Abstract

With any form of phase distortions there is the need to qualitatively characterize beam quality. Three different qualitative criteria are most commonly used for this purpose, each of them describing the beam with one ratio: the overlapping integral, the Strehl ratio, and the M2 parameter. We have analyzed the interrelation of the above- mentioned criteria in the three most common types of beam quality degradation: thermal lens, electronic self-focusing, and spherical aberration. Approximate analytical expressions for all three criteria and three types of beam distortion are derived for Gaussian and super-Gaussian intensity profiles. The efficiency of characterizing those beams by various criteria is discussed.

© 2007 Optical Society of America

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    [CrossRef]

2006 (1)

M. R. Duparre, B. Luedge, and S. Schroeter, "ETALONs for pure and composite transversal modes," in Laser Beam Control and Applications, A. Gisen and D. Nickel, eds., Proc. SPIE 6101, 61011C (2006).
[CrossRef]

2005 (4)

A. K. Poteomkin and E. A. Khazanov, "Calculation of the M2 factor of the laser beam by the method of moments," Quantum Electron. 35, 1042-1044 (2005).
[CrossRef]

B. J. Neubert and B. Eppich, "Influences on the beam propagation ratio M2," Opt. Commun. 250, 241-251 (2005).
[CrossRef]

A. K. Poteomkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaykin, "Compact 100 J/100 GW Nd:Phosphate laser for optical parametric chirped pulse amplifier pumping," Quantum Electron. 35, 302-310 (2005).
[CrossRef]

V. N. Mahajan, "Strehl ratio of Gaussian beam," J. Opt. Soc. Am. A 22, 1824-1833 (2005).
[CrossRef]

2004 (2)

S. Stallinga, "Strehl ratio for focusing into biaxially birefringent media," J. Opt. Soc. Am. A 21, 2406-2413 (2004).
[CrossRef]

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

2002 (2)

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, "Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers," Class. Quantum Grav. 19, 1793-1801 (2002).
[CrossRef]

S. Amano and T. Mochizuki, "Propagation characteristics of a diffracted M2 beam," Appl. Opt. 41, 6325-6331 (2002).
[CrossRef] [PubMed]

2000 (2)

E. A. Khazanov, "Characteristic features of the operation of different designs of the Faraday isolator for high average laser-radiation power," Quantum Electron. 30, 147-151 (2000).
[CrossRef]

A. van den Bos, "Aberration and the Strehl ratio," J. Opt. Soc. Am. A 17, 356-358 (2000).
[CrossRef]

1999 (1)

1998 (1)

1997 (3)

1993 (1)

1992 (3)

C. Pare and P.-A. Belanger, "Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments," Opt. Quantum Electron. 24, S1051-S1070 (1992).
[CrossRef]

N. Reng and B. Eppich, "Definition and measurements of high-power laser beam parameters," Opt. Quantum Electron. 24, S973-S992 (1992).
[CrossRef]

A. Parent, M. Morin, and P. Lavigne, "Propogation of super-Gaussian field distribution," Opt. Quantum Electron. 24, S1071-S1079 (1992).
[CrossRef]

1990 (1)

A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2-14 (1990).

1985 (1)

1983 (1)

1982 (1)

1981 (1)

W. W. Simmons, J. T. Hunt, and W. E. Warren, "Light propagation through large laser systems," IEEE J. Quantum Electron. QE-17, 1727-1744 (1981).
[CrossRef]

1974 (2)

1971 (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, "Averaged description of wave beams in linear and nonlinear media," Izv. Vyssh. Uchebn. Zaved. , Radiofiz 14, 1353-1363 (1971).

1902 (1)

K. Strehl, "Uber Luftschlieren und Zonenfehler," Z. Instrumentenkd. 22, 213-217 (1902).

Amano, S.

Amin, R. S.

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, "Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers," Class. Quantum Grav. 19, 1793-1801 (2002).
[CrossRef]

Andreev, N. F.

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

Belanger, P.-A.

