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).

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

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]

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.

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]

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.

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

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]

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).

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.

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]

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.

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]

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.

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

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]

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 J. Betswieser, "Summary of H1 g factor measurements," LIGO Internal Document G050030-00-W (LIGO, 2005).

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 recent measurements of g factor changes induced by thermal loading in the H1 interferometer," LIGO Internal Document G050111-00-W (LIGO, 2005).

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).

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).

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.

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

[CrossRef]

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).

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]

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).

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. 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).

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).

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]

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]

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

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).

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]

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

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

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]

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).

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]

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.

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).

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.

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).

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).

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]

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]

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]

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).

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]

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]

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]

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]

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

[CrossRef]

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.