Abstract

Compensation of thermally induced depolarization in laser active elements at small birefringence without additional phase elements was proposed and observed experimentally. Requirements to the crystals were formulated. An order of magnitude reduction of depolarization degree was obtained experimentally. A further modification of the scheme was developed.

© 2013 Optical Society of America

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References

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  1. W. Koechner, Solid-State Laser Engineering (Springer-Verlag, 1999).
  2. J. F. Nye, Physical Properties of Crystals (Oxford University Press, 1964).
  3. J. D. Foster and L. M. Osterink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys.41(9), 3656–3663 (1970).
    [CrossRef]
  4. G. A. Massey, “Criterion for selection of cw laser host materials to increase available power in the fundamental mode,” Appl. Phys. Lett.17(5), 213–215 (1970).
    [CrossRef]
  5. W. Koechner, “Absorbed pump power, thermal profile and stresses in a cw pumped Nd:YAG crystal,” Appl. Opt.9(6), 1429–1434 (1970).
    [CrossRef] [PubMed]
  6. W. Koechner and D. K. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron.6(9), 557–566 (1970).
    [CrossRef]
  7. M. A. Karr, “Nd:YAlG laser cavity loss due to an internal Brewster polarizer,” Appl. Opt.10(4), 893–895 (1971).
    [CrossRef] [PubMed]
  8. L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “On the problem of depolarization of linearly polarized light by a YAG:Nd3+ laser rod under conditions of thermally induced birefringence conditions,” Soviet J. Quantum Electron.10(3), 350–351 (1980).
    [CrossRef]
  9. L. N. Soms and A. A. Tarasov, “Thermal deformation in color-center laser active elements,” Soviet J. Quantum Electron.9(12), 1506–1508 (1979).
    [CrossRef]
  10. I. Shoji and T. Taira, “Intrinsic reduction of the depolarization loss in solid-state lasers by use of a (110)-cut Y3Al5O12 crystal,” Appl. Phys. Lett.80(17), 3048–3050 (2002).
    [CrossRef]
  11. I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and I. A. Ivanov, “Influence of the orientation of a crystal on thermal polarization effects in high-power solid-state lasers,” JETP Lett.81(3), 90–94 (2005).
    [CrossRef]
  12. I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Reduction of thermally induced depolarization of laser radiation in [110] oriented cubic crystals,” Opt. Express17(7), 5496–5501 (2009).
    [CrossRef] [PubMed]
  13. E. Khazanov, N. Andreev, O. Palashov, A. Poteomkin, A. Sergeev, O. Mehl, and D. H. Reitze, “Effect of terbium gallium garnet crystal orientation on the isolation ratio of a Faraday isolator at high average power,” Appl. Opt.41(3), 483–492 (2002).
    [CrossRef] [PubMed]
  14. E. A. Khazanov, “Thermally induced birefringence in Nd:YAG ceramics,” Opt. Lett.27(9), 716–718 (2002).
    [CrossRef] [PubMed]
  15. I. Snetkov, A. Vyatkin, O. Palashov, and E. Khazanov, “Drastic reduction of thermally induced depolarization in CaF₂ crystals with [111] orientation,” Opt. Express20(12), 13357–13367 (2012).
    [CrossRef] [PubMed]
  16. W. Koechner and D. K. Rice, “Birefringence of YAG: Nd laser rods as a function of growth direction,” J. Opt. Soc. Am.61(6), 758–766 (1971).
    [CrossRef]
  17. A. G. Vyatkin and E. A. Khazanov, “Thermally induced depolarization in sesquioxide class m3 single crystals,” J. Opt. Soc. Am. B28(4), 805–811 (2011).
    [CrossRef]
  18. W. C. Scott and M. de Wit, “Birefringence compensation and TEM00 mode enhancement in a Nd:YAG laser,” Appl. Phys. Lett.18(1), 3–4 (1971).
    [CrossRef]
  19. G. Giuliani and P. Ristori, “Polarization flip cavities: a new approach to laser resonators,” Opt. Commun.35(1), 109–112 (1980).
    [CrossRef]
  20. N. F. Andreev, N. G. Bondarenko, I. V. Eremina, S. V. Kuznetsov, O. V. Palashov, G. A. Pasmanik, and E. A. Khazanov, “Single-mode YAG:Nd laser with a stimulated Brillouin scattering mirror and conversion of radiation to the second and fourth harmonics,” Soviet J. Quantum Electron.21(10), 1045–1051 (1991).
    [CrossRef]
  21. E. A. Khazanov, “A new Faraday rotator for high average power lasers,” Quantum Electron.31(4), 351–356 (2001).
    [CrossRef]
  22. W. A. Clarkson, N. S. Felgate, and D. C. Hanna, “Simple method for reducing the depolarization loss resulting from thermally induced birefringence in solid-state lasers,” Opt. Lett.24(12), 820–822 (1999).
    [CrossRef] [PubMed]
  23. E. Khazanov, A. Poteomkin, and E. Katin, “Compensating for birefringence in active elements of solid-state lasers: novel method,” J. Opt. Soc. Am. B19(4), 667–671 (2002).
    [CrossRef]
  24. E. Khazanov, “Use of parallel axicon for compensation of birefringence in active elements of solid-state lasers,” Proc. SPIE 4632, 155–163 (2002) (Laser and Beam Control Technologies, ed. S. Basu and J. F. Riker).
  25. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am.31(7), 488–503 (1941).
    [CrossRef]
  26. S. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill, 1951).
  27. F. W. Quelle., “Thermal distortion of diffraction-limited optical elements,” Appl. Opt.5(4), 633–637 (1966).
    [CrossRef] [PubMed]
  28. M. J. Weber, Handbook of optical materials (CRC Press, 2003).
  29. K. Veerabhadra Rao and T. S. Narasimhamurty, “Photoelastic constants of CaF2 and BaF2,” J. Phys. Chem. Solids31, 876–878 (1969).
  30. R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” Appl. Phys. (Berl.)38, 5149–5153 (1967).
  31. R. E. Joiner, J. Marburger, and W. H. Steier, “Elimination of stress-induced birefringence effects in single-crystal high-power laser windows,” Appl. Phys. Lett.30(9), 485–486 (1977).
    [CrossRef]
  32. W. Martienssen and H. Warlimont, Handbook of Condensed Matter and Materials Data (Springer, 2006).
  33. M. S. Kochetkova, M. A. Martyanov, A. K. Poteomkin, and E. A. Khazanov, “Propagation of laser radiation in a medium with thermally induced birefringence and cubic nonlinearity,” Opt. Express18(12), 12839–12851 (2010).
    [CrossRef] [PubMed]
  34. M. S. Kuzmina, M. A. Martyanov, A. K. Poteomkin, E. A. Khazanov, and A. A. Shaykin, “Theoretical and experimental study of laser radiation propagating in a medium with thermally induced birefringence and cubic nonlinearity,” Opt. Express19(22), 21977–21988 (2011).
    [CrossRef] [PubMed]