C. Pare and P.-A. Belanger, "Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments," Opt. Quantum Electron. 24, S1051-S1070 (1992).
[CrossRef]

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Bowersox, R. D.

Chen, Y. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[CrossRef]

Duparre, M. R.

M. R. Duparre, B. Luedge, and S. Schroeter, "ETALONs for pure and composite transversal modes," in Laser Beam Control and Applications, A. Gisen and D. Nickel, eds., Proc. SPIE 6101, 61011C (2006).
[CrossRef]

Eppich, B.

B. J. Neubert and B. Eppich, "Influences on the beam propagation ratio M2," Opt. Commun. 250, 241-251 (2005).
[CrossRef]

N. Reng and B. Eppich, "Definition and measurements of high-power laser beam parameters," Opt. Quantum Electron. 24, S973-S992 (1992).
[CrossRef]

Gardner, P. J.

Guagliardo, D.

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, "Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers," Class. Quantum Grav. 19, 1793-1801 (2002).
[CrossRef]

Herloski, R.

Herman, R. M.

Huang, T. M.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[CrossRef]

Hunt, J. T.

W. W. Simmons, J. T. Hunt, and W. E. Warren, "Light propagation through large laser systems," IEEE J. Quantum Electron. QE-17, 1727-1744 (1981).
[CrossRef]

Ivanov, I.

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

Kao, C. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[CrossRef]

Katin, E. V.

A. K. Poteomkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaykin, "Compact 100 J/100 GW Nd:Phosphate laser for optical parametric chirped pulse amplifier pumping," Quantum Electron. 35, 302-310 (2005).
[CrossRef]

Khazanov, E. A.

A. K. Poteomkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaykin, "Compact 100 J/100 GW Nd:Phosphate laser for optical parametric chirped pulse amplifier pumping," Quantum Electron. 35, 302-310 (2005).
[CrossRef]

A. K. Poteomkin and E. A. Khazanov, "Calculation of the M2 factor of the laser beam by the method of moments," Quantum Electron. 35, 1042-1044 (2005).
[CrossRef]

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

E. A. Khazanov, "Characteristic features of the operation of different designs of the Faraday isolator for high average laser-radiation power," Quantum Electron. 30, 147-151 (2000).
[CrossRef]

Kirsanov, A. V.

A. K. Poteomkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaykin, "Compact 100 J/100 GW Nd:Phosphate laser for optical parametric chirped pulse amplifier pumping," Quantum Electron. 35, 302-310 (2005).
[CrossRef]

Lavigne, P.

A. Parent, M. Morin, and P. Lavigne, "Propogation of super-Gaussian field distribution," Opt. Quantum Electron. 24, S1071-S1079 (1992).
[CrossRef]

Lowenthal, D. D.

Luchinin, G. A.

A. K. Poteomkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaykin, "Compact 100 J/100 GW Nd:Phosphate laser for optical parametric chirped pulse amplifier pumping," Quantum Electron. 35, 302-310 (2005).
[CrossRef]

Luedge, B.

M. R. Duparre, B. Luedge, and S. Schroeter, "ETALONs for pure and composite transversal modes," in Laser Beam Control and Applications, A. Gisen and D. Nickel, eds., Proc. SPIE 6101, 61011C (2006).
[CrossRef]

Luke, T. E.

Lundock, R.

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, "Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers," Class. Quantum Grav. 19, 1793-1801 (2002).
[CrossRef]

Mahajan, V. N.

Mal'shakov, A. N.

A. K. Poteomkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaykin, "Compact 100 J/100 GW Nd:Phosphate laser for optical parametric chirped pulse amplifier pumping," Quantum Electron. 35, 302-310 (2005).
[CrossRef]

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

Martyanov, M. A.

A. K. Poteomkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaykin, "Compact 100 J/100 GW Nd:Phosphate laser for optical parametric chirped pulse amplifier pumping," Quantum Electron. 35, 302-310 (2005).
[CrossRef]

Matveev, A. Z.

