Abstract

The computer-aided design of high quality mono-mode, continuous-wave solid-state lasers requires fast, flexible and accurate simulation algorithms. Therefore in this work a model for the calculation of the transversal dominant mode structure is introduced. It is based on the generalization of the scalar Fox and Li algorithm to a fully-vectorial light representation. To provide a flexible modeling concept of different resonator geometries containing various optical elements, rigorous and approximative solutions of Maxwell’s equations are combined in different subdomains of the resonator. This approach allows the simulation of plenty of different passive intracavity components as well as active media. For the numerically efficient simulation of nonlinear gain, thermal lensing and stress-induced birefringence effects in solid-state active crystals a semi-analytical vectorial beam propagation method is discussed in detail. As a numerical example the beam quality and output power of a flash-lamp-pumped Nd:YAG laser are improved. To that end we compensate the influence of stress-induced birefringence and thermal lensing by an aspherical mirror and a 90° quartz polarization rotator.

© 2015 Optical Society of America

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References

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  1. J. R. Leger, D. Chen, and Z. Wang, “Diffractive optical element for mode shaping of a Nd:YAG laser,” Opt. Lett. 19(2), 108–110 (1994).
    [Crossref] [PubMed]
  2. O. Svelto, Principles of Lasers (Springer, 2010).
    [Crossref]
  3. W. W. Rigrod, “Saturation effects in high gain lasers,” Appl. Phys. 36(8), 2487–2490 (1965).
    [Crossref]
  4. A. G. Fox and T. Li, “Effects of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron.  2(12), 774–783 (1966).
    [Crossref]
  5. W. Koechner, “Thermal lensing in a Nd:YAG laser rod,” Appl. Opt. 9(11), 2548–2553 (1970).
    [Crossref] [PubMed]
  6. J. D. Foster and L. M. Osterink, “Thermal effects in a Nd: YAG laser,” Appl. Phys. 41(9), 3656–3663 (1970).
    [Crossref]
  7. T. Taira, A. Mukai, Y. Nozawa, and T. Kobayashi, “Single-mode oscillation of laser-diode-pumped Nd:YVO4 microchip lasers,” Opt. Lett. 16(24), 1955–1957 (1991).
    [Crossref] [PubMed]
  8. D. S. Kliger and J. W. Lewis, Polarized Light in Optics and Spectroscopy (Elsevier, 1990).
  9. Y.-Z. Huang, W.-H. Guo, and Q.-M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron. 37(1), 100–107 (2001).
    [Crossref]
  10. P. Nyakas, “Full-vectorial three-dimensional finite element optical simulation of vertical-cavity surface-emitting lasers,” J. Lightw. Technol. 25(9), 2427–2434 (2007).
    [Crossref]
  11. A. Christ, N. Kuster, M. Streiff, A. Witzig, and W. Fichtner, “Correction of the numerical reflection coefficient of the finite-difference time-domain method for efficient simulation of vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 20(7), 1401–1408 (2003).
    [Crossref]
  12. M. Streiff, A. Witzig, M. Pfeiffer, P. Royo, and W. Fichtner, “A comprehensive VCSEL device simulator,” IEEE J. Sel. Topics Quantum Electron. 9(3), 879–891 (2003).
    [Crossref]
  13. J. Pomplun, S. Burger, F. Schmidt, A. Schliwa, D. Bimberg, A. Pietrzak, H. Wenzel, and G. Erbert, “Finite element simulation of the optical modes of semiconductor lasers,” Phys. Status Solidi B 247(4), 846–853 (2010).
    [Crossref]
  14. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966).
    [Crossref] [PubMed]
  15. A. E. Siegman, Lasers (University Science Books, 1986).
  16. C. N. Kurtz and W. Streifer, “Guided waves in inhomogeneous focusing media, part I: formulation, solution for quadratic inhomogeneity,” IEEE Trans. Microw. Theory Techn. 17(1), 11–15 (1969).
    [Crossref]
  17. W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. 22(2), 226–230 (1977).
    [Crossref]
  18. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain,” Appl. Opt. 4(12), 1562–1569 (1965).
    [Crossref]
  19. T. Y. Fan and R. L. Byer, “Diode laser-pumped solid-state lasers,” IEEE J. Quantum Electron. 24(6), 895–912 (1988).
    [Crossref]
  20. D. G. Hall, R. J. Smith, and R. R. Rice, “Pump-size effects in Nd:YAG lasers,” Appl. Opt. 19(18), 3041–3043 (1980).
    [Crossref] [PubMed]
  21. L. W. Casperson, “Laser power calculations: sources of error,” Appl. Opt. 19(3), 422–431 (1980).
    [Crossref] [PubMed]
  22. P. F. Moulton, “An investigation of the Co:MgF2 laser system,” IEEE J. Quantum Electron. 21(10), 1582–1588 (1985).
    [Crossref]
  23. J. Junghans, M. Keller, and H. Weber, “Laser resonators with polarizing elements - eigenstates and eigenvalues of polarization,” Appl. Opt. 13(12), 2793–2798 (1974).
    [Crossref] [PubMed]
  24. T. Graupeter, R. Hartmann, and C. Pflaum, “Calculations of eigenpolarization in Nd:YAG laser rods due to thermally induced birefringence,” IEEE J. Quantum Electron. 50(12), 1035–1043 (2014).
    [Crossref]
  25. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
    [Crossref]
  26. A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51(1), 80–89 (1963).
