain, “Wave-front distortion in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 43, 3117–3130 (1991).
[CrossRef]

Science

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

Other

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

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986); errata URL: http://www-ee.stanford.edu/~ siegman/lasers_book_errata.txt.

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

See http://www.mathworks.com/products/matlab/.

The Melody/MATLAB software package can be downloaded from the LIGO Software Tools for Advanced Interferometer Configurations Internet web site at the URL http://www.phys.ufl.edu/LIGO/LIGO/STAIC.html.

See http://www.ligo.caltech.edu/gari/COCAsBuilt.htm.

C. M. Will, Theory and Experiment in Gravitational Physics, revised ed. (Cambridge University, Cambridge, England, 1993).

D. G. Blair, ed., The Detection of Gravitational Waves (Cambridge University, Cambridge, England, 1993).

P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, Singapore, 1994).

A. Giazotto, “The VIRGO experiment: status of the art,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, and 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, and 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, and F. Ronga, eds. (World Scientific, Singapore, 1995), p. 112.

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Figures (21)

Fig. 1
Fig. 1

Schematic configuration of the initial LIGO detector, a power-recycled Fabry–Perot Michelson interferometer (PRFPMI).

Fig. 2
Fig. 2

Fractional error in |K00,00| obtained by comparing the exact value given by Eq. (19) with the TEM00 element of the approximate propagator K=C A, where A is provided by Eq. (12) and C by Eq. (14).

Fig. 3
Fig. 3

Schematic diagram of a singly coated mirror and the corresponding coordinate system used by Hello and Vinet.18 Both the mirror and the incident power distribution are assumed to be azimuthally symmetric. The total incident power P can be absorbed in the coating and/or the substrate, resulting in a total temperature distribution T(r, z) and a corresponding thermal lens in the substrate.

Fig. 4
Fig. 4

Schematic diagram of field distributions in a mirror of thickness h, consisting of a substrate (SS) with applied high-reflectance (HR) and antireflecting (AR) coatings.

Fig. 5
Fig. 5

Schematic diagram of the generalized mirror transfer matrix given by Eq. (36). The HR-coated surface of the mirror is on the left. The labels z< and z> denote the input and output reference planes, respectively, near the HR surface, while ζ< and ζ> are the input and output reference planes near the AR surface.

Fig. 6
Fig. 6

Interaction of the two primary propagation paths, parallel (∥) and perpendicular (⊥), with the five optical absorption regions that cause thermal focusing in the beam-splitter substrate: SS, SS, HR, AR, and AR.

Fig. 7
Fig. 7

Schematic diagram of the generalized beam-splitter transfer matrix given by Eq. (41). The HR surface of the mirror is on the upper left.

Fig. 8
Fig. 8

Schematic diagram of the enhancement, reflection, and transmission operators of a Fabry–Perot interferometer, defined by Eq. (50), Eq. (55), and Eq. (56), respectively.

Fig. 9
Fig. 9

Schematic diagram of the reflection and transmission operators of a Michelson interferometer, defined by Eq. (63). Note that the internal reflection operators RI-(Δωq) and RII-(Δωq) may each represent either the reflection operator of a mirror, given by Eq. (38a), or that of a Fabry–Perot interferometer, given by Eq. (56a). The labels z< and z> denote the input and output reference planes, respectively, near the HR surface of the beam splitter, while ζ< and ζ> are the input and output reference planes near the beam-splitter AR surface.

Fig. 10
Fig. 10

Schematic diagram of the enhancement, reflection, and transmission operators of a power-recycled Fabry–Perot Michelson interferometer, defined by Eq. (70) and Eq. (72), respectively.

Fig. 11
Fig. 11

Comparison between the predictions of computer code based on our model (Melody) and those of the FFT model in the case of simple aperturing. The power enhancement (or “gain”) at the parallel/in-line FPI reference plane z1> in Fig. 8 is plotted as a function of the aperture of M3. The “clip approximation” represents the result expected by assuming that the loss arises from simple single-pass aperture clipping at M3.

