Abstract

Laser-induced thermal lens in optical components causes wavefront distortion of the laser beam and may affect performance and stability of optical systems such as high-power lasers. The bulging of the heated area, the temperature dependence of the refractive index, and the photoelastic effects are responsible for phase shifts damaging beam quality. The theoretical background for laser-induced beam distortion is well understood and applies only for axially symmetric thermal loadings, with the assumptions that the stresses follow thin-disk or long-rod approximations. This, in fact, limits the overall applications of this model. In this work, we developed an unified theoretical model for the optical path change in optical materials regardless of its thickness. The modeling is based on the solution of the thermoelastic equation and has a real description of the surface deformation caused in the optical element. In the appropriated limits, as expected, the model retrieves the thin-disk and the long-rod type distributions. Furthermore, we provided time-dependent radial expressions for the temperature, surface displacement, and stresses. The theory presented in this paper provides simple analytical tools for designing laser systems, and complements previous work allowing one to access optical distortions of materials ranging from thin-disk to long-rod-like distributions.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2011 (2)

2008 (2)

N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C. Malacarne, A. C. Bento, and M. L. Baesso, “Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials,” Opt. Lett. 33, 1464–1466 (2008).
[CrossRef]

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

2007 (1)

N. G. C. Astrath, L. C. Malacarne, P. R. B. Pedreira, A. C. Bento, M. L. Baesso, and J. Shen, “Time-resolved thermal mirror for nanoscale surface displacement detection in low absorbing solids,” Appl. Phys. Lett. 91, 191908 (2007).
[CrossRef]

2006 (1)

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

2005 (2)

C. A. Klein, “Analytical stress modeling of high-energy laser windows: Application to fusion-cast calcium fluoride windows,” J. Appl. Phys. 98, 043103 (2005).
[CrossRef]

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

2004 (1)

2002 (1)

L. B. Glebov, “Intrinsic laser-induced breakdown of silicate glasses,” Proc. SPIE 4679, 321–331 (2002).
[CrossRef]

2001 (1)

C. A. Klein, “Describing beam-aberration effects induced by laser-light transmitting components: a short account of Raytheon’s contribution,” Proc. SPIE 4376, 24–34 (2001).
[CrossRef]

1994 (1)

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin-film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[CrossRef]

1991 (1)

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

1990 (1)

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990).
[CrossRef]

1985 (1)

1971 (1)

M. Sparks, “Optical distortion by heated windows in high-power laser systems,” J. Appl. Phys. 42, 5029–5046 (1971).
[CrossRef]

Anjos, V.

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

Astrath, F. B. G.

Astrath, N. G. C.

L. C. Malacarne, N. G. C. Astrath, G. V. B. Lukasievcz, E. K. Lenzi, M. L. Baesso, and S. E. Bialkowski, “Time-resolved thermal lens and thermal mirror spectroscopy with sample-fluid heat coupling: A complete model for material characterization,” Appl. Spectrosc. 65, 99–104 (2011).
[CrossRef]

N. G. C. Astrath, L. C. Malacarne, V. S. Zanuto, M. P. Belancon, R. S. Mendes, M. L. Baesso, and C. Jacinto, “Finite-size effect on the surface deformation thermal mirror method,” J. Opt. Soc. Am. B 28, 1735–1739 (2011).
[CrossRef]

N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C. Malacarne, A. C. Bento, and M. L. Baesso, “Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials,” Opt. Lett. 33, 1464–1466 (2008).
[CrossRef]

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

N. G. C. Astrath, L. C. Malacarne, P. R. B. Pedreira, A. C. Bento, M. L. Baesso, and J. Shen, “Time-resolved thermal mirror for nanoscale surface displacement detection in low absorbing solids,” Appl. Phys. Lett. 91, 191908 (2007).
[CrossRef]

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

Baesso, M. L.