    [Crossref]
  27. D. Asoubar, S. Zhang, F. Wyrowski, and M. Kuhn, “Laser resonator modeling by field tracing: a flexible approach for fully vectorial transversal eigenmode calculation,” J. Opt. Soc. Am. B 31(11), 2565–2573 (2014).
    [Crossref]
  28. F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
    [Crossref]
  29. L. Thylen and D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. 21(15), 2751–2754 (1982).
    [Crossref] [PubMed]
  30. J. A. Fleck and M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73(7), 920–926 (1983).
    [Crossref]
  31. L. Xu, P. Huang, J. Chrostowski, and S. K. Chaudhuri, “Full-vectorial beam propagation method for anisotropic waveguides,” J. Lightw. Technol. 12(11), 1926–1931 (1994).
    [Crossref]
  32. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
    [Crossref]
  33. J. Saijonmaa and D. Yevick, “Beam-propagation analysis of loss in bent optical waveguides and fibers,” J. Opt. Soc. Am. 73(12), 1785–1791 (1983).
    [Crossref]
  34. A. E. Siegman and E. A. Sziklas, “Mode calculations in unstable resonators with flowing saturable gain. 1: Hermite-Gaussian expansion,” Appl. Opt. 13(12), 2775–2792 (1974).
    [Crossref] [PubMed]
  35. E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: fast Fourier transform method,” Appl. Opt. 14(8), 1874–1889 (1975).
    [Crossref] [PubMed]
  36. B. A. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley Interscience, 2007).
  37. Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
    [Crossref]
  38. Meta Numerics, Library for advanced scientific computation in the .NET Framework, www.meta-numerics.net , accessed (2/18/2015).
  39. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the LambertW function,” Adv. Comput. Math. 5(1), 329–359 (1996).
    [Crossref]
  40. A. Hoorfar and M. Hassani, “Inequalities on the Lambert W function and hyperpower function,” J. Inequal. Pure and Appl. Math 9(2), 5–9 (2008).
  41. N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts, Applications (Springer Science, 2005).
  42. C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd : YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
    [Crossref]
  43. U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state laser rods,” IEEE J. Quantum Electron. 24(11), 2253–2263 (1988).
    [Crossref]
  44. S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd: YAG lasers to high average powers,” IEEE J. Quantum Electron. 28(4), 997–1009 (1992).
    [Crossref]
  45. W. Koechner, Solid-State Laser Engineering (Springer-Verlag, 1996).
    [Crossref]
  46. M. Schmid, T. Graf, and H. P. Weber, “Analytical model of the temperature distribution and the thermally induced birefringence in laser rods with cylindrically symmetric heating,” J. Opt. Soc. Am. B 17(8), 1398–1404 (2000).
    [Crossref]
  47. S. Chenais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and absolute temperature mapping and heat transfer measurements in diode-end-pumped Yb:YAG,” Appl. Phys. B 79(2), 221–224 (2004).
    [Crossref]
  48. J. Didierjean, S. Forget, S. Chenais, F. Druon, F. Balembois, P. Georges, K. Altmann, and C. Pflaum, “High-resolution absolute temperature mapping of laser crystals in diode-end-pumped configuration,” Proc. SPIE 5707, 370–379 (2005).
    [Crossref]
  49. J. D. Foster and L. M. Osterink, “Index of refraction and expansion thermal coefficients of Nd:YAG,” Appl. Opt. 7(12), 2428–2429 (1968).
    [Crossref] [PubMed]
  50. J. L. Blows, J. M. Dawes, and T. Omatsu, “Thermal lensing measurements in line-focus end-pumped neodymium yttrium aluminium garnet using holographic lateral shearing interferometry,” Appl. Phys. 83(6), 2901–2906 (1998).
    [Crossref]
  51. 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]
  52. R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2), 245–254 (1999).
    [Crossref]
  53. M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36(5), 620–626 (2000).
    [Crossref]
  54. R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” Appl. Phys. 38(13), 5149–5153 (1967).
    [Crossref]
  55. W. R. Cook, D. F. Nelson, and K. Vedam, “High frequency properties of dielectric crystals: piezooptic and electrooptic constants,” in Landolt-Börnstein - Group III Condensed Matter Vol. III/30A, D. F. Nelson, ed. (Springer, 1996).
  56. A. S. Andrushchak, Y. V. Bobitski, M. V. Kaidan, B. G. Mytsyk, A. V. Kityk, and W. Schranz, “Two-fold interferometric measurements of piezooptic constants: application to β-BaB2O4 crystals,” Opt. Laser Technol. 37(4), 319–328 (2005).
    [Crossref]
  57. 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]
  58. LightTrans GmbH, LightTrans VirtualLab Advanced, www.lighttrans.com , accessed (4/26/2015).
  59. D. Asoubar, M. Kuhn, and F. Wyrowski, “Fully vectorial laser resonator modeling by vector extrapolation methods,” Proc. SPIE 9342, 934214 (2015).
    [Crossref]
  60. Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, “A novel approach for compensation of birefringence in cylindrical Nd:YAG rods,” Opt. Quant. Electron. 28(1), 57–69 (1996).
    [Crossref]