2012 (1)

2011 (2)

2010 (1)

2009 (1)

2005 (1)

I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and I. A. Ivanov, “Influence of the orientation of a crystal on thermal polarization effects in high-power solid-state lasers,” JETP Lett.81(3), 90–94 (2005).
[CrossRef]

2002 (4)

2001 (1)

E. A. Khazanov, “A new Faraday rotator for high average power lasers,” Quantum Electron.31(4), 351–356 (2001).
[CrossRef]

1999 (1)

1991 (1)

N. F. Andreev, N. G. Bondarenko, I. V. Eremina, S. V. Kuznetsov, O. V. Palashov, G. A. Pasmanik, and E. A. Khazanov, “Single-mode YAG:Nd laser with a stimulated Brillouin scattering mirror and conversion of radiation to the second and fourth harmonics,” Soviet J. Quantum Electron.21(10), 1045–1051 (1991).
[CrossRef]

1980 (2)

G. Giuliani and P. Ristori, “Polarization flip cavities: a new approach to laser resonators,” Opt. Commun.35(1), 109–112 (1980).
[CrossRef]

L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “On the problem of depolarization of linearly polarized light by a YAG:Nd3+ laser rod under conditions of thermally induced birefringence conditions,” Soviet J. Quantum Electron.10(3), 350–351 (1980).
[CrossRef]

1979 (1)

L. N. Soms and A. A. Tarasov, “Thermal deformation in color-center laser active elements,” Soviet J. Quantum Electron.9(12), 1506–1508 (1979).
[CrossRef]