N. G. C. Astrath, L. C. Malacarne, V. S. Zanuto, M. P. Belancon, R. S. Mendes, M. L. Baesso, and C. Jacinto, “Finite-size effect on the surface deformation thermal mirror method,” J. Opt. Soc. Am. B 28, 1735–1739 (2011).
[CrossRef]

L. C. Malacarne, N. G. C. Astrath, G. V. B. Lukasievcz, E. K. Lenzi, M. L. Baesso, and S. E. Bialkowski, “Time-resolved thermal lens and thermal mirror spectroscopy with sample-fluid heat coupling: A complete model for material characterization,” Appl. Spectrosc. 65, 99–104 (2011).
[CrossRef]

N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C. Malacarne, A. C. Bento, and M. L. Baesso, “Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials,” Opt. Lett. 33, 1464–1466 (2008).
[CrossRef]

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

N. G. C. Astrath, L. C. Malacarne, P. R. B. Pedreira, A. C. Bento, M. L. Baesso, and J. Shen, “Time-resolved thermal mirror for nanoscale surface displacement detection in low absorbing solids,” Appl. Phys. Lett. 91, 191908 (2007).
[CrossRef]

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin-film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[CrossRef]

Barton, M. A.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Bass, M.

W. Koechner and M. Bass, Solid-State Lasers: A Graduate Text (Springer, 2003).

Belancon, M. P.

N. G. C. Astrath, L. C. Malacarne, V. S. Zanuto, M. P. Belancon, R. S. Mendes, M. L. Baesso, and C. Jacinto, “Finite-size effect on the surface deformation thermal mirror method,” J. Opt. Soc. Am. B 28, 1735–1739 (2011).
[CrossRef]

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

Bell, M. J. V.

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

Bento, A. C.

N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C. Malacarne, A. C. Bento, and M. L. Baesso, “Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials,” Opt. Lett. 33, 1464–1466 (2008).
[CrossRef]

N. G. C. Astrath, L. C. Malacarne, P. R. B. Pedreira, A. C. Bento, M. L. Baesso, and J. Shen, “Time-resolved thermal mirror for nanoscale surface displacement detection in low absorbing solids,” Appl. Phys. Lett. 91, 191908 (2007).
[CrossRef]

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

Bialkowski, S. E.

Billingsley, G.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Blair, D. G.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Brooks, A.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Carslaw, H. S.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, 1959), Vol. 1, p. 78.

Catunda, T.

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

Danzmann, K.

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

Degallaix, J.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Fan, X.

Fan, Y.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Gandra, F. G.

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

Glebov, L. B.

L. B. Glebov, “Intrinsic laser-induced breakdown of silicate glasses,” Proc. SPIE 4679, 321–331 (2002).
[CrossRef]

Gray, M. B.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Herloski, R.

Hosken, D. J.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Jacinto, C.

N. G. C. Astrath, L. C. Malacarne, V. S. Zanuto, M. P. Belancon, R. S. Mendes, M. L. Baesso, and C. Jacinto, “Finite-size effect on the surface deformation thermal mirror method,” J. Opt. Soc. Am. B 28, 1735–1739 (2011).
[CrossRef]

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

Jaeger, J. C.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, 1959), Vol. 1, p. 78.

Ju, L.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Klein, C. A.

C. A. Klein, “Analytical stress modeling of high-energy laser windows: Application to fusion-cast calcium fluoride windows,” J. Appl. Phys. 98, 043103 (2005).
[CrossRef]

C. A. Klein, “Describing beam-aberration effects induced by laser-light transmitting components: a short account of Raytheon’s contribution,” Proc. SPIE 4376, 24–34 (2001).
[CrossRef]

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990).
[CrossRef]

Koechner, W.

W. Koechner and M. Bass, Solid-State Lasers: A Graduate Text (Springer, 2003).

Lenzi, E. K.

Lima, S. M.

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

Lukasievcz, G. V. B.

Malacarne, L. C.

McClelland, D. E.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Medina, A. N.

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

Mendes, R. S.

N. G. C. Astrath, L. C. Malacarne, V. S. Zanuto, M. P. Belancon, R. S. Mendes, M. L. Baesso, and C. Jacinto, “Finite-size effect on the surface deformation thermal mirror method,” J. Opt. Soc. Am. B 28, 1735–1739 (2011).
[CrossRef]

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

Mow Lowry, C. M.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Mudge, D.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Munch, J.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Nowacki, W.

W. Nowacki, Thermoelasticity (Pergamon, 1982), Vol. 3, p. 11.

Pedreira, P. R. B.