2015 (1)

D. Asoubar, M. Kuhn, and F. Wyrowski, “Fully vectorial laser resonator modeling by vector extrapolation methods,” Proc. SPIE 9342, 934214 (2015).
[Crossref]

2014 (2)

T. Graupeter, R. Hartmann, and C. Pflaum, “Calculations of eigenpolarization in Nd:YAG laser rods due to thermally induced birefringence,” IEEE J. Quantum Electron. 50(12), 1035–1043 (2014).
[Crossref]

D. Asoubar, S. Zhang, F. Wyrowski, and M. Kuhn, “Laser resonator modeling by field tracing: a flexible approach for fully vectorial transversal eigenmode calculation,” J. Opt. Soc. Am. B 31(11), 2565–2573 (2014).
[Crossref]

2011 (1)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
[Crossref]

2010 (1)

J. Pomplun, S. Burger, F. Schmidt, A. Schliwa, D. Bimberg, A. Pietrzak, H. Wenzel, and G. Erbert, “Finite element simulation of the optical modes of semiconductor lasers,” Phys. Status Solidi B 247(4), 846–853 (2010).
[Crossref]

2008 (1)

A. Hoorfar and M. Hassani, “Inequalities on the Lambert W function and hyperpower function,” J. Inequal. Pure and Appl. Math 9(2), 5–9 (2008).

2007 (1)

P. Nyakas, “Full-vectorial three-dimensional finite element optical simulation of vertical-cavity surface-emitting lasers,” J. Lightw. Technol. 25(9), 2427–2434 (2007).
[Crossref]

2005 (2)

J. Didierjean, S. Forget, S. Chenais, F. Druon, F. Balembois, P. Georges, K. Altmann, and C. Pflaum, “High-resolution absolute temperature mapping of laser crystals in diode-end-pumped configuration,” Proc. SPIE 5707, 370–379 (2005).
[Crossref]

A. S. Andrushchak, Y. V. Bobitski, M. V. Kaidan, B. G. Mytsyk, A. V. Kityk, and W. Schranz, “Two-fold interferometric measurements of piezooptic constants: application to β-BaB2O4 crystals,” Opt. Laser Technol. 37(4), 319–328 (2005).
[Crossref]

2004 (2)

S. Chenais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and absolute temperature mapping and heat transfer measurements in diode-end-pumped Yb:YAG,” Appl. Phys. B 79(2), 221–224 (2004).
[Crossref]

Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
[Crossref]

2003 (2)

2001 (1)

Y.-Z. Huang, W.-H. Guo, and Q.-M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron. 37(1), 100–107 (2001).
[Crossref]

2000 (2)

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36(5), 620–626 (2000).
[Crossref]

M. Schmid, T. Graf, and H. P. Weber, “Analytical model of the temperature distribution and the thermally induced birefringence in laser rods with cylindrically symmetric heating,” J. Opt. Soc. Am. B 17(8), 1398–1404 (2000).
[Crossref]

1999 (1)

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2), 245–254 (1999).
[Crossref]

1998 (1)

J. L. Blows, J. M. Dawes, and T. Omatsu, “Thermal lensing measurements in line-focus end-pumped neodymium yttrium aluminium garnet using holographic lateral shearing interferometry,” Appl. Phys. 83(6), 2901–2906 (1998).
[Crossref]

1996 (2)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the LambertW function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, “A novel approach for compensation of birefringence in cylindrical Nd:YAG rods,” Opt. Quant. Electron. 28(1), 57–69 (1996).
[Crossref]

