Abstract

Thermal tuning of optical Fabry–Perot cavities has been proposed to reduce the parametric gain of high-frequency parametric instabilities in gravitational wave detectors. We investigate the performance achievable for such tuning obtained by thermal actuation of the mirrors (also called test masses) of the arm cavities. We show that for test mass dimensions used in advanced detectors, when circularly symmetric heating is applied to the rear side of the mirror, the steady-state tuning performance is almost independent of the heating pattern and depends only on the heating power. We derived the optimal time-dependent heating required to achieve the fastest possible actuation in sapphire and fused-silica substrates. Our simulations show that sapphire mechanical deformation response to heating is 15 times faster than that of fused silica, although sapphire requires three times more heating power to obtain the same radius of curvature change.

© 2007 Optical Society of America

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  1. V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Analysis of parametric oscillatory instability in power recycled LIGO interferometer," Phys. Lett. A 305, 111-124 (2002).
    [CrossRef]
  2. P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, 1984).
  3. G. Heinzel, K. Strain, J. Mizuno, K. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rudiger, and K. Danzmann, "Experimental demonstration of a suspended dual recycling interferometer for gravitational wave detection," Phys. Rev. Lett. 81, 5493-5496 (1998).
    [CrossRef]
  4. D. Shoemaker, "Advanced LIGO," in Fifth Edoardo Amaldi Conference on Gravitational Waves, Tirrenia, Italy, July 6-11, 2003.
  5. V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett. A 287, 331-338 (2001).
    [CrossRef]
  6. See http://www.ligo.caltech.edu/advLIGO/.
  7. V. B. Braginsky, A. Gurkovsky, S. E. Strigin, and S. P. Vyatchaninl, "Analysis of parametric oscillatory instability in signal recycled LIGO interferometer," submitted to the LIGO Scientific Collaboration Review Comittee.
  8. C. Zhao, L. Ju, J. Degallaix, S. Gras, and D. G. Blair, "Parametric instabilities and their control in advanced interferometer gravitational-wave detectors," Phys. Rev. Lett. 94, 121102 (2005).
    [CrossRef] [PubMed]
  9. J. M. Hill and J. N. Dewynne, Heat Conduction (Blackwell Scientific, 1987), section entitled "Radial flow in solid circular cylinders and sphere."
  10. A. E. Siegman, Lasers (University Science, 1986).
  11. For this calculation we used the software FINESSE with the specification of Advanced LIGO with a Schnupp asymmetry of 40 cm.
  12. H. Lück, A. Freise, S. Goler, K. Kawabe, and K. Danzmann, "Thermal correction of the radii of curvature of mirrors for GEO 600," Class. Quantum Grav. 21, S985-S989 (2004).
    [CrossRef]
  13. R. Lawrence, "Active wavefront correction in laser interferometric gravitational wave detectors," Ph.D thesis (MIT, 2003).
  14. S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).
  15. R. Savage, M. Rakhmanov, K. Kawabe, and J. Betswieser, "Summary of recent measurements of g factor changes induced by thermal loading in the H1 interferometer," LIGO Internal Document G050111-00-W (LIGO, 2005).
  16. H. Lück, S. Hild, S. Goler, and the GEO 600 Team, "Thermal compensation of the radius of curvature of GEO600 mirrors," in Fifth Edoardo Amaldi Conference on Gravitational Waves, Tirrenia, Italy, July 6-11, 2003.
  17. The temperature is a linear function of the heating sources only for a small increase of temperature (less than 10 K). In this condition of a small increase, the heating loss due to radiation can be approximated as proportional to the temperature.
  18. L. Ju, C. Zhao, S. Gras, J. Degallaix, D. Blair, J. Munch, and D. H. Reitze, "Comparison of parametric instabilities for different test mass materials in advanced gravitational wave interferometers," Phys. Lett. A 355, 419-426 (2006).
    [CrossRef]
  19. J. Degallaix, "Thermal lensing time constant for sapphire and fused silica substrates," Australian International Gravitation Observatory, Military Road, Gingin, Australia; internal presentation, available on demand, 2004.
  20. R. Savage, M. Rakhmanov, K. Kawabe, and S. Waldman, "Measurement of thermally induced test mass surface curvature changes in a LIGO 4-km interferometer," LIGO Internal Document G050362-00-W (LIGO, 2005).
  21. R. Savage, M. Rakhmanov, K. Kawabe, and J. Betswieser, "Summary of H1 g factor measurements," LIGO Internal Document G050030-00-W (LIGO, 2005).
  22. S. Gras, C. Zhao, L. Ju, and D. G. Blair, "Preliminary investigation on a passive method for parametric instability control in advanced gravitational wave detectors," J. Phys.: Conf. Ser. 32, 251-258 (2006).
    [CrossRef]
  23. D. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," SIAM J. Appl. Math. 11, 431-441 (1963).
    [CrossRef]