N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C. Malacarne, A. C. Bento, and M. L. Baesso, “Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials,” Opt. Lett. 33, 1464–1466 (2008).
[CrossRef]

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

N. G. C. Astrath, L. C. Malacarne, P. R. B. Pedreira, A. C. Bento, M. L. Baesso, and J. Shen, “Time-resolved thermal mirror for nanoscale surface displacement detection in low absorbing solids,” Appl. Phys. Lett. 91, 191908 (2007).
[CrossRef]

Peng, Y.

Rohling, J. H.

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

Rüdiger, A.

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

Sato, F.

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

Schilling, R.

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

Shen, J.

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C. Malacarne, A. C. Bento, and M. L. Baesso, “Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials,” Opt. Lett. 33, 1464–1466 (2008).
[CrossRef]

N. G. C. Astrath, L. C. Malacarne, P. R. B. Pedreira, A. C. Bento, M. L. Baesso, and J. Shen, “Time-resolved thermal mirror for nanoscale surface displacement detection in low absorbing solids,” Appl. Phys. Lett. 91, 191908 (2007).
[CrossRef]

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin-film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[CrossRef]

Sheng, Z.

Slagmolen, B. J. J.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Snook, R. D.

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin-film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[CrossRef]

Sparks, M.

M. Sparks, “Optical distortion by heated windows in high-power laser systems,” J. Appl. Phys. 42, 5029–5046 (1971).
[CrossRef]

Veitch, P. J.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Winkler, W.

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

Zanuto, V. S.

Zhang, H.

Zhao, C.

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Zhou, J.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

N. G. C. Astrath, L. C. Malacarne, P. R. B. Pedreira, A. C. Bento, M. L. Baesso, and J. Shen, “Time-resolved thermal mirror for nanoscale surface displacement detection in low absorbing solids,” Appl. Phys. Lett. 91, 191908 (2007).
[CrossRef]

Appl. Spectrosc. (1)

J. Appl. Phys. (4)

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin-film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[CrossRef]

C. A. Klein, “Analytical stress modeling of high-energy laser windows: Application to fusion-cast calcium fluoride windows,” J. Appl. Phys. 98, 043103 (2005).
[CrossRef]

M. Sparks, “Optical distortion by heated windows in high-power laser systems,” J. Appl. Phys. 42, 5029–5046 (1971).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Eng. (1)

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

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

Phys. Rev. B (1)

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[CrossRef]

Proc. SPIE (2)

C. A. Klein, “Describing beam-aberration effects induced by laser-light transmitting components: a short account of Raytheon’s contribution,” Proc. SPIE 4376, 24–34 (2001).
[CrossRef]

L. B. Glebov, “Intrinsic laser-induced breakdown of silicate glasses,” Proc. SPIE 4679, 321–331 (2002).
[CrossRef]

Other (3)

W. Koechner and M. Bass, Solid-State Lasers: A Graduate Text (Springer, 2003).

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, 1959), Vol. 1, p. 78.

W. Nowacki, Thermoelasticity (Pergamon, 1982), Vol. 3, p. 11.

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Figures (4)

Fig. 1.
Fig. 1.

Schematic diagram of the optical element geometry.

Fig. 2.
Fig. 2.

Radial temperature (open circles) and surface displacement (continuous lines) profiles for the CaF2 windows glass.

Fig. 3.
Fig. 3.

Time dependence of the optical path S(r,t)/(l0Q0) for ω=50μm using the physical parameters of CaF2 glass windows [6]. Solid lines were calculated using Eq. (36) and Eqs. (37) and (38) for plane-stress (open squares) and plane-strain (open circles) approximations, respectively.

Fig. 4.
Fig. 4.

Time and radial evolution to the optical path S(r,t)/(l0Q0)(×108) for the CaF2 [6] glass windows for l0=2mm.

