Abstract

We describe an adaptive optical system for use as a tunable focusing element. The system provides adaptive beam shaping via controlled thermal lensing in the optical elements. The system is agile, remotely controllable, touch free, and vacuum compatible; it offers a wide dynamic range, aberration-free focal length tuning, and can provide both positive and negative lensing effects. Focusing is obtained through dynamic heating of an optical element by an external pump beam. The system is especially suitable for use in interferometric gravitational wave interferometers employing high laser power, allowing for in situ control of the laser modal properties and compensation for thermal lensing of the primary laser. Using CO2 laser heating of fused-silica substrates, we demonstrate a focal length variable from infinity to 4.0  m, with a slope of 0.082 diopter∕W of absorbed heat. For on-axis operation, no higher-order modes are introduced by the adaptive optical element. Theoretical modeling of the induced optical path change and predicted thermal lens agrees well with measurement.

© 2007 Optical Society of America

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  1. J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejer, R. L. Byer, D. Clubley, S. Yoshida, and D. Reitze, "Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack-Hartmann wavefront sensor," Appl. Opt. 40, 366-368 (2001).
    [CrossRef]
  2. R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, "Adaptive thermal compensation of test masses in advanced LIGO," Class. Quantum Grav. 19, 1803-1812 (2002).
    [CrossRef]
  3. J. Degallaix, C. Zhao, L. Ju, and D. Blair, "Thermal lensing compensation for AIGO high optical power test facility," Class. Quantum Grav. 21, S903-S908 (2004).
    [CrossRef]
  4. V. Quetschke, J. Gleason, M. Rakhmanov, J. Lee, L. Zhang, K. Yoshiki Franzen, C. Leidel, G. Mueller, R. Amin, D. B. Tanner, and D. H. Reitze, "Adaptive control of laser modal properties," Opt. Lett. 31, 217-219 (2006).
    [CrossRef] [PubMed]
  5. P. Hello and J. Vinet, "Analytical models of thermal aberrations in massive mirrors heated by high power laser beams," J. Phys. (France) 51, 1267-1282 (1990).
    [CrossRef]
  6. P. Hello and J. Vinet, "Analytical models of transient thermoelastic deformations of mirrors heated by high power CW laser beams," J. Phys. (France) 51, 2243-2261 (1990).
    [CrossRef]
  7. J. D. Foster and L. M. Osterink, "Thermal effects in Nd:YAG laser," Appl. Opt. 41, 3656-3663 (1970).
  8. C. E. Greninger, "Thermally induced wavefront distortions in laser windows," Appl. Opt. 41, 549-552 (1986).
  9. 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]
  10. 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]
  11. R. G. Beausoleil, E. K. Gustafson, M. M. Fejer, E. D'Ambrosio, W. Kells, and J. Camp, "Model of thermal wave-front distortion in interferometric gravitational-wave detectors. I. Thermal focusing," J. Opt. Soc. Am. B 20, 1247-1268 (2003).
    [CrossRef]
  12. A. Weinstein, "Advanced LIGO optical configuration and prototyping effort," Class. Quantum Grav. 19, 1575-1584 (2002).
    [CrossRef]
  13. R. Lawrence, D. Ottaway, M. Zucker, and P. Fritschel, "Active correction of thermal lensing through external radiative thermal actuation," Opt. Lett. 29, 2635-2637 (2004).
    [CrossRef] [PubMed]
  14. E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
    [CrossRef]
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    [CrossRef]
  16. Beam Scan, Model XYFIR, Photon Inc., http://www.photon-inc.com.
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    [CrossRef] [PubMed]

2006 (1)

2004 (3)

J. Degallaix, C. Zhao, L. Ju, and D. Blair, "Thermal lensing compensation for AIGO high optical power test facility," Class. Quantum Grav. 21, S903-S908 (2004).
[CrossRef]

R. Lawrence, D. Ottaway, M. Zucker, and P. Fritschel, "Active correction of thermal lensing through external radiative thermal actuation," Opt. Lett. 29, 2635-2637 (2004).
[CrossRef] [PubMed]

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

2003 (1)

2002 (2)