A. K. Poteomkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaykin, "Compact 100 J/100 GW Nd:Phosphate laser for optical parametric chirped pulse amplifier pumping," Quantum Electron. 35, 302-310 (2005).
[CrossRef]

McFeron, D.

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, "Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers," Class. Quantum Grav. 19, 1793-1801 (2002).
[CrossRef]

Mochizuki, T.

Morin, M.

A. Parent, M. Morin, and P. Lavigne, "Propogation of super-Gaussian field distribution," Opt. Quantum Electron. 24, S1071-S1079 (1992).
[CrossRef]

Mueller, G.

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, "Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers," Class. Quantum Grav. 19, 1793-1801 (2002).
[CrossRef]

Neubert, B. J.

B. J. Neubert and B. Eppich, "Influences on the beam propagation ratio M2," Opt. Commun. 250, 241-251 (2005).
[CrossRef]

Palashov, O. V.

A. K. Poteomkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaykin, "Compact 100 J/100 GW Nd:Phosphate laser for optical parametric chirped pulse amplifier pumping," Quantum Electron. 35, 302-310 (2005).
[CrossRef]

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

Pare, C.

C. Pare and P.-A. Belanger, "Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments," Opt. Quantum Electron. 24, S1051-S1070 (1992).
[CrossRef]

Parent, A.

A. Parent, M. Morin, and P. Lavigne, "Propogation of super-Gaussian field distribution," Opt. Quantum Electron. 24, S1071-S1079 (1992).
[CrossRef]

Petrishchev, V. A.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, "Averaged description of wave beams in linear and nonlinear media," Izv. Vyssh. Uchebn. Zaved. , Radiofiz 14, 1353-1363 (1971).

Poteomkin, A. K.

A. K. Poteomkin and E. A. Khazanov, "Calculation of the M2 factor of the laser beam by the method of moments," Quantum Electron. 35, 1042-1044 (2005).
[CrossRef]

A. K. Poteomkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaykin, "Compact 100 J/100 GW Nd:Phosphate laser for optical parametric chirped pulse amplifier pumping," Quantum Electron. 35, 302-310 (2005).
[CrossRef]

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

Reitze, D. H.

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, "Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers," Class. Quantum Grav. 19, 1793-1801 (2002).
[CrossRef]

Reng, N.

N. Reng and B. Eppich, "Definition and measurements of high-power laser beam parameters," Opt. Quantum Electron. 24, S973-S992 (1992).
[CrossRef]

Roggemann, M. C.

Santarsiero, M.

Schroeter, S.

M. R. Duparre, B. Luedge, and S. Schroeter, "ETALONs for pure and composite transversal modes," in Laser Beam Control and Applications, A. Gisen and D. Nickel, eds., Proc. SPIE 6101, 61011C (2006).
[CrossRef]

Sergeev, A. M.

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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. A.

A. K. Poteomkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaykin, "Compact 100 J/100 GW Nd:Phosphate laser for optical parametric chirped pulse amplifier pumping," Quantum Electron. 35, 302-310 (2005).
[CrossRef]

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

Siegman, A. E.

A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2-14 (1990).

Simmons, W. W.

W. W. Simmons, J. T. Hunt, and W. E. Warren, "Light propagation through large laser systems," IEEE J. Quantum Electron. QE-17, 1727-1744 (1981).
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Stallinga, S.

Strehl, K.

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S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, "Averaged description of wave beams in linear and nonlinear media," Izv. Vyssh. Uchebn. Zaved. , Radiofiz 14, 1353-1363 (1971).

Tanner, D. B.

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, "Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers," Class. Quantum Grav. 19, 1793-1801 (2002).
[CrossRef]

van den Bos, A.

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S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, "Averaged description of wave beams in linear and nonlinear media," Izv. Vyssh. Uchebn. Zaved. , Radiofiz 14, 1353-1363 (1971).

Wang, C. L.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[CrossRef]

Wang, S. C.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[CrossRef]

Warren, W. E.

W. W. Simmons, J. T. Hunt, and W. E. Warren, "Light propagation through large laser systems," IEEE J. Quantum Electron. QE-17, 1727-1744 (1981).
[CrossRef]

Welsh, B. M.