Equations (43)

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S(r,t)=pathn(r,z,t)dz.
S(r,t)=20b(r,t)ns(r,z,t)dz+2b(r,t)znf(r,z¯,t)dz¯,
Δl=2|uz(r+ur,l0/2,t)|2|uz(r,l0/2,t)|.
Δn=Δnth+Δnst.
S(r,t)=S0+Sthf(r,t)+Sths(r,t)+Sst(r,t)+Sexp(r,t),
Δnthi=(niT)thTi(r,z,t).
Sthf(r,t)=2(nfT)thl0zTf(r,z¯,t)dz¯,
Sths(r,t)=(nsT)th0l0Ts(r,z,t)dz.
Δnsts=12(n0s)3[qσrr+q(σϕϕ+σzz)],
Δnsts=12(n0s)3[qσϕϕ+q(σrr+σzz)]
Sst(r,t)=Sst+(r,t)+Sst(r,t)
Sst+(r,t)=(n0s)340l0[(q+q)(σrr+σϕϕ)+2qσzz]dz,
Sst(r,t)=(n0s)340l0[(qq)(σrrσϕϕ)]dz.
Sexp(r,t)=(n0sn0f)Δl(r,z,t)|z=0.
tT(r,z,t)kcρ2T(r,z,t)=Q0e2r2ω2,
2Ψ(r,z,t)=1+ν1ναTT(r,z,t),
22ψ(r,z,t)=0.
uz=Ψz+112ν[2(1ν)22z2]ψ
σzz=E1+ν(2r2+1rr)Ψ+E1ν2ν2[(2ν)22z2]ψr,
σrr=E1+ν(2z2+1rr)Ψ+E1ν2ν2[ν22r2]ψz,
σϕϕ=E1+ν(2r2+2z2)Ψ+E1ν2ν2[ν21rr]ψz,
T(r,t)=0T(α,t)J0(αr)αdα.
T(α,t)=Q0ω2e18ω2α2(1eDtα24Dα2)
Ψ(r,z,t)=1+ν1ναT0T(α,t)α1J0(αr)dα,
ψ(r,z,t)=αT0f(α,z)α2T(α,t)J0(αr)dα,
f(α,z)=ν1+2ν2l0α+sinh(l0α)e(l0+z)α2(ν1)[el0α(l0αzα2ν)+e(l0+2z)α(l0α+zα2ν)+e2zα(zα+2ν)+e2l0α(zα+2ν)].
Δl=4(1+ν)αT0h(l0,α)T(α,t)J0(rα)dα,
σr=EαT(ν1)0(1r4νh(l0,α)rl0α)J1(rα)T(α,t)dα,
σz=EαT(ν1)0(14h(l0,α)l0α)J0(rα)T(α,t)αdα,
σϕ=EαT(ν1)0(14νh(l0,α)l0α)J0(rα)T(α,t)αdαEαT(ν1)0(1r4νh(l0,α)rl0α)J1(rα)T(α,t)dα,
h(l0,α)=cosh(l0α)1sinh(l0α)+l0α,
σi=1l00l0σii(r,z,t)dz.
Sst+(r,t)=ϑ0(q+3q)J0(rα)T(α,t)αdαϑ0[qν+q(2+ν)]h(l0,α)l0/4J0(rα)T(α,t)dα,
Sst(r,t)=ϑ0(qq)[ανh(l0,α)l0/4]J2(rα)T(α,t)dα,
Sexp(r,t)=4(n0s1)(1+ν)αT0h(l0,α)T(α,t)J0(rα)dα.
S(r,t)=0l0{(nsT)th+(n0s)3EαT4(1ν)[(q+3q)4[qν+q(2+ν)]h(l0,α)l0α]+4(n0s1)(1+ν)αTh(l0,α)l0α}T(α,t)J0(rα)αdα.
S0(r,t)=l0χ00T(α,t)J0(rα)αdα,
S(r,t)=l0χ0T(α,t)J0(rα)αdα,
χ0=(nsT)th+(n0s)3EαT4(q+q)+(n0s1)(1+ν)αT
χ=(nsT)th+(n0s)3EαT4(1ν)(q+3q).
ϕ(r,t)=2πλ[S(r,t)S(0,t)],
Sg=|02π0er2/ω2eiϕ(r,t)rdrdϕ|2|02π0er2/ω2rdrdϕ|2.
I(t)=|0exp[(1+iV)giϕ(g,t)]dg|2|0exp[(1+iV)g]dg|2,

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