2006 (2)

L. Ju, C. Zhao, S. Gras, J. Degallaix, D. Blair, J. Munch, and D. H. Reitze, "Comparison of parametric instabilities for different test mass materials in advanced gravitational wave interferometers," Phys. Lett. A 355, 419-426 (2006).
[CrossRef]

S. Gras, C. Zhao, L. Ju, and D. G. Blair, "Preliminary investigation on a passive method for parametric instability control in advanced gravitational wave detectors," J. Phys.: Conf. Ser. 32, 251-258 (2006).
[CrossRef]

2005 (1)

C. Zhao, L. Ju, J. Degallaix, S. Gras, and D. G. Blair, "Parametric instabilities and their control in advanced interferometer gravitational-wave detectors," Phys. Rev. Lett. 94, 121102 (2005).
[CrossRef] [PubMed]

2004 (1)

H. Lück, A. Freise, S. Goler, K. Kawabe, and K. Danzmann, "Thermal correction of the radii of curvature of mirrors for GEO 600," Class. Quantum Grav. 21, S985-S989 (2004).
[CrossRef]

2002 (1)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Analysis of parametric oscillatory instability in power recycled LIGO interferometer," Phys. Lett. A 305, 111-124 (2002).
[CrossRef]

2001 (1)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett. A 287, 331-338 (2001).
[CrossRef]

1998 (1)

G. Heinzel, K. Strain, J. Mizuno, K. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rudiger, and K. Danzmann, "Experimental demonstration of a suspended dual recycling interferometer for gravitational wave detection," Phys. Rev. Lett. 81, 5493-5496 (1998).
[CrossRef]

1963 (1)

D. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," SIAM J. Appl. Math. 11, 431-441 (1963).
[CrossRef]

Ballmer, S.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).

Betswieser, J.

R. Savage, M. Rakhmanov, K. Kawabe, and J. Betswieser, "Summary of H1 g factor measurements," LIGO Internal Document G050030-00-W (LIGO, 2005).

R. Savage, M. Rakhmanov, K. Kawabe, and J. Betswieser, "Summary of recent measurements of g factor changes induced by thermal loading in the H1 interferometer," LIGO Internal Document G050111-00-W (LIGO, 2005).

Blair, D.

L. Ju, C. Zhao, S. Gras, J. Degallaix, D. Blair, J. Munch, and D. H. Reitze, "Comparison of parametric instabilities for different test mass materials in advanced gravitational wave interferometers," Phys. Lett. A 355, 419-426 (2006).
[CrossRef]

Blair, D. G.

S. Gras, C. Zhao, L. Ju, and D. G. Blair, "Preliminary investigation on a passive method for parametric instability control in advanced gravitational wave detectors," J. Phys.: Conf. Ser. 32, 251-258 (2006).
[CrossRef]

C. Zhao, L. Ju, J. Degallaix, S. Gras, and D. G. Blair, "Parametric instabilities and their control in advanced interferometer gravitational-wave detectors," Phys. Rev. Lett. 94, 121102 (2005).
[CrossRef] [PubMed]

Braginsky, V. B.

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Analysis of parametric oscillatory instability in power recycled LIGO interferometer," Phys. Lett. A 305, 111-124 (2002).
[CrossRef]

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett. A 287, 331-338 (2001).
[CrossRef]

V. B. Braginsky, A. Gurkovsky, S. E. Strigin, and S. P. Vyatchaninl, "Analysis of parametric oscillatory instability in signal recycled LIGO interferometer," submitted to the LIGO Scientific Collaboration Review Comittee.

Danzmann, K.