A. Weinstein, "Advanced LIGO optical configuration and prototyping effort," Class. Quantum Grav. 19, 1575-1584 (2002).
[CrossRef]

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, "Adaptive thermal compensation of test masses in advanced LIGO," Class. Quantum Grav. 19, 1803-1812 (2002).
[CrossRef]

2001 (1)

1994 (1)

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]

1991 (1)

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 (2)

P. Hello and J. Vinet, "Analytical models of thermal aberrations in massive mirrors heated by high power laser beams," J. Phys. (France) 51, 1267-1282 (1990).
[CrossRef]

P. Hello and J. Vinet, "Analytical models of transient thermoelastic deformations of mirrors heated by high power CW laser beams," J. Phys. (France) 51, 2243-2261 (1990).
[CrossRef]

1986 (1)

C. E. Greninger, "Thermally induced wavefront distortions in laser windows," Appl. Opt. 41, 549-552 (1986).

1984 (1)

1970 (1)

J. D. Foster and L. M. Osterink, "Thermal effects in Nd:YAG laser," Appl. Opt. 41, 3656-3663 (1970).

1961 (1)

Amin, R.

V. Quetschke, J. Gleason, M. Rakhmanov, J. Lee, L. Zhang, K. Yoshiki Franzen, C. Leidel, G. Mueller, R. Amin, D. B. Tanner, and D. H. Reitze, "Adaptive control of laser modal properties," Opt. Lett. 31, 217-219 (2006).
[CrossRef] [PubMed]

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

Anderson, D. Z.

Andreev, N. F.

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

Beausoleil, R. G.

Blair, D.

J. Degallaix, C. Zhao, L. Ju, and D. Blair, "Thermal lensing compensation for AIGO high optical power test facility," Class. Quantum Grav. 21, S903-S908 (2004).
[CrossRef]

Byer, R. L.

Camp, J.

Clubley, D.

D'Ambrosio, E.

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]

Degallaix, J.

J. Degallaix, C. Zhao, L. Ju, and D. Blair, "Thermal lensing compensation for AIGO high optical power test facility," Class. Quantum Grav. 21, S903-S908 (2004).
[CrossRef]

Fejer, M. M.

Foster, J. D.

J. D. Foster and L. M. Osterink, "Thermal effects in Nd:YAG laser," Appl. Opt. 41, 3656-3663 (1970).

Franzen, K. Yoshiki

Fritschel, P.

R. Lawrence, D. Ottaway, M. Zucker, and P. Fritschel, "Active correction of thermal lensing through external radiative thermal actuation," Opt. Lett. 29, 2635-2637 (2004).
[CrossRef] [PubMed]

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, "Adaptive thermal compensation of test masses in advanced LIGO," Class. Quantum Grav. 19, 1803-1812 (2002).
[CrossRef]

Gleason, J.

Greninger, C. E.

C. E. Greninger, "Thermally induced wavefront distortions in laser windows," Appl. Opt. 41, 549-552 (1986).

Gustafson, E. K.

Hello, P.

P. Hello and J. Vinet, "Analytical models of thermal aberrations in massive mirrors heated by high power laser beams," J. Phys. (France) 51, 1267-1282 (1990).
[CrossRef]

P. Hello and J. Vinet, "Analytical models of transient thermoelastic deformations of mirrors heated by high power CW laser beams," J. Phys. (France) 51, 2243-2261 (1990).
[CrossRef]

Hennawi, J.

Ivanov, I. A.

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

Ju, L.

J. Degallaix, C. Zhao, L. Ju, and D. Blair, "Thermal lensing compensation for AIGO high optical power test facility," Class. Quantum Grav. 21, S903-S908 (2004).
[CrossRef]

Kells, W.

Khazanov, E.

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

Lawrence, R.

R. Lawrence, D. Ottaway, M. Zucker, and P. Fritschel, "Active correction of thermal lensing through external radiative thermal actuation," Opt. Lett. 29, 2635-2637 (2004).
[CrossRef] [PubMed]

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, "Adaptive thermal compensation of test masses in advanced LIGO," Class. Quantum Grav. 19, 1803-1812 (2002).
[CrossRef]

Lee, J.