Wiggins, T. A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Zelenogorsky, V. V.

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

Appl. Opt. (5)

Class. Quantum Grav. (1)

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, "Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers," Class. Quantum Grav. 19, 1793-1801 (2002).
[CrossRef]

IEEE J. Quantum Electron. (3)

E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. 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]

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[CrossRef]

W. W. Simmons, J. T. Hunt, and W. E. Warren, "Light propagation through large laser systems," IEEE J. Quantum Electron. QE-17, 1727-1744 (1981).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, "Averaged description of wave beams in linear and nonlinear media," Izv. Vyssh. Uchebn. Zaved. , Radiofiz 14, 1353-1363 (1971).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

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[CrossRef]

Proc. SPIE (1)

M. R. Duparre, B. Luedge, and S. Schroeter, "ETALONs for pure and composite transversal modes," in Laser Beam Control and Applications, A. Gisen and D. Nickel, eds., Proc. SPIE 6101, 61011C (2006).
[CrossRef]

Quantum Electron. (3)

A. K. Poteomkin and E. A. Khazanov, "Calculation of the M2 factor of the laser beam by the method of moments," Quantum Electron. 35, 1042-1044 (2005).
[CrossRef]

E. A. Khazanov, "Characteristic features of the operation of different designs of the Faraday isolator for high average laser-radiation power," Quantum Electron. 30, 147-151 (2000).
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A. K. Poteomkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaykin, "Compact 100 J/100 GW Nd:Phosphate laser for optical parametric chirped pulse amplifier pumping," Quantum Electron. 35, 302-310 (2005).
[CrossRef]

Z. Instrumentenkd. (1)

K. Strehl, "Uber Luftschlieren und Zonenfehler," Z. Instrumentenkd. 22, 213-217 (1902).

Other (2)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2-14 (1990).

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Figures (14)

Fig. 1
Fig. 1

(Color online) Dependences R o p t ( p i ) (solid line) and R e x ( p i ) (pluses) after optimization of overlapping integral χ 5 at m = 4 .

Fig. 2
Fig. 2

(Color online) Dependences R o p t ( B ) (solid line) and R e x t ( B ) (pluses) after optimization of overlapping integral χ at m = 8 .

Fig. 3
Fig. 3

(Color online) Dependences R o p t ( V ) (solid line) and R e x ( V ) (pluses) after optimization of overlapping integral χ at m = 2 .

Fig. 4
Fig. 4

(Color online) Dependences of the overlapping integral χ on parameter p i at m = 4 : without compensation, plotted by formula (23) χ 0 ( p i ) (dashed line) and numerically χ ( p i ) (circles); with compensation, plotted numerically χ max ( p i , R e x ) , (crosses), by approximated formula (26) χ 0 max ( p i , R o p t ) (solid curve).

Fig. 5
Fig. 5

(Color online) Dependences of the overlapping integral χ on parameter B at m = 8 : without compensation, plotted by formula (23) χ 0 ( B ) (dashed curve) and numerically χ ( B ) (circles); with compensation, plotted numerically χ max ( B , R e x ) (crosses), by approximated formula (26) χ 0 max ( B , R o p t ) (solid curve).

Fig. 6
Fig. 6

(Color online) Dependences of the overlapping integral χ on parameter V at m = 2 : without compensation, plotted by formula (23) χ 0 ( V ) (dashed curve) and numerically χ ( V ) (circles); with compensation, plotted numerically χ max ( V , R e x ) (crosses), by approximated formula (26) χ 0 max ( V , R o p t ) (solid curve).

Fig. 7
Fig. 7

(Color online) Dependences of series expansion coefficients of R o p t on parameter m for thermal distortions ξ t ( m ) (solid curve), η t ( m ) (diamonds); electronic self-focusing ξ e ( m ) (dashed curve), η e ( m ) (triangles); spherical aberrations ξ s ( m ) (dashed–dotted curve), η s ( m ) (boxes); plotted according to formulas (27) and (36).