Fig. 12
Fig. 12

Comparison between the predictions of Melody and those of the FFT model in the case of unstable curvature reduction. The gain at the recycling mirror reference plane z5> in Fig. 10 is plotted as a function of the curvature of M2. Note that, in the absence of thermal loading, the recycling cavity becomes unstable for the resonant sidebands at a radius of curvature of 14480 m.

Fig. 13
Fig. 13

Initial LIGO recycled powers as a function of the PRM radius of curvature. The total TEM00 input laser power is 6.5 W, with 5.8 W delivered by the carrier.

Fig. 14
Fig. 14

Initial LIGO antisymmetric (“dark”) port output powers as a function of the PRM radius of curvature. The total TEM00 input laser power is 6.5 W, with 5.8 W delivered by the carrier.

Fig. 15
Fig. 15

Optimum initial LIGO PRM radius of curvature as a function of total TEM00 input laser power.

Fig. 16
Fig. 16

Optimized initial LIGO recycled sideband powers as a function of total TEM00 input laser power. At each value of the input power, the PRM radius of curvature shown in Fig. 15 has been used.

Fig. 17
Fig. 17

Optimized initial LIGO antisymmetric (“dark”) port output sideband powers as a function of total TEM00 input laser power. At each value of the input power, the PRM radius of curvature shown in Fig. 15 has been used.

Fig. 18
Fig. 18

Initial LIGO recycled powers as a function of total TEM00 input laser power.

Fig. 19
Fig. 19

Initial LIGO antisymmetric (“dark”) port output powers as a function of total TEM00 input laser power.

Fig. 20
Fig. 20

Initial LIGO PRM and beam-splitter microdisplacements as a function of total TEM00 input laser power.

Fig. 21
Fig. 21

Initial LIGO effective (i.e., quadratic) thermal focal lengths as a function of total TEM00 input laser power.

Tables (5)

Tables Icon

Table 1 Material Constants for Fused Silica Used in Our Simulations at λ0 = 1.0642 μm and T0 = 300 K

Tables Icon

Table 2 Total Absorbed TEM00 Powers for Each of the Three Regions Shown in Fig. 4 , Summed Over the Lowest-Order Transverse Modes of all Active Sidebands

Tables Icon

Table 3 Total Absorbed TEM00 Powers for Each of the Five Absorption Regions Shown in Fig. 6 , Summed Over the Lowest-Order Transverse Modes of all Active Sidebands a

Tables Icon

Table 4 High-Reflection (hr) and Antireflection (ar) Loss Parameters for the Initial LIGO Optical Elements Used in Our Simulations28 a

Tables Icon

Table 5 Physical Parameters for the Initial LIGO Optical Elements Used in Our Simulations28 a

Equations (103)