1977 (1)

R. E. Joiner, J. Marburger, and W. H. Steier, “Elimination of stress-induced birefringence effects in single-crystal high-power laser windows,” Appl. Phys. Lett.30(9), 485–486 (1977).
[CrossRef]

1971 (3)

1970 (4)

J. D. Foster and L. M. Osterink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys.41(9), 3656–3663 (1970).
[CrossRef]

G. A. Massey, “Criterion for selection of cw laser host materials to increase available power in the fundamental mode,” Appl. Phys. Lett.17(5), 213–215 (1970).
[CrossRef]

W. Koechner and D. K. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron.6(9), 557–566 (1970).
[CrossRef]

W. Koechner, “Absorbed pump power, thermal profile and stresses in a cw pumped Nd:YAG crystal,” Appl. Opt.9(6), 1429–1434 (1970).
[CrossRef] [PubMed]

1969 (1)

K. Veerabhadra Rao and T. S. Narasimhamurty, “Photoelastic constants of CaF2 and BaF2,” J. Phys. Chem. Solids31, 876–878 (1969).

1967 (1)

R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” Appl. Phys. (Berl.)38, 5149–5153 (1967).

1966 (1)

1941 (1)

Andreev, N.

Andreev, N. F.

N. F. Andreev, N. G. Bondarenko, I. V. Eremina, S. V. Kuznetsov, O. V. Palashov, G. A. Pasmanik, and E. A. Khazanov, “Single-mode YAG:Nd laser with a stimulated Brillouin scattering mirror and conversion of radiation to the second and fourth harmonics,” Soviet J. Quantum Electron.21(10), 1045–1051 (1991).
[CrossRef]

Bondarenko, N. G.

N. F. Andreev, N. G. Bondarenko, I. V. Eremina, S. V. Kuznetsov, O. V. Palashov, G. A. Pasmanik, and E. A. Khazanov, “Single-mode YAG:Nd laser with a stimulated Brillouin scattering mirror and conversion of radiation to the second and fourth harmonics,” Soviet J. Quantum Electron.21(10), 1045–1051 (1991).
[CrossRef]

Clarkson, W. A.

de Wit, M.

W. C. Scott and M. de Wit, “Birefringence compensation and TEM00 mode enhancement in a Nd:YAG laser,” Appl. Phys. Lett.18(1), 3–4 (1971).
[CrossRef]

Dixon, R. W.

R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” Appl. Phys. (Berl.)38, 5149–5153 (1967).

Eremina, I. V.

N. F. Andreev, N. G. Bondarenko, I. V. Eremina, S. V. Kuznetsov, O. V. Palashov, G. A. Pasmanik, and E. A. Khazanov, “Single-mode YAG:Nd laser with a stimulated Brillouin scattering mirror and conversion of radiation to the second and fourth harmonics,” Soviet J. Quantum Electron.21(10), 1045–1051 (1991).
[CrossRef]

Felgate, N. S.

Foster, J. D.

J. D. Foster and L. M. Osterink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys.41(9), 3656–3663 (1970).
[CrossRef]

Giuliani, G.

G. Giuliani and P. Ristori, “Polarization flip cavities: a new approach to laser resonators,” Opt. Commun.35(1), 109–112 (1980).
[CrossRef]

Hanna, D. C.

Ivanov, I. A.

I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and I. A. Ivanov, “Influence of the orientation of a crystal on thermal polarization effects in high-power solid-state lasers,” JETP Lett.81(3), 90–94 (2005).
[CrossRef]

Joiner, R. E.

R. E. Joiner, J. Marburger, and W. H. Steier, “Elimination of stress-induced birefringence effects in single-crystal high-power laser windows,” Appl. Phys. Lett.30(9), 485–486 (1977).
[CrossRef]

Jones, R. C.

Karr, M. A.

Katin, E.

Khazanov, E.

Khazanov, E. A.