Fig. 8
Fig. 8

(Color online) Dependences of series expansion coefficients of quality criteria χ, S, and μ on parameter m for thermal distortions: overlapping integral χ without compensation a 1 t ( m ) (solid curve), formula (24), and with compensation a 2 t ( m ) (crosses), formula (28); Strehl ratio S without compensation b 1 t ( m ) (bold solid curve), formula (33), and with compensation b 2 t ( m ) (pluses), formula (37); relation μ c t ( m ) (diamonds), formula (40).

Fig. 9
Fig. 9

(Color online) Dependences of series expansion coefficients of quality criteria χ, S, and μ on parameter m for electronic self-focusing: overlapping integral χ without compensation a 1 e ( m ) (solid curve), formula (24), and with compensation a 2 e ( m ) (crosses), formula (28); Strehl ratio S without compensation b 1 e ( m ) (bold solid curve), formula (33), and with compensation b 2 e ( m ) (pluses), formula (37); relation μ c e ( m ) (diamonds), formula (40).

Fig. 10
Fig. 10

(Color online) Dependences of series expansion coefficients of quality criteria χ, S, and μ on parameter m for spherical aberrations: overlapping integral χ without compensation a 1 s ( m ) (solid curve), formula (24), and with compensation a 2 s ( m ) (crosses), formula (28); Strehl ratio S without compensation b 1 s ( m ) (bold solid curve), formula (33), and with compensation b 2 s ( m ) (pluses), formula (37); relation μ c s ( m ) (diamonds), formula (40).

Fig. 11
Fig. 11

(Color online) Numerical dependences of quality criteria χ, S, and μ on parameter p i at m = 4 : overlapping integral χ without compensation χ ( p i ) (thin curve) and with compensation χ max ( p i , R e x ) (crosses); Strehl ratio S without compensation S ( p i ) (thick curve) and with compensation S max ( p i , R e x ) (pluses); relation μ ( p i ) (diamonds).

Fig. 12
Fig. 12

(Color online) Numerical dependences of quality criteria χ, S, and μ on parameter B at m = 8 : overlapping integral χ without compensation χ ( B ) (thin curve) and with compensation χ max ( B , R e x ) (crosses); Strehl ratio S without compensation S(B) (thick curve) and with compensation S max ( B , R e x ) (pluses); relation μ ( B ) (diamonds).

Fig. 13
Fig. 13

(Color online) Numerical dependences of quality criteria χ, S, and μ on parameter V at m = 2: overlapping integral χ without compensation χ(V) (thin curve) and with compensation χ max (V, R ex ) (crosses); Strehl ratio S without compensation S(V) (thick curve) and with compensation S max (V, R ex ) (pluses); relation μ(V) (diamonds).

Fig. 14
Fig. 14

(Color online) Numerical dependences of quality criteria χ, S, and μ on parameter B at m = 0.5: overlapping integral χ without compensation χ(B) (thin curve) and with compensation χ max (B, R ex ) (crosses); Strehl ratio S without compensation S(B) (thick curve) and with compensation S max (B, R ex ) (pluses); relation μ(B) (diamonds).

Tables (1)

Tables Icon

Table 1 Values of Series Expansion Coefficients of Quality Criteria χ, S , and μ at Various m a

Equations (61)