Leidel, C.

Low, F. J.

Mal'shakov, A.

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

Mansell, J. D.

Marfuta, P.

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, "Adaptive thermal compensation of test masses in advanced LIGO," Class. Quantum Grav. 19, 1803-1812 (2002).
[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]

Mueller, G.

V. Quetschke, J. Gleason, M. Rakhmanov, J. Lee, L. Zhang, K. Yoshiki Franzen, C. Leidel, G. Mueller, R. Amin, D. B. Tanner, and D. H. Reitze, "Adaptive control of laser modal properties," Opt. Lett. 31, 217-219 (2006).
[CrossRef] [PubMed]

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[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]

Osterink, L. M.

J. D. Foster and L. M. Osterink, "Thermal effects in Nd:YAG laser," Appl. Opt. 41, 3656-3663 (1970).

Ottaway, D.

Palashov, O.

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

Poteomkin, A. K.

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

Quetschke, V.

Rakhmanov, M.

Reitze, D.

Reitze, D. H.

V. Quetschke, J. Gleason, M. Rakhmanov, J. Lee, L. Zhang, K. Yoshiki Franzen, C. Leidel, G. Mueller, R. Amin, D. B. Tanner, and D. H. Reitze, "Adaptive control of laser modal properties," Opt. Lett. 31, 217-219 (2006).
[CrossRef] [PubMed]

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

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]

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]

Sergeev, A.

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

Shaykin, A.

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

Shoemaker, D.

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, "Adaptive thermal compensation of test masses in advanced LIGO," Class. Quantum Grav. 19, 1803-1812 (2002).
[CrossRef]

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]

Tanner, D. B.

V. Quetschke, J. Gleason, M. Rakhmanov, J. Lee, L. Zhang, K. Yoshiki Franzen, C. Leidel, G. Mueller, R. Amin, D. B. Tanner, and D. H. Reitze, "Adaptive control of laser modal properties," Opt. Lett. 31, 217-219 (2006).
[CrossRef] [PubMed]

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

Vinet, J.

P. Hello and J. Vinet, "Analytical models of transient thermoelastic deformations of mirrors heated by high power CW laser beams," J. Phys. (France) 51, 2243-2261 (1990).
[CrossRef]

P. Hello and J. Vinet, "Analytical models of thermal aberrations in massive mirrors heated by high power laser beams," J. Phys. (France) 51, 1267-1282 (1990).
[CrossRef]

Weinstein, A.

A. Weinstein, "Advanced LIGO optical configuration and prototyping effort," Class. Quantum Grav. 19, 1575-1584 (2002).
[CrossRef]

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]

Yoshida, S.

Zelenogorsky, V.

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

Zhang, L.

Zhao, C.

J. Degallaix, C. Zhao, L. Ju, and D. Blair, "Thermal lensing compensation for AIGO high optical power test facility," Class. Quantum Grav. 21, S903-S908 (2004).
[CrossRef]

Zucker, M.

R. Lawrence, D. Ottaway, M. Zucker, and P. Fritschel, "Active correction of thermal lensing through external radiative thermal actuation," Opt. Lett. 29, 2635-2637 (2004).
[CrossRef] [PubMed]

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, "Adaptive thermal compensation of test masses in advanced LIGO," Class. Quantum Grav. 19, 1803-1812 (2002).
[CrossRef]

Appl. Opt. (4)

Class. Quantum Grav. (3)

A. Weinstein, "Advanced LIGO optical configuration and prototyping effort," Class. Quantum Grav. 19, 1575-1584 (2002).
[CrossRef]

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, "Adaptive thermal compensation of test masses in advanced LIGO," Class. Quantum Grav. 19, 1803-1812 (2002).
[CrossRef]

J. Degallaix, C. Zhao, L. Ju, and D. Blair, "Thermal lensing compensation for AIGO high optical power test facility," Class. Quantum Grav. 21, S903-S908 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. Khazanov, N. F. Andreev, A. Mal'shakov, O. Palashov, A. K. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. A. Ivanov, R. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, "Compensation of thermally induced modal distortions in Faraday isolators," IEEE J. Quantum Electron. 40, 1500-1510 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. (2)

P. Hello and J. Vinet, "Analytical models of thermal aberrations in massive mirrors heated by high power laser beams," J. Phys. (France) 51, 1267-1282 (1990).
[CrossRef]

P. Hello and J. Vinet, "Analytical models of transient thermoelastic deformations of mirrors heated by high power CW laser beams," J. Phys. (France) 51, 2243-2261 (1990).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

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]

Phys. Rev. A (1)

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]

Other (1)

Beam Scan, Model XYFIR, Photon Inc., http://www.photon-inc.com.