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E(r, t)Re{eE(r, t)exp[i(k0z-ω0t)]}.
E(r, t)qEq(r, t)exp[i(kqz-ωqt)],
Eq(r, t)=mnEmnq(z, t)umn(r).
Gmn,mn=exp(iΔφmn)δmmδnn,
Ωqq=exp(iΔωqτ)δqq,
E(z2)=GE(z1)Ω.
umn(x, y)=12m+nm!n!2πw21/2Hm2xwHn2yw×expi k2R-1w2(x2+y2),
E(x, y, z>)=AdxdyK(x, y; x, y)E(x, y, z<),
K(x, y; x, y)=-exp-i πλ02RM (x2+y2)×δ(x-x)δ(y-y),
Kmn,mn=- dxdyA dxdyK(x, y; x, y)×umn(x, y, z>)um,n(x, y, z<)=Adxdy exp-i πλ02RM (x2+y2)×umn(x, y, z>)umn(x, y, z<).
KCA,
Amn,mn=Adxdyumn*(x, y)umn(x, y)δm,mδn,n-exp(-α)Imn,mn(α),
I(α)=10012 α012 α01+α0000001+α00012 α001+12 α+34 α2014 α(α-2)00001+α+12 α2012 α0014 α(α-2)01+12 α+34 α2.
Cexp(iγc),
γπw2λ01RF-1RM,
cmn,m,n2w2- dxdy(x2+y2)×|umn(x, y)umn(x, y)|.
cmn,mn=Xm,m2δn,n+δm,mXn,n2,
Xm,m212 (1+2m)δm,m+12(m+1)(m+2)δm,m+2+12(m+1)(m+2)δm,m+2.
K00,00=1-exp[-(1+iγ)α]1+iγ.
Ts(r, z)=P αsa2kTkpkζk2×1-2τAkcoshζkzaJ0ζkra,
Tc(r, z)=P acakTkpk×Akcoshζkza-BksinhζkzaJ0ζkra,
Ak=12[ζksinh(γk)+τ cosh(γk)],
Bk=12[ζkcosh(γk)+τ sinh(γk)],
ζJ1(ζ)-τJ0(ζ)=0,
I(r)kpkJ0ζkra.
pk=2ζk2(ζk2+τ2)J02(ζk)1a20adr r J0ζkraI(r).
pk=1πa2ζk2(ζk2+τ2)J02(ζk)exp(-βk),
f=πw2αshPkTdη/dT,
ϕ(r)2πλ0dηdT-h/2h/2dzT(r, z),
ϕs(r)=Ps2πa2λ0kTdηdTkpkζk2×1-2τAkγksinh(γk)J0ζkra,
ϕc(r)=Pc2πa2λ0kTdηdTkpkζk2Aksinh(γk)J0ζkra,
Smn,mn=-dxdy exp[iΔϕ(r)]×umn(x, y)umn(x, y)
Φs;mn,mn=2πa2λ0kTdηdTkpkζk21-τγk 2Aksinh(γk)×exp(-βk)Ik;mn,mn,
Φc;mn,mn=2πa2λ0kTdηdTkpkζk2Aksinh(γk)×exp(-βk)Ik;mn,mn,
Ik;mn,mnexp(βk)- dxdyJ0ζkra-1×umn(x, y)umn(x, y).
Ik(βk)=100-12 βk0-12 βk01-βk0000001-βk000-12 βk001-2βk+34 βk2014 βk200001-2βk+12 βk20-12 βk0014 βk201-2βk+34 βk2.
S=exp{i[PsΦs+(Ph+Pa)Φc]},
ts=exp[-(αs+σs)h/2]1-(aar+sar),
E-E+=T-R-R+T+ F-F+,
T-=itts(C-C+)1/2SA,
T+=ittsS(C+C-)1/2A,
R-=-r exp(+i2kΔz)C-A,
R+=-rts2exp(-i2kΔz)SC+SA.
Amn,mn=Edxdyumn(x, y)umn(x, y),
S=exp{i[PsΦs+PsΦs+PhΦc+PaΦc+PaΦc]},
S=exp{i[PsΦs+PsΦs+PhΦc+PaΦc+PaΦc]}.
E-E+E+E-=T-R-00R+T+0000T+R+00R-T- F-F+F+F-,
T-=T+=ittsSA,
T-=T+=ittsSA,