H. Lück, A. Freise, S. Goler, K. Kawabe, and K. Danzmann, "Thermal correction of the radii of curvature of mirrors for GEO 600," Class. Quantum Grav. 21, S985-S989 (2004).
[CrossRef]

G. Heinzel, K. Strain, J. Mizuno, K. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rudiger, and K. Danzmann, "Experimental demonstration of a suspended dual recycling interferometer for gravitational wave detection," Phys. Rev. Lett. 81, 5493-5496 (1998).
[CrossRef]

Degallaix, J.

L. Ju, C. Zhao, S. Gras, J. Degallaix, D. Blair, J. Munch, and D. H. Reitze, "Comparison of parametric instabilities for different test mass materials in advanced gravitational wave interferometers," Phys. Lett. A 355, 419-426 (2006).
[CrossRef]

C. Zhao, L. Ju, J. Degallaix, S. Gras, and D. G. Blair, "Parametric instabilities and their control in advanced interferometer gravitational-wave detectors," Phys. Rev. Lett. 94, 121102 (2005).
[CrossRef] [PubMed]

J. Degallaix, "Thermal lensing time constant for sapphire and fused silica substrates," Australian International Gravitation Observatory, Military Road, Gingin, Australia; internal presentation, available on demand, 2004.

Dewynne, J. N.

J. M. Hill and J. N. Dewynne, Heat Conduction (Blackwell Scientific, 1987), section entitled "Radial flow in solid circular cylinders and sphere."

Freise, A.

H. Lück, A. Freise, S. Goler, K. Kawabe, and K. Danzmann, "Thermal correction of the radii of curvature of mirrors for GEO 600," Class. Quantum Grav. 21, S985-S989 (2004).
[CrossRef]

Frolov, V.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).

Goler, S.

H. Lück, A. Freise, S. Goler, K. Kawabe, and K. Danzmann, "Thermal correction of the radii of curvature of mirrors for GEO 600," Class. Quantum Grav. 21, S985-S989 (2004).
[CrossRef]

H. Lück, S. Hild, S. Goler, and the GEO 600 Team, "Thermal compensation of the radius of curvature of GEO600 mirrors," in Fifth Edoardo Amaldi Conference on Gravitational Waves, Tirrenia, Italy, July 6-11, 2003.

Gras, S.

L. Ju, C. Zhao, S. Gras, J. Degallaix, D. Blair, J. Munch, and D. H. Reitze, "Comparison of parametric instabilities for different test mass materials in advanced gravitational wave interferometers," Phys. Lett. A 355, 419-426 (2006).
[CrossRef]

S. Gras, C. Zhao, L. Ju, and D. G. Blair, "Preliminary investigation on a passive method for parametric instability control in advanced gravitational wave detectors," J. Phys.: Conf. Ser. 32, 251-258 (2006).
[CrossRef]

C. Zhao, L. Ju, J. Degallaix, S. Gras, and D. G. Blair, "Parametric instabilities and their control in advanced interferometer gravitational-wave detectors," Phys. Rev. Lett. 94, 121102 (2005).
[CrossRef] [PubMed]

Gurkovsky, A.

V. B. Braginsky, A. Gurkovsky, S. E. Strigin, and S. P. Vyatchaninl, "Analysis of parametric oscillatory instability in signal recycled LIGO interferometer," submitted to the LIGO Scientific Collaboration Review Comittee.

Heinzel, G.

G. Heinzel, K. Strain, J. Mizuno, K. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rudiger, and K. Danzmann, "Experimental demonstration of a suspended dual recycling interferometer for gravitational wave detection," Phys. Rev. Lett. 81, 5493-5496 (1998).
[CrossRef]

Hild, S.

H. Lück, S. Hild, S. Goler, and the GEO 600 Team, "Thermal compensation of the radius of curvature of GEO600 mirrors," in Fifth Edoardo Amaldi Conference on Gravitational Waves, Tirrenia, Italy, July 6-11, 2003.

Hill, J. M.

J. M. Hill and J. N. Dewynne, Heat Conduction (Blackwell Scientific, 1987), section entitled "Radial flow in solid circular cylinders and sphere."

Ju, L.