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Figures (13)

Fig. 1
Fig. 1

(Color online) Non-uniform-intensity beam incident on an optical element. The power absorbed in the coating or the surface creates thermal aberrations on the surface due to the thermal expansion coefficient α T and due to the d n / d T in the substrate.

Fig. 2
Fig. 2

(Color online) Dini series representation of a top-hat beam and a Gaussian beam using the first 40 terms of the expansion.

Fig. 3
Fig. 3

(Color online) Thermal aberration in the substrate mirror due to 0.5   ppm absorption in the coating for the case of Table 1 data plotted as a solid curve. The approximation using the 12th degree polynomial is plotted as circles, and the optimal solution is plotted as a dotted–dashed curves.

Fig. 4
Fig. 4

Geometry of the incident, reflected, and transmitted electric fields from an optical material with thermal lensing. In case of reflection, the thermal aberrations consist of surface deformation as shown in (a) while in transmission, thermal deformations represent the sum of surface and substrate aberrations as depicted in (b).

Fig. 5
Fig. 5

The amplitude 1D coupling coefficient [Eq. (17)] for the case of substrate thermal lensing in LIGO test mass. The maximum occurs at a thermal ROC of 6.4 km indicating the optimal ROC associated with the thermal aberrations.

Fig. 6
Fig. 6

(Color online) Thermal lens created through top-hat and Gaussian beam plotted using dashed curve and solid curve, respectively. The dotted line is an ideal thermal lens of 6.4 km focal length.

Fig. 7
Fig. 7

Experimental arrangement of the adaptive optical system. SMF, single-mode fiber; L, lens; M, mirror; PD, photodetector; QPD, quad photodetector; GL, graded index lens; BS, beam splitter; and RC, scanning Fabry–Perot analyzer cavity.

Fig. 8
Fig. 8

(Color online) Beam profile data and the corresponding Gaussian fit at [ 0 , 1 , 2 , 3 ] W absorbed power.

Fig. 9
Fig. 9

(Color online) Calculated and measured back focal distance as a function of absorbed power (left axis) and the corresponding focal length of the thermal lens (right axis) as a function of the absorbed power. The measured data are shown as symbols.

Fig. 10
Fig. 10

(Color online) Measured values of lens power as a function of absorbed power is shown as symbols, with error bars. A linear fit to the data gives a slope of 0.082   diopter / W .

Fig. 11
Fig. 11

(Color online) Prediction of higher-order losses in the experimental setup of Fig. 7 (left axis) and the corresponding focal length (right axis) as a function of absorbed power.

Fig. 12
Fig. 12

Deviation of the probe beam from its base position at room temperature as a function of absorbed power. The solid line is the theoretical prediction using thick lens approximation. The slope gives 0.16   mrad / W deviation at 18.5° incidence angle of the probe beam.

Fig. 13
Fig. 13

(Color online) Response time of the demonstrated system as a function of absorbed power. The slope is 2.715   s / W over an essential characteristic time delay of 52.916 s.