M. S. Kuzmina, M. A. Martyanov, A. K. Poteomkin, E. A. Khazanov, and A. A. Shaykin, “Theoretical and experimental study of laser radiation propagating in a medium with thermally induced birefringence and cubic nonlinearity,” Opt. Express19(22), 21977–21988 (2011).
[CrossRef] [PubMed]

A. G. Vyatkin and E. A. Khazanov, “Thermally induced depolarization in sesquioxide class m3 single crystals,” J. Opt. Soc. Am. B28(4), 805–811 (2011).
[CrossRef]

M. S. Kochetkova, M. A. Martyanov, A. K. Poteomkin, and E. A. Khazanov, “Propagation of laser radiation in a medium with thermally induced birefringence and cubic nonlinearity,” Opt. Express18(12), 12839–12851 (2010).
[CrossRef] [PubMed]

I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Reduction of thermally induced depolarization of laser radiation in [110] oriented cubic crystals,” Opt. Express17(7), 5496–5501 (2009).
[CrossRef] [PubMed]

I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and I. A. Ivanov, “Influence of the orientation of a crystal on thermal polarization effects in high-power solid-state lasers,” JETP Lett.81(3), 90–94 (2005).
[CrossRef]

E. A. Khazanov, “Thermally induced birefringence in Nd:YAG ceramics,” Opt. Lett.27(9), 716–718 (2002).
[CrossRef] [PubMed]

E. A. Khazanov, “A new Faraday rotator for high average power lasers,” Quantum Electron.31(4), 351–356 (2001).
[CrossRef]

N. F. Andreev, N. G. Bondarenko, I. V. Eremina, S. V. Kuznetsov, O. V. Palashov, G. A. Pasmanik, and E. A. Khazanov, “Single-mode YAG:Nd laser with a stimulated Brillouin scattering mirror and conversion of radiation to the second and fourth harmonics,” Soviet J. Quantum Electron.21(10), 1045–1051 (1991).
[CrossRef]

Kochetkova, M. S.

Koechner, W.

Kuzmina, M. S.

Kuznetsov, S. V.

N. F. Andreev, N. G. Bondarenko, I. V. Eremina, S. V. Kuznetsov, O. V. Palashov, G. A. Pasmanik, and E. A. Khazanov, “Single-mode YAG:Nd laser with a stimulated Brillouin scattering mirror and conversion of radiation to the second and fourth harmonics,” Soviet J. Quantum Electron.21(10), 1045–1051 (1991).
[CrossRef]

Marburger, J.

R. E. Joiner, J. Marburger, and W. H. Steier, “Elimination of stress-induced birefringence effects in single-crystal high-power laser windows,” Appl. Phys. Lett.30(9), 485–486 (1977).
[CrossRef]

Martyanov, M. A.

Massey, G. A.

G. A. Massey, “Criterion for selection of cw laser host materials to increase available power in the fundamental mode,” Appl. Phys. Lett.17(5), 213–215 (1970).
[CrossRef]

Mehl, O.

Mukhin, I. B.

I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Reduction of thermally induced depolarization of laser radiation in [110] oriented cubic crystals,” Opt. Express17(7), 5496–5501 (2009).
[CrossRef] [PubMed]

I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and I. A. Ivanov, “Influence of the orientation of a crystal on thermal polarization effects in high-power solid-state lasers,” JETP Lett.81(3), 90–94 (2005).
[CrossRef]

Narasimhamurty, T. S.

K. Veerabhadra Rao and T. S. Narasimhamurty, “Photoelastic constants of CaF2 and BaF2,” J. Phys. Chem. Solids31, 876–878 (1969).

Osterink, L. M.

J. D. Foster and L. M. Osterink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys.41(9), 3656–3663 (1970).
[CrossRef]

Palashov, O.

Palashov, O. V.

I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Reduction of thermally induced depolarization of laser radiation in [110] oriented cubic crystals,” Opt. Express17(7), 5496–5501 (2009).
[CrossRef] [PubMed]

I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and I. A. Ivanov, “Influence of the orientation of a crystal on thermal polarization effects in high-power solid-state lasers,” JETP Lett.81(3), 90–94 (2005).
[CrossRef]

N. F. Andreev, N. G. Bondarenko, I. V. Eremina, S. V. Kuznetsov, O. V. Palashov, G. A. Pasmanik, and E. A. Khazanov, “Single-mode YAG:Nd laser with a stimulated Brillouin scattering mirror and conversion of radiation to the second and fourth harmonics,” Soviet J. Quantum Electron.21(10), 1045–1051 (1991).
[CrossRef]

Pasmanik, G. A.