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χ = | Ω E i n E o u t * d S | 2 Ω | E o u t | 2 d S Ω | E i n | 2 d S ,
E o u t ( r ) = N E i n ( r ) + E 0 ( r ) ,
Ω E i n E 0 * d S = 0 .
Ω | E o u t | 2 d S = | N | 2 Ω | E i n | 2 d S + Ω | E 0 | 2 d S ,
χ = | N | 2 .
S = I F o u t ( 0 ) I F i n ( 0 ) .
M 2 = 2 π σ 0 σ f ,
σ 0 2 = 0 | E ( r ) | 2 r 3 d r 0 | E ( r ) | 2 r d r , σ f 2 = 0 | u ( k ) | 2 k 3 d k 0 | u ( k ) | 2 k d k ,
σ 2 ( z ) = σ 0 2 + θ 2 z 2 ,
θ = M 2 k σ 0 .
μ = M i n 4 M o u t 4 .
E i n ( r ) = E exp ( r 2 m 2 a 2 m ) .
φ 1 ( r ) = p i m 2 Γ ( 1 / m ) 0 r 2 / a 2 [ 0 z exp ( t m ) d t z ] d z = p i u t ( t ) ,
p i = P L L α P λ κ ,
P = d n d T 1 L d L d T n 0 3 4 1 + v 1 v ( p 11 + p 12 )
t = r 2 / a 2 ,
φ e ( r ) = B exp ( r 2 m a 2 m ) = B u e ( t ) ,
B = k γ 0 L I 0 ( z ) d z
φ s ( r ) = V r 4 a 4 = V u s ( t ) ,
V = k a 4 2 f 2 4 ( G 1 f 1 + G 2 f 2 ) ,
E o u t ( r ) = E exp ( r 2 m 2 a 2 m ) exp [ i q u ( t ) ] ,
exp ( ± i k r 2 2 F ) = exp [ ± i R ( r a ) 2 ] ,
R = k a 2 2 F .
χ = m 2 Γ ( 1 / m ) 2 | 0 exp ( t m ) exp ( i q u ( t ) ) d t | 2 .
χ 0 = χ ( q 0 ) = 1 q 2 a 1 ( m ) ,
a 1 t ( m ) = m Γ ( 1 / m ) 0 exp ( t m ) u t 2 ( t ) d t m 2 Γ ( 1 / m ) 2 [ 0 exp ( t m ) u t ( t ) d t ] 2 ,
a 1 e ( m ) = 3 1 / m 4 1 / m ,
a 1 s ( m ) = Γ ( 5 / m ) Γ ( 1 / m ) Γ ( 3 / m ) 2 Γ ( 1 / m ) 2 .
R o p t = q ξ ( m ) ,
χ 0 max = χ max ( q 0 ) = 1 q 2 a 2 ( m ) ,
ξ t ( m ) = m Γ ( 2 / m ) Γ ( 1 / m ) Γ ( 3 / m ) Γ ( 2 / m ) 2 × 0 u t ( t ) [ 1 t Γ ( 1 / m ) Γ ( 2 / m ) ] exp ( t m ) d t ,
ξ e ( m ) = Γ ( 2 / m ) Γ ( 1 / m ) ( Γ ( 1 / m ) Γ ( 3 / m ) Γ ( 2 / m ) 2 ) [ 2 1 / m 4 1 / m ] ,
ξ s ( m ) = Γ ( 2 / m ) Γ ( 1 / m ) Γ ( 3 / m ) Γ ( 2 / m ) 2 × [ Γ ( 3 / m ) Γ ( 1 / m ) Γ ( 2 / m ) Γ ( 4 / m ) ] ,
a 2 t ( m ) = a 1 t ( m ) m 2 Γ ( 2 / m ) 2 Γ ( 1 / m ) 2 ( Γ ( 1 / m ) Γ ( 3 / m ) Γ ( 2 / m ) 2 ) × { 0 u t ( t ) [ 1 t Γ ( 1 / m ) Γ ( 2 / m ) ] exp ( t m ) d t } 2 ,
a 2 e ( m ) = a 1 e ( m ) Γ ( 2 / m ) 2 Γ ( 1 / m ) Γ ( 3 / m ) Γ ( 2 / m ) 2 × ( 2 1 / m 4 1 / m ) 2 ,