R-=R-=-r exp[+i2k(z+Δz)]A,
R+=-rts2exp[-i2k(z+Δz)]SSA,
R+=-rts2exp[-i2k(z+Δz)]SSA.
F1q-=exp(iΔωqτ13)G13(R3-F3q++T3-Fq-),
F3q+=exp(iΔωqτ31)G31(R1-F1q-+T1-Fq+),
[1-Gˆ(Δωq)R^-]F1q-F3q+=Gˆ(Δωq)T^-Fq-Fq+,
T^-=T3-00T1-,R^-=0R3-R1-0,
Gˆ(Δω)=exp(iΔωτ13)G1300exp(iΔωτ31)G31.
Hˆ(Δω)=[1-Gˆ(Δω)R^-]-1Gˆ(Δω)T^-
F1q-F3q+=H1-(Δωq)H1+(Δωq)H3-(Δωq)H3+(Δωq) Fq-Fq+,
H1-(Δω)=[1-exp(i2Δωτ13)G13R3-G31R1-]-1×exp(iΔωτ13)G13T3-,
H1+(Δω)=[1-exp(i2Δωτ13)G13R3-G31R1-]-1×exp(i2Δωτ13)G13R3-G31T1-,
H3-(Δω)=[1-exp(i2Δωτ31)G31R1-G13R3-]-1×exp(i2Δωτ31)G31R1-G13T3-,
H3+(Δω)=[1-exp(i2Δωτ31)G31R1-G13R3-]-1×exp(iΔωτ31)G31T1-.
Eq-=T1+F1q-+R1+Fq+,
Eq+=T3+F3q++R3+Fq-,
Eq-Eq+=[R^++T^+Hˆ(Δωq)]Fq-Fq+,
T^+=T1+00T3+,R^+=0R1+R3+0.
Eq-Eq+=TFPI-(Δωq)RFPI-(Δωq)RFPI+(Δωq)TFPI+(Δωq) Fq-Fq+,
TFPI-(Δω)=T1+H1-(Δω),
TFPI+(Δω)=T3+H3+(Δω),
RFPI-(Δω)=R1++T1+H1+(Δω),
RFPI+(Δω)=R3++T3+H3-(Δω).
KFPI(Δz1)=exp(-i2kΔz1)G13R3-G31R1-,
Δz1=ϕ02k,
F0+=1-|λ0|λ0*(T1-)-1R1-E0.
E-Eq-,F-Fq+,
E+Eq+,F+Eq-.
Fq-=K616(Δωq)Eq+,
Fq+=K626(Δωq)Eq-,
K616(Δω)exp(i2Δωτ61)G61RI-(Δω)G16,
K626(Δω)exp(i2Δωτ62)G62RII-(Δω)G26.
Eq-Eq+=TMI-(Δωq)RMI-(Δωq)RMI+(Δωq)TMI+(Δωq) Fq-Fq+,
TMI-(Δω)RMI-(Δω)RMI+(Δω)TMI+(Δω)
=T6-R6-R6+T6+×K616(Δω)00K626(Δω) T6+R6+R6-R6- 0110.
R6-=R6-=ir6exp(+i2kΔz6)A6,
R6+=-ir6ts2exp(-i2kΔz6)S6S6A6,
R6+=-ir6ts2exp(-i2kΔz6)S6S6A6.
E0+=[D1(Δz6)+D2(Δz6)]F0+,
D1(Δz6)=exp(+i2kΔz6)R6+K616(0)T6+,
D2(Δz6)=exp(-i2kΔz6)T6+K626(0)R6-.
P0+(Δz6)=12 E0+E0+=12 F0+D1(Δz6)D2(Δz6)F0+c.c.=|F0+D1(0)D2(0)F0| cos(22kΔz6-ϕ),
Δz6=ϕ22k.
F5q-=H5+(Δωq)Fq+,
F6q+=H6+(Δωq)Fq+,
H5+(Δω)=[1-exp(i2Δωτ56)G56RMI-(Δω)G65R5-]-1×exp(i2Δωτ56)G56RMI-(Δω)G65T5-,
H6+(Δω)=[1-exp(i2Δωτ65)G65R5-G56RMI-(Δω)]-1×exp(iΔωτ65)G65T5-,
Eq-=RIFO-(Δωq)Fq+,
Eq+=TIFO+(Δωq)Fq+,
TIFO+(Δω)=TMI+(Δω)H6+(Δω),
RIFO-(Δω)=R5++T5+H5+(Δω).
KIFO(Δz5, Δz6)=exp(-i2kΔz5)G56[T6-K616(0)T6++exp(-i22kΔz6)R6-K626(0)R6-]×G65R5-,
Δz5=ϕ02k,
F0+=1-|λ0|λ0*(T5-)-1R5-E0,

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