N. F. Andreev, N. G. Bondarenko, I. V. Eremina, S. V. Kuznetsov, O. V. Palashov, G. A. Pasmanik, and E. A. Khazanov, “Single-mode YAG:Nd laser with a stimulated Brillouin scattering mirror and conversion of radiation to the second and fourth harmonics,” Soviet J. Quantum Electron.21(10), 1045–1051 (1991).
[CrossRef]

Poteomkin, A.

Poteomkin, A. K.

Quelle, F. W.

Reitze, D. H.

Rice, D. K.

W. Koechner and D. K. Rice, “Birefringence of YAG: Nd laser rods as a function of growth direction,” J. Opt. Soc. Am.61(6), 758–766 (1971).
[CrossRef]

W. Koechner and D. K. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron.6(9), 557–566 (1970).
[CrossRef]

Ristori, P.

G. Giuliani and P. Ristori, “Polarization flip cavities: a new approach to laser resonators,” Opt. Commun.35(1), 109–112 (1980).
[CrossRef]

Scott, W. C.

W. C. Scott and M. de Wit, “Birefringence compensation and TEM00 mode enhancement in a Nd:YAG laser,” Appl. Phys. Lett.18(1), 3–4 (1971).
[CrossRef]

Sergeev, A.

Shashkin, V. V.

L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “On the problem of depolarization of linearly polarized light by a YAG:Nd3+ laser rod under conditions of thermally induced birefringence conditions,” Soviet J. Quantum Electron.10(3), 350–351 (1980).
[CrossRef]

Shaykin, A. A.

Shoji, I.

I. Shoji and T. Taira, “Intrinsic reduction of the depolarization loss in solid-state lasers by use of a (110)-cut Y3Al5O12 crystal,” Appl. Phys. Lett.80(17), 3048–3050 (2002).
[CrossRef]

Snetkov, I.

Soms, L. N.

L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “On the problem of depolarization of linearly polarized light by a YAG:Nd3+ laser rod under conditions of thermally induced birefringence conditions,” Soviet J. Quantum Electron.10(3), 350–351 (1980).
[CrossRef]

L. N. Soms and A. A. Tarasov, “Thermal deformation in color-center laser active elements,” Soviet J. Quantum Electron.9(12), 1506–1508 (1979).
[CrossRef]

Steier, W. H.

R. E. Joiner, J. Marburger, and W. H. Steier, “Elimination of stress-induced birefringence effects in single-crystal high-power laser windows,” Appl. Phys. Lett.30(9), 485–486 (1977).
[CrossRef]

Taira, T.

I. Shoji and T. Taira, “Intrinsic reduction of the depolarization loss in solid-state lasers by use of a (110)-cut Y3Al5O12 crystal,” Appl. Phys. Lett.80(17), 3048–3050 (2002).
[CrossRef]

Tarasov, A. A.

L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “On the problem of depolarization of linearly polarized light by a YAG:Nd3+ laser rod under conditions of thermally induced birefringence conditions,” Soviet J. Quantum Electron.10(3), 350–351 (1980).
[CrossRef]

L. N. Soms and A. A. Tarasov, “Thermal deformation in color-center laser active elements,” Soviet J. Quantum Electron.9(12), 1506–1508 (1979).
[CrossRef]

Veerabhadra Rao, K.

K. Veerabhadra Rao and T. S. Narasimhamurty, “Photoelastic constants of CaF2 and BaF2,” J. Phys. Chem. Solids31, 876–878 (1969).

Vyatkin, A.

Vyatkin, A. G.

Appl. Opt. (4)

Appl. Phys. (Berl.) (1)

R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” Appl. Phys. (Berl.)38, 5149–5153 (1967).