Tables (1)

Tables Icon

Table 1 Nominal Values for Advanced LIGO Cavity Mirrors (Test Masses)

Equations (25)

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

T s ( r , z ) = P α s a 2 k t k p k ζ k 2 × [ 1 2 τ A k cosh ( ζ k z a ) ] J 0 ( ζ k r a ) ,
T c ( r , z ) = P a c a k t k p k × [ A k cosh ( ζ k z a ) B k sinh ( ζ k z a ) ] × J 0 ( ζ k r a ) ,
A k = 1 2 [ ζ k sinh ( γ k ) + τ cosh ( γ k ) ] ,
B k = 1 2 [ ζ k cosh ( γ k ) + τ sinh ( γ k ) ] ,
ζ J 1 ( ζ ) τ J 0 ( ζ ) = 0.
p k = 2 ζ k 2 ( ζ k 2 + τ 2 ) J 0 2 ( ζ k ) 1 a 2 0 a r J 0 ( ζ k r a ) I ( r ) d r .
ϕ ( r ) 2 π λ 0 d n d T h / 2 h / 2 T ( r , z ) d z .
ϕ s ( r ) = α s h P 2 π a 2 λ 0 k T d n d t k P k ζ k 2 × [ 1 2 τ A k γ k sinh ( γ k ) ] × J 0 ( ζ k r a ) ,
ϕ c ( r ) = a c P 2 π a 2 λ 0 k T d n d t k P k ζ k × 2 A k sinh ( γ k ) J 0 ( ζ k r a ) .
u c ( r , ± h / 2 ) = ± a c P α T ( 1 + v ) a 2 k T k p k ζ k 2 × sinh ( γ k ) γ k × [ sinh ( γ k ) γ k + sinh ( γ k ) cosh ( γ k ) τ A k ] × [ J 0 ( ζ k r a ) 1 ] ,    
u s ( r , ± h / 2 ) = ± α s h P α T ( 1 + v ) a 2 k T k p k ζ k 2 × [ A k cosh ( γ k ) B k sinh ( γ k ) ] ×[ J 0 ( ζ k r a ) 1 ] 3 2 α s h P × α T ( 1 v ) a 2 k T k r 2 p k a 2 ζ k 2 B k × [ sinh ( γ k ) γ k cosh ( γ k ) ] J 1 ( ζ k ) .
F = σ T 4 T e x t 4 ,
F = 4 σ T e x t 3 Δ T .
s ( x ) = m = 0 M A m x m .
E 1 ( x , z ) = ( 2 π ) 1 / 4 1 w ( z ) exp x 2 [ 1 w 2 ( z ) + i π λ R ( z ) ] .
E 2 ( x , z ) = ( 2 π ) 1 / 4 1 w ( z ) exp x 2 [ 1 w 2 ( z ) + i π λ R ( z ) i 2 π λ R 1 + i π λ 4 s ( x ) x 2 ] .
E 3 ( x , z ) = ( 2 π ) 1 / 4 1 w ( z ) exp x 2 [ 1 w 2 ( z ) + i π λ R ( z ) i 2 π λ R 1 + i π λ 4 A o p t ] ,
I ( A o p t ) = E 2 ( x , z ) × E 3 * ( x , z ) d x = ( 2 π ) 1 / 2 1 w ( z ) exp x 2 [ 2 w 2 ( z ) + i π λ 4 s ( x ) x 2 i π λ 4 A o p t ] d x .
I ( A ) = ( 2 π ) 1 / 2 1 w ( z ) exp x 2 [ 2 w 2 ( z ) ] × exp i [ 4 π λ 1 s ( x ) π λ 4 A o p t x 2 ] d x .
I ( A ) = 2 ( 2 π ) 1 / 2 1 w ( z ) 0 exp x 2 [ 2 w 2 ( z ) ] [ cos { 4 π λ [ s ( x ) A o p t x 2 ] } + i sin { 4 π λ [ s ( x ) A o p t x 2 ] } ] d x .
FOM M = [ d n d T + α T ( 1 + ν ) × ( n 1 ) α T n 3 4 ( 1 + ν ) ( 1 ν ) × ( p 11 + p 12 ) ] .
[ A B C D ] = [ 1 ( h / n ) ( P 1 ) h / n ( P 1 + P 1 P 1 P 2 h / n ) 1 ( h / n ) ( P 1 ) ] ,
[ x 2 tan θ 2 ] = [ A D B C ] [ x 1 tan θ 1 ] ,
τ P B D = C ρ r e 2 κ β P .
τ P B D = C ρ r e 2 κ C ρ r c 2 κ 2 P = C ρ r e 2 κ + β r P .

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