a 2 s ( m ) = a 1 s ( m ) Γ ( 2 / m ) 2 [ Γ ( 1 / m ) Γ ( 3 / m ) Γ ( 2 / m ) 2 ] × [ Γ ( 3 / m ) Γ ( 1 / m ) Γ ( 2 / m ) Γ ( 4 / m ) ] 2 .
R o p t ( m = 1 ) = p i 4 .
S = | Ω E o u t d S | 2 | Ω E i n d S | 2 .
S = m 2 Γ ( 1 / m ) 2 | 0 exp ( t m ) exp ( i q u ( t 2 m ) d t ) | 2 .
S 0 = S ( q 0 ) = 1 q 2 b 1 ( m ) ,
b 1 t ( m ) = m Γ ( 1 / m ) 0 exp ( t m ) u t 2 ( t 2 m ) d t m 2 Γ ( 1 / m ) 2 × [ 0 exp ( t m ) u t ( t 2 m ) d t ] 2 ,
b 1 e ( m ) = 5 1 / m 9 1 / m ,
b 1 s ( m ) = 16 1 / m a 1 s ( m ) = 16 1 / m [ Γ ( 5 / m ) Γ ( 1 / m ) Γ ( 3 / m ) 2 Γ ( 1 / m ) 2 ] .
R o p t = q η ( m ) ,
S 0 max = S max ( q 0 ) = 1 q 2 b 2 ( m ) ,
η t ( m ) = 1 2 m m Γ ( 2 / m ) Γ ( 1 / m ) Γ ( 3 / m ) Γ ( 2 / m ) 2 × 0 u t ( t 2 m ) [ 1 t Γ ( 1 / m ) Γ ( 2 / m ) ] exp ( t m ) d t ,
η e ( m ) = 1 2 m Γ ( 2 / m ) Γ ( 1 / m ) Γ ( 1 / m ) Γ ( 3 / m ) Γ ( 2 / m ) 2 ( 3 1 / m 9 1 / m ) ,
η s ( m ) = 2 m ξ s ( m ) = 2 m Γ ( 2 / m ) Γ ( 1 / m ) Γ ( 3 / m ) Γ ( 2 / m ) 2 × [ Γ ( 3 / m ) Γ ( 1 / m ) Γ ( 2 / m ) Γ ( 4 / m ) ] ,
b 2 t ( m ) = b 1 t ( m ) m 2 Γ ( 2 / m ) 2 Γ ( 1 / m ) 2 [ Γ ( 1 / m ) Γ ( 3 / m ) Γ ( 2 / m ) 2 ] × { 0 u t ( t 2 m ) [ 1 t Γ ( 1 / m ) Γ ( 2 / m ) ] exp ( t m ) d t } 2 ,
b 2 e ( m ) = b 1 e Γ ( 2 / m ) 2 Γ ( 1 / m ) Γ ( 3 / m ) Γ ( 2 / m ) 2 ( 2 1 / m 4 1 / m ) 2 ,
b 2 s ( m ) = 16 1 / m a 2 s ( m ) = b 1 s ( m ) 16 1 / m Γ ( 2 / m ) 2 [ Γ ( 1 / m ) Γ ( 3 / m ) Γ ( 2 / m ) 2 ] × [ Γ ( 3 / m ) Γ ( 1 / m ) Γ ( 2 / m ) Γ ( 4 / m ) ] 2 .
R o p t = p i 6 .
M 2 = k A σ 0 2 b 2 / 4 ,
A = 1 k 2 0 ( E ) 2 r d r 0 E 2 r d r + 0 ( φ / k ) 2 E 2 r d r 0 E 2 r d r ,
b = 2 k 0 ( r φ ) E 2 r d r 0 E 2 r d r .
M 2 = k σ 0 A ( φ c o r r ) .
μ = 1 1 + q 2 c ( m ) ,
c t ( m ) = m Γ ( 1 / m ) 2 0 [ 0 t exp ( y m ) d y ] 2 exp ( t m ) d t t 1 4 Γ ( 1 / m ) 2 m 2 Γ ( 2 / m ) ,
c e ( m ) = 4 9 4 1 / m Γ ( 1 / m ) 2 m 2 Γ ( 2 / m ) ,
c s ( m ) = 16 m 2 Γ ( 2 / m ) ( Γ ( 2 / m ) Γ ( 4 / m ) Γ ( 3 / m ) 2 ) .
μ μ 0 = μ ( q 0 ) = 1 q 2 c ( m ) .

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