S. Gras, C. Zhao, L. Ju, and D. G. Blair, "Preliminary investigation on a passive method for parametric instability control in advanced gravitational wave detectors," J. Phys.: Conf. Ser. 32, 251-258 (2006).
[CrossRef]

L. Ju, C. Zhao, S. Gras, J. Degallaix, D. Blair, J. Munch, and D. H. Reitze, "Comparison of parametric instabilities for different test mass materials in advanced gravitational wave interferometers," Phys. Lett. A 355, 419-426 (2006).
[CrossRef]

C. Zhao, L. Ju, J. Degallaix, S. Gras, and D. G. Blair, "Parametric instabilities and their control in advanced interferometer gravitational-wave detectors," Phys. Rev. Lett. 94, 121102 (2005).
[CrossRef] [PubMed]

Kawabe, K.

H. Lück, A. Freise, S. Goler, K. Kawabe, and K. Danzmann, "Thermal correction of the radii of curvature of mirrors for GEO 600," Class. Quantum Grav. 21, S985-S989 (2004).
[CrossRef]

R. Savage, M. Rakhmanov, K. Kawabe, and S. Waldman, "Measurement of thermally induced test mass surface curvature changes in a LIGO 4-km interferometer," LIGO Internal Document G050362-00-W (LIGO, 2005).

R. Savage, M. Rakhmanov, K. Kawabe, and J. Betswieser, "Summary of H1 g factor measurements," LIGO Internal Document G050030-00-W (LIGO, 2005).

R. Savage, M. Rakhmanov, K. Kawabe, and J. Betswieser, "Summary of recent measurements of g factor changes induced by thermal loading in the H1 interferometer," LIGO Internal Document G050111-00-W (LIGO, 2005).

Kells, W.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).

Lawrence, R.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).

R. Lawrence, "Active wavefront correction in laser interferometric gravitational wave detectors," Ph.D thesis (MIT, 2003).

Lück, H.

H. Lück, A. Freise, S. Goler, K. Kawabe, and K. Danzmann, "Thermal correction of the radii of curvature of mirrors for GEO 600," Class. Quantum Grav. 21, S985-S989 (2004).
[CrossRef]

H. Lück, S. Hild, S. Goler, and the GEO 600 Team, "Thermal compensation of the radius of curvature of GEO600 mirrors," in Fifth Edoardo Amaldi Conference on Gravitational Waves, Tirrenia, Italy, July 6-11, 2003.

Marquardt, D.

D. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," SIAM J. Appl. Math. 11, 431-441 (1963).
[CrossRef]

Mason, K.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).

Mizuno, J.

G. Heinzel, K. Strain, J. Mizuno, K. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rudiger, and K. Danzmann, "Experimental demonstration of a suspended dual recycling interferometer for gravitational wave detection," Phys. Rev. Lett. 81, 5493-5496 (1998).
[CrossRef]

Moreno, G.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).

Munch, J.

L. Ju, C. Zhao, S. Gras, J. Degallaix, D. Blair, J. Munch, and D. H. Reitze, "Comparison of parametric instabilities for different test mass materials in advanced gravitational wave interferometers," Phys. Lett. A 355, 419-426 (2006).
[CrossRef]

Ottaway, D.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).

Rakhmanov, M.

R. Savage, M. Rakhmanov, K. Kawabe, and S. Waldman, "Measurement of thermally induced test mass surface curvature changes in a LIGO 4-km interferometer," LIGO Internal Document G050362-00-W (LIGO, 2005).

R. Savage, M. Rakhmanov, K. Kawabe, and J. Betswieser, "Summary of H1 g factor measurements," LIGO Internal Document G050030-00-W (LIGO, 2005).

R. Savage, M. Rakhmanov, K. Kawabe, and J. Betswieser, "Summary of recent measurements of g factor changes induced by thermal loading in the H1 interferometer," LIGO Internal Document G050111-00-W (LIGO, 2005).

Reitze, D. H.

L. Ju, C. Zhao, S. Gras, J. Degallaix, D. Blair, J. Munch, and D. H. Reitze, "Comparison of parametric instabilities for different test mass materials in advanced gravitational wave interferometers," Phys. Lett. A 355, 419-426 (2006).
[CrossRef]

Rudiger, A.

G. Heinzel, K. Strain, J. Mizuno, K. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rudiger, and K. Danzmann, "Experimental demonstration of a suspended dual recycling interferometer for gravitational wave detection," Phys. Rev. Lett. 81, 5493-5496 (1998).
[CrossRef]

Saulson, P. R.

P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, 1984).

Savage, R.

R. Savage, M. Rakhmanov, K. Kawabe, and S. Waldman, "Measurement of thermally induced test mass surface curvature changes in a LIGO 4-km interferometer," LIGO Internal Document G050362-00-W (LIGO, 2005).