Appl. Phys. Lett. (4)

R. E. Joiner, J. Marburger, and W. H. Steier, “Elimination of stress-induced birefringence effects in single-crystal high-power laser windows,” Appl. Phys. Lett.30(9), 485–486 (1977).
[CrossRef]

W. C. Scott and M. de Wit, “Birefringence compensation and TEM00 mode enhancement in a Nd:YAG laser,” Appl. Phys. Lett.18(1), 3–4 (1971).
[CrossRef]

G. A. Massey, “Criterion for selection of cw laser host materials to increase available power in the fundamental mode,” Appl. Phys. Lett.17(5), 213–215 (1970).
[CrossRef]

I. Shoji and T. Taira, “Intrinsic reduction of the depolarization loss in solid-state lasers by use of a (110)-cut Y3Al5O12 crystal,” Appl. Phys. Lett.80(17), 3048–3050 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. Koechner and D. K. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron.6(9), 557–566 (1970).
[CrossRef]

J. Appl. Phys. (1)

J. D. Foster and L. M. Osterink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys.41(9), 3656–3663 (1970).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (2)

J. Phys. Chem. Solids (1)

K. Veerabhadra Rao and T. S. Narasimhamurty, “Photoelastic constants of CaF2 and BaF2,” J. Phys. Chem. Solids31, 876–878 (1969).

JETP Lett. (1)

I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and I. A. Ivanov, “Influence of the orientation of a crystal on thermal polarization effects in high-power solid-state lasers,” JETP Lett.81(3), 90–94 (2005).
[CrossRef]

Opt. Commun. (1)

G. Giuliani and P. Ristori, “Polarization flip cavities: a new approach to laser resonators,” Opt. Commun.35(1), 109–112 (1980).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Quantum Electron. (1)

E. A. Khazanov, “A new Faraday rotator for high average power lasers,” Quantum Electron.31(4), 351–356 (2001).
[CrossRef]

Soviet J. Quantum Electron. (3)

N. F. Andreev, N. G. Bondarenko, I. V. Eremina, S. V. Kuznetsov, O. V. Palashov, G. A. Pasmanik, and E. A. Khazanov, “Single-mode YAG:Nd laser with a stimulated Brillouin scattering mirror and conversion of radiation to the second and fourth harmonics,” Soviet J. Quantum Electron.21(10), 1045–1051 (1991).
[CrossRef]

L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “On the problem of depolarization of linearly polarized light by a YAG:Nd3+ laser rod under conditions of thermally induced birefringence conditions,” Soviet J. Quantum Electron.10(3), 350–351 (1980).
[CrossRef]

L. N. Soms and A. A. Tarasov, “Thermal deformation in color-center laser active elements,” Soviet J. Quantum Electron.9(12), 1506–1508 (1979).
[CrossRef]

Other (6)

W. Koechner, Solid-State Laser Engineering (Springer-Verlag, 1999).

J. F. Nye, Physical Properties of Crystals (Oxford University Press, 1964).

E. Khazanov, “Use of parallel axicon for compensation of birefringence in active elements of solid-state lasers,” Proc. SPIE 4632, 155–163 (2002) (Laser and Beam Control Technologies, ed. S. Basu and J. F. Riker).

M. J. Weber, Handbook of optical materials (CRC Press, 2003).

W. Martienssen and H. Warlimont, Handbook of Condensed Matter and Materials Data (Springer, 2006).

S. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill, 1951).

Supplementary Material (5)

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Figures (12)

Fig. 1
Fig. 1

Initial polarization (a) and calculated depolarized beam profiles for CaF2 (ξ = −0.47) with [001] orientation (α = β = 0) for χ = 0 (b) and χ = 22.5° (c), and with [011] orientation (α = 0, β = π/4) for χ = 6° (d) and χ = 18° (e).

Fig. 2
Fig. 2

Single element with double length (a) and schemes for compensation of depolarization with counterrotation (b) and with λ/2 plates (c). The rightmost λ/2 plate in (c) is presented for simplicity of the further statement.

Fig. 3
Fig. 3

Set of [0MN] [α = 0; (a)] and [MMN] [α = π/4; (b)] orientations of the crystals with depolarization compensation in the scheme with counterrotation (grey).

Fig. 4
Fig. 4

Domain of existence of compensation in the scheme with counterrotation (all shades of blue for any ξd) and of profile rotation of the depolarized beam for ξd = D(ξ‒1): all shades of blue for D = 0, medium blue and dark blue for D = 1, dark blue for D = 2 on a cube in crystallographic axes (a,b,c) at different ξ (Media 1). a) ξ = −0.001, b) ξ = −0.1, c) ξ = −0.3, d) ξ = −0.47 (CaF2), e) ξ = −0.8, f) ξ = −2, g) ξ = −3, h) ξ = −5, i) ξ = −30. Markers denote simplest orientations: [001] (filled squares), [011] (filled circles), [111] (filled triangles), as well as [[C]] orientation for D = 0 (open diamonds with dots), D = 1 (open circles with dots), and D = 2 (crosses). [[C]] orientation is close to [011] and therefore is not shown in (a) and does not exist for D = 1 and D = 2 in (h) and (i).