R. Savage, M. Rakhmanov, K. Kawabe, and J. Betswieser, "Summary of H1 g factor measurements," LIGO Internal Document G050030-00-W (LIGO, 2005).

R. Savage, M. Rakhmanov, K. Kawabe, and J. Betswieser, "Summary of recent measurements of g factor changes induced by thermal loading in the H1 interferometer," LIGO Internal Document G050111-00-W (LIGO, 2005).

Schilling, R.

G. Heinzel, K. Strain, J. Mizuno, K. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rudiger, and K. Danzmann, "Experimental demonstration of a suspended dual recycling interferometer for gravitational wave detection," Phys. Rev. Lett. 81, 5493-5496 (1998).
[CrossRef]

Shoemaker, D.

D. Shoemaker, "Advanced LIGO," in Fifth Edoardo Amaldi Conference on Gravitational Waves, Tirrenia, Italy, July 6-11, 2003.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Skeldon, K.

G. Heinzel, K. Strain, J. Mizuno, K. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rudiger, and K. Danzmann, "Experimental demonstration of a suspended dual recycling interferometer for gravitational wave detection," Phys. Rev. Lett. 81, 5493-5496 (1998).
[CrossRef]

Smith, M.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).

Strain, K.

G. Heinzel, K. Strain, J. Mizuno, K. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rudiger, and K. Danzmann, "Experimental demonstration of a suspended dual recycling interferometer for gravitational wave detection," Phys. Rev. Lett. 81, 5493-5496 (1998).
[CrossRef]

Strigin, S. E.

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Analysis of parametric oscillatory instability in power recycled LIGO interferometer," Phys. Lett. A 305, 111-124 (2002).
[CrossRef]

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett. A 287, 331-338 (2001).
[CrossRef]

V. B. Braginsky, A. Gurkovsky, S. E. Strigin, and S. P. Vyatchaninl, "Analysis of parametric oscillatory instability in signal recycled LIGO interferometer," submitted to the LIGO Scientific Collaboration Review Comittee.

Vorvick, C.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).

Vyatchanin, S. P.

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Analysis of parametric oscillatory instability in power recycled LIGO interferometer," Phys. Lett. A 305, 111-124 (2002).
[CrossRef]

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett. A 287, 331-338 (2001).
[CrossRef]

Vyatchaninl, S. P.

V. B. Braginsky, A. Gurkovsky, S. E. Strigin, and S. P. Vyatchaninl, "Analysis of parametric oscillatory instability in signal recycled LIGO interferometer," submitted to the LIGO Scientific Collaboration Review Comittee.

Waldman, S.

R. Savage, M. Rakhmanov, K. Kawabe, and S. Waldman, "Measurement of thermally induced test mass surface curvature changes in a LIGO 4-km interferometer," LIGO Internal Document G050362-00-W (LIGO, 2005).

Willems, P.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).

Willke, B.

G. Heinzel, K. Strain, J. Mizuno, K. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rudiger, and K. Danzmann, "Experimental demonstration of a suspended dual recycling interferometer for gravitational wave detection," Phys. Rev. Lett. 81, 5493-5496 (1998).
[CrossRef]

Winkler, W.

G. Heinzel, K. Strain, J. Mizuno, K. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rudiger, and K. Danzmann, "Experimental demonstration of a suspended dual recycling interferometer for gravitational wave detection," Phys. Rev. Lett. 81, 5493-5496 (1998).
[CrossRef]

Zhao, C.

S. Gras, C. Zhao, L. Ju, and D. G. Blair, "Preliminary investigation on a passive method for parametric instability control in advanced gravitational wave detectors," J. Phys.: Conf. Ser. 32, 251-258 (2006).
[CrossRef]

L. Ju, C. Zhao, S. Gras, J. Degallaix, D. Blair, J. Munch, and D. H. Reitze, "Comparison of parametric instabilities for different test mass materials in advanced gravitational wave interferometers," Phys. Lett. A 355, 419-426 (2006).
[CrossRef]

C. Zhao, L. Ju, J. Degallaix, S. Gras, and D. G. Blair, "Parametric instabilities and their control in advanced interferometer gravitational-wave detectors," Phys. Rev. Lett. 94, 121102 (2005).
[CrossRef] [PubMed]

Zucker, M.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).