Fig. 5
Fig. 5

Integral depolarization degree γ (solid curves) and its quadratic term γ(2) (dotted curves) as a function of angle χ at p = 5 in CaF2 (ξ = −0.47) with [001] orientation for a single element of double length (black curves) and in the scheme with counterrotation (red curves).

Fig. 6
Fig. 6

Integral depolarization degree γ as a function of dimensionless heat power in a) BaF2, b) SrF2, c) CaF2, d) KCl with [001] orientation (solid curves) and [111] orientation (dashed curves) for a single element of double length [Fig. 2(a), black curves] and in schemes with λ/2 plates [Fig. 2(b), red curves] and with counterrotation [Fig. 2(c), blue curves]. The horizontal line is a guide for the eye indicating a typical value of “cold” depolarization degree.

Fig. 7
Fig. 7

Schematic of the experiment: 1 – calcite wedge, 2 – beam absorber, 3, 4 – two halves of investigated sample, 5 – quartz wedge, 6 – Glan prism, 7 – CCD camera.

Fig. 8
Fig. 8

a) Integral depolarization degree in CaF2 versus angle χ for the transmitted radiation power of 70 W for a double-length crystal – black circles, and in the scheme with counterrotation – red squares: experiment (symbols) and theory for [001] orientation (curves). b) Experimental plots for minimal integral depolarization degree versus transmitted radiation power (symbols) and their approximation by power functions (lines) in the same cases.

Fig. 9
Fig. 9

Experimental [(a)–(c)] and calculated at p = 1 [(d)–(i), Media 2] depolarized beam profiles in the scheme with counterrotation for CaF2 (ξ = −0.47), a) χ = 30°, b) χ = 12°, c) χ = 17° (the angle is close to optimum); d) χ = 13° (γ = 7.4·10−4), e) χ = 15° (γ = 2.6·10−4), f) χ = 17° (γ = 5·10−5); g)–i) same as (d)–(f) but both crystals are additionally tilted by 1° to one side. (a)–(c) are scaled to their maximum values. White dots in (d)–(i) are marks on the crystals.

Fig. 10
Fig. 10

Depolarized beam profiles as a function of crystal angle of turn χ for a single element of double length [(a),(c),(e)] and in the scheme with counterrotation [(b),(d),(f)] for CaF2 (ξ = −0.47) in [001] [(a),(b), Media 3] and [011] [(c),(d), Media 4] orientations, and for a crystal with ξ = + 0.47 in [001] [(e),(f), Media 5] orientation. White dots are marks on the crystals.

Fig. 11
Fig. 11

Modifications of the depolarization compensation scheme with λ/2 plates. The following power orders are compensated: a) second, b) second and fourth, с) from second to sixth, d) from second to eighth (see also Fig. 12). Redundant plates are omitted.

Fig. 12
Fig. 12

The depolarization degree γ as a function of dimensionless heat power p in the cases of a single active element (black curves) and compensation schemes with λ/2 plates [Fig. 2(b), red curves] and with counterrotation [Fig. 2(c), blue curves]; a) CaF2 (ξ = −0.47), b) KCl (ξ = −1.2); crystal orientation is [001] in all cases. The number of active elements is indicated in circles. The horizontal line is a guide for the eye indicating a typical value of “cold” depolarization degree.

Equations (22)

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Γ= | E out | 2 | E out | 2 , γ= S Γ | E out | 2 dS S | E out | 2 dS ,
J i =sin δ i 2 ( cot( δ i /2 )+jcos2 Ψ i jsin2 Ψ i jsin2 Ψ i cot( δ i /2 )jcos2 Ψ i ),
J Σ = J 1,2,,( N1 ),N = J N J N1 J 2 J 1 .
Γ= | J Σ12 | 2 = | J Σ21 | 2 .
Γ small Γ (2) = 1 4 ( i=1 N δ i sin2 Ψ i ) 2 .
tan2 Ψ 0 i = D i / C i , δ i = δ 0i D i / ( sin2 Ψ 0 i ), δ 0i = 1 2 k 0 L i n 0i 3 ,
C i =Δ B 11 Δ B 22 = A 1 ( ξ, ξ d ; α i , β i )Σ( r )+ +[ A 2 ( ξ; α i , β i )cos2 φ 0 + A 3 ( ξ, ξ d ; α i , β i )sin2 φ 0 ]Δ( r ), D i =2Δ B 12 = B 1 ( ξ, ξ d ; α i , β i )Σ( r )+ +[ B 2 ( ξ, ξ d ; α i , β i )cos2 φ 0 + B 3 ( ξ; α i , β i )sin2 φ 0 ]Δ( r ),
Σ( r )= π S [ 1 2 ( σ rr + σ φφ ) σ zz ], Δ( r )= π S ( σ rr σ φφ ), π S = π aaaa 1 2 ( π aabb + π bbaa ).
A 1 ( α,β )=( 1ξ ) a 1 ( α,β )+ ξ d a 2 ( α,β ), B 1 ( α,β )=( 1ξ ) b 1 ( α,β )+ ξ d b 2 ( α,β ), A 2 ( α,β )=ξ+( 1ξ ) a 3 ( α,β ), B 2 ( α,β )=( 1ξ ) c 1 ( α,β )+ ξ d c 2 ( α,β ), A 3 ( α,β )=( 1ξ ) c 1 ( α,β ) ξ d c 2 ( α,β ), B 3 ( α,β )=ξ+( 1ξ ) b 3 ( α,β ),
ξ= 2 π abab / π S , ξ d = ( π aabb π bbaa ) / π S
a 1 = [ sin 2 2α( 1 cos 4 β ) sin 2 2β ] / 2, b 1 = 1 4 sinβsin2βsin4α, a 2 =cos2αcos2β, b 2 =sin2αcosβ ( 13 cos 2 β ) /2 , a 3 =[ 1 ( sin 2 2α ) /4 ] sin 4 β+ cos 2 2α cos 2 β, b 3 = sin 2 2α cos 2 β, c 1 =sin4αcosβ ( 1+ cos 2 β ) /4 , c 2 = 3 4 sin2αcosβsi n 2 β
p disk = α T E P Σ k 0 n 0 3 π S / ( 8πκ ) , p rod = p disk / ( 1ν ) ,
φ= φ 0 + χ i , Ψ i = Ψ 0 i + χ i ,
Γ (2) = 1 4 [ i=1 N δ 0i V i ] 2 , V i = D i cos2 χ i + C i sin2 χ i .
Γ (2) = δ 0 2 4 ( V 1 + V 2 ) 2 , V 1 + V 2 =Δ( r )sin2φ{ 2ξ+( 1ξ )[ ( a 3 + b 3 )+( b 3 a 3 )cos4χ+2 c 1 sin4χ ] } .
( ξ ξ1 ) 2 ( ξ ξ1 )( a 3 + b 3 )+( a 3 b 3 c 1 2 )0
ξ[ 4+( ξ1 ) sin 2 2β ]0.
[ 1( 1ξ ) sin 2 β ][ ξ+ 3 4 ( 1ξ ) sin 4 β ]0
tan2 φ ¯ = ( A 2 + B 3 )+( B 3 A 2 )cos4χ+( A 3 + B 2 )sin4χ ( B 2 A 3 )+( A 3 + B 2 )cos4χ+( B 3 A 2 )sin4χ .
c 2 2 ( ξ d ξ1 ) 2 + ( ξ ξ1 ) 2 ( ξ ξ1 )( a 3 + b 3 )+( a 3 b 3 c 1 2 )0.
J[ 1 ]=sin δ 2 ( cot( δ/2 )+jcos2Ψ jsin2Ψ jsin2Ψ cot( δ/2 )jcos2Ψ ),
J[ 2 ]=( 1 0 0 1 )J[ 1 ]( 1 0 0 1 )=sin δ 2 ( cot( δ/2 )+jcos2Ψ jsin2Ψ jsin2Ψ cot( δ/2 )jcos2Ψ ).

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