Class. Quantum Grav. (1)

H. Lück, A. Freise, S. Goler, K. Kawabe, and K. Danzmann, "Thermal correction of the radii of curvature of mirrors for GEO 600," Class. Quantum Grav. 21, S985-S989 (2004).
[CrossRef]

J. Phys.: Conf. Ser. (1)

S. Gras, C. Zhao, L. Ju, and D. G. Blair, "Preliminary investigation on a passive method for parametric instability control in advanced gravitational wave detectors," J. Phys.: Conf. Ser. 32, 251-258 (2006).
[CrossRef]

Phys. Lett. A (3)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett. A 287, 331-338 (2001).
[CrossRef]

L. Ju, C. Zhao, S. Gras, J. Degallaix, D. Blair, J. Munch, and D. H. Reitze, "Comparison of parametric instabilities for different test mass materials in advanced gravitational wave interferometers," Phys. Lett. A 355, 419-426 (2006).
[CrossRef]

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Analysis of parametric oscillatory instability in power recycled LIGO interferometer," Phys. Lett. A 305, 111-124 (2002).
[CrossRef]

Phys. Rev. Lett. (2)

G. Heinzel, K. Strain, J. Mizuno, K. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rudiger, and K. Danzmann, "Experimental demonstration of a suspended dual recycling interferometer for gravitational wave detection," Phys. Rev. Lett. 81, 5493-5496 (1998).
[CrossRef]

C. Zhao, L. Ju, J. Degallaix, S. Gras, and D. G. Blair, "Parametric instabilities and their control in advanced interferometer gravitational-wave detectors," Phys. Rev. Lett. 94, 121102 (2005).
[CrossRef] [PubMed]

SIAM J. Appl. Math. (1)

D. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," SIAM J. Appl. Math. 11, 431-441 (1963).
[CrossRef]

Other (15)

See http://www.ligo.caltech.edu/advLIGO/.

V. B. Braginsky, A. Gurkovsky, S. E. Strigin, and S. P. Vyatchaninl, "Analysis of parametric oscillatory instability in signal recycled LIGO interferometer," submitted to the LIGO Scientific Collaboration Review Comittee.

J. M. Hill and J. N. Dewynne, Heat Conduction (Blackwell Scientific, 1987), section entitled "Radial flow in solid circular cylinders and sphere."

A. E. Siegman, Lasers (University Science, 1986).

For this calculation we used the software FINESSE with the specification of Advanced LIGO with a Schnupp asymmetry of 40 cm.

D. Shoemaker, "Advanced LIGO," in Fifth Edoardo Amaldi Conference on Gravitational Waves, Tirrenia, Italy, July 6-11, 2003.

P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, 1984).

J. Degallaix, "Thermal lensing time constant for sapphire and fused silica substrates," Australian International Gravitation Observatory, Military Road, Gingin, Australia; internal presentation, available on demand, 2004.

R. Savage, M. Rakhmanov, K. Kawabe, and S. Waldman, "Measurement of thermally induced test mass surface curvature changes in a LIGO 4-km interferometer," LIGO Internal Document G050362-00-W (LIGO, 2005).

R. Savage, M. Rakhmanov, K. Kawabe, and J. Betswieser, "Summary of H1 g factor measurements," LIGO Internal Document G050030-00-W (LIGO, 2005).

R. Lawrence, "Active wavefront correction in laser interferometric gravitational wave detectors," Ph.D thesis (MIT, 2003).

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, "Thermal compensation description," LIGO Internal Document T050064-00-R (LIGO, 2005).

R. Savage, M. Rakhmanov, K. Kawabe, and J. Betswieser, "Summary of recent measurements of g factor changes induced by thermal loading in the H1 interferometer," LIGO Internal Document G050111-00-W (LIGO, 2005).

H. Lück, S. Hild, S. Goler, and the GEO 600 Team, "Thermal compensation of the radius of curvature of GEO600 mirrors," in Fifth Edoardo Amaldi Conference on Gravitational Waves, Tirrenia, Italy, July 6-11, 2003.

The temperature is a linear function of the heating sources only for a small increase of temperature (less than 10 K). In this condition of a small increase, the heating loss due to radiation can be approximated as proportional to the temperature.

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Figures (7)

Fig. 1
Fig. 1

Mirror deformation characterized by 1 R heated as a function of the surface area heated. The percentage of the surface heated is relative to the total area of the rear side of the mirror. R heated represents the radius of curvature (positive for a concave mirror) of the heated mirror, which is assumed to be initially flat. By conduction finite-element modeling, we determine the mechanical deformation of the mirror reflective surface when heat is applied on the rear side. For this plot, an Advanced LIGO sapphire test mass was heated with 10 W . In the insets, we have included schematic views of the test mass with the different heating patterns represented by dark color.

Fig. 2
Fig. 2

Cross section of the sapphire mirror temperature profile with exaggerated mechanical deformation for both annular (left) and central heating (right). The gray scale at the bottom represents the substrate temperature scale in kelvins. The heating power is 10 W , and the room temperature is 300 K . The reflective surface of the mirror on top of the figure is always curved in the same direction independently of the heating pattern.

Fig. 3
Fig. 3

Mirror deformation characterized by 1 R heated function of the thickness-to-radius ratio of the test mass for both central and annular heating. For each heating pattern, we display values obtained by heating the rear side of the test mass, using three different values for the percentage of the surface heated: 25%, 50%, and 75% as defined for the horizontal axis of Fig. 1. For this plot we consider a sapphire test mass heated with 10 W . The radius of the test mass is 13.7 cm .

Fig. 4
Fig. 4

Shift of the frequency difference between the optical TEM 10 mode and the TEM 00 mode as a function of the heating power. For this plot, only one mirror in the cavity is heated. As expected from Eqs. (6, 10), the relationship between the frequency shift and the heating power is almost linear.

Fig. 5
Fig. 5

Evolution of the sapphire test mass radius of curvature for 10 W central heating power. The mirror deformations are plotted for different central heating areas. In the plot legend, the surface heating area percentage is relative to the total area ( π R 2 ) of the rear of the mirror. Initially, the mirror has a curvature of 2076 m , and the steady-state value is 2054 m . When only a small area of the test mass is heated, a hot spot is created on the rear side of the test mass. This hot spot generates quickly a strong gradient of temperature at the rear of the test mass, creating the quick and strong bending of the reflective side of the mirror. ETM, end test mass, RofC, radius of curvature.

Fig. 6
Fig. 6

Test mass deformation evolution for sapphire (left) and fused silica (right) when optimized time-dependent heating is used. The step input response corresponds to the case of constant heating power. For both plots a central area of 7 cm diameter has been heated on the rear of the test mass, this corresponding to the 5% heated area in Fig. 5. The optimized heating has greatly reduced the transient time of the mechanical deformations. In these plots, the horizontal axis time scale for sapphire is 1 order of magnitude smaller than that of fused silica.

Fig. 7
Fig. 7

Zoom on the deformations presented in Fig. 6. Owing to the optimized time-dependent heating, the characteristic time of the exponential response has been reduced by a factor of 2 for both sapphire and fused-silica substrates. The characteristic time is calculated by fitting an exponential function to the mechanical deformations from the beginning of the response ( time = 0 ) to the maximum of the deformations of the mirror (equivalent to the minimum in the radius of curvature of the mirror). The optimized time-dependent heating reduced the characteristic time, thanks to the combination of two effects: first, the maximum of the deformations is reached earlier, and, second, the amplitude of the maximum of the deformations is smaller than in the case of a step heating.

Tables (2)

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Table 1 Test Mass Dimension and Cavity Parameters

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Table 2 Material Constants Used for Sapphire and Fused-Silica Test Mass Substrates a

Equations (10)

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

R = R 0 Λ 1 + ( Λ ω 1 δ 1 ) 2 > 1 ,
R 0 = 4 P circ Q 1 Q m m d c ω m 2 .
Δ ω 1 = ω m ( ω 0 ω 1 ) < δ 1 .
Δ ω 1 δ 1 .
Δ T ( r ) = P heat 2 π R l r 2 2 k R .
R heated = 4 π k R 2 α P heat ,
1 R h = 1 R m + 1 R heated .
ω m n = π c d [ q + ( m + n + 1 ) cos 1 g 1 g 2 π ] .
Δ f = c 2 d [ ( m + n ) cos 1 1 d R m π ] .
δ f = c 2 d [ ( m + n ) 2 π d ( 1 ( 1 d R m ) 2 ) 1 2 ] 1 R heated .

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