1994 (3)

J. R. Leger, D. Chen, and Z. Wang, “Diffractive optical element for mode shaping of a Nd:YAG laser,” Opt. Lett. 19(2), 108–110 (1994).
[Crossref] [PubMed]

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd : YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[Crossref]

L. Xu, P. Huang, J. Chrostowski, and S. K. Chaudhuri, “Full-vectorial beam propagation method for anisotropic waveguides,” J. Lightw. Technol. 12(11), 1926–1931 (1994).
[Crossref]

1992 (1)

S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd: YAG lasers to high average powers,” IEEE J. Quantum Electron. 28(4), 997–1009 (1992).
[Crossref]

1991 (1)

1988 (2)

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state laser rods,” IEEE J. Quantum Electron. 24(11), 2253–2263 (1988).
[Crossref]

T. Y. Fan and R. L. Byer, “Diode laser-pumped solid-state lasers,” IEEE J. Quantum Electron. 24(6), 895–912 (1988).
[Crossref]

1985 (1)

P. F. Moulton, “An investigation of the Co:MgF2 laser system,” IEEE J. Quantum Electron. 21(10), 1582–1588 (1985).
[Crossref]

1983 (2)

1982 (1)

1980 (2)

1977 (1)

W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. 22(2), 226–230 (1977).
[Crossref]

1975 (1)

1974 (2)

1970 (4)

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]

W. Koechner, “Thermal lensing in a Nd:YAG laser rod,” Appl. Opt. 9(11), 2548–2553 (1970).
[Crossref] [PubMed]

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

1969 (1)

C. N. Kurtz and W. Streifer, “Guided waves in inhomogeneous focusing media, part I: formulation, solution for quadratic inhomogeneity,” IEEE Trans. Microw. Theory Techn. 17(1), 11–15 (1969).
[Crossref]

1968 (1)

1967 (1)

R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” Appl. Phys. 38(13), 5149–5153 (1967).
[Crossref]

1966 (2)

A. G. Fox and T. Li, “Effects of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron.  2(12), 774–783 (1966).
[Crossref]

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966).
[Crossref] [PubMed]

1965 (2)

1963 (1)

A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51(1), 80–89 (1963).
[Crossref]

1961 (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Altmann, K.

J. Didierjean, S. Forget, S. Chenais, F. Druon, F. Balembois, P. Georges, K. Altmann, and C. Pflaum, “High-resolution absolute temperature mapping of laser crystals in diode-end-pumped configuration,” Proc. SPIE 5707, 370–379 (2005).
[Crossref]

Andrushchak, A. S.

A. S. Andrushchak, Y. V. Bobitski, M. V. Kaidan, B. G. Mytsyk, A. V. Kityk, and W. Schranz, “Two-fold interferometric measurements of piezooptic constants: application to β-BaB2O4 crystals,” Opt. Laser Technol. 37(4), 319–328 (2005).
[Crossref]

Asoubar, D.

Balembois, F.

J. Didierjean, S. Forget, S. Chenais, F. Druon, F. Balembois, P. Georges, K. Altmann, and C. Pflaum, “High-resolution absolute temperature mapping of laser crystals in diode-end-pumped configuration,” Proc. SPIE 5707, 370–379 (2005).
[Crossref]

S. Chenais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and absolute temperature mapping and heat transfer measurements in diode-end-pumped Yb:YAG,” Appl. Phys. B 79(2), 221–224 (2004).
[Crossref]

Bimberg, D.

J. Pomplun, S. Burger, F. Schmidt, A. Schliwa, D. Bimberg, A. Pietrzak, H. Wenzel, and G. Erbert, “Finite element simulation of the optical modes of semiconductor lasers,” Phys. Status Solidi B 247(4), 846–853 (2010).
[Crossref]

Blows, J. L.

J. L. Blows, J. M. Dawes, and T. Omatsu, “Thermal lensing measurements in line-focus end-pumped neodymium yttrium aluminium garnet using holographic lateral shearing interferometry,” Appl. Phys. 83(6), 2901–2906 (1998).
[Crossref]

Bobitski, Y. V.

A. S. Andrushchak, Y. V. Bobitski, M. V. Kaidan, B. G. Mytsyk, A. V. Kityk, and W. Schranz, “Two-fold interferometric measurements of piezooptic constants: application to β-BaB2O4 crystals,” Opt. Laser Technol. 37(4), 319–328 (2005).
[Crossref]

Bowers, M. S.

S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd: YAG lasers to high average powers,” IEEE J. Quantum Electron. 28(4), 997–1009 (1992).
[Crossref]

Buoncristiani, A. M.

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state laser rods,” IEEE J. Quantum Electron. 24(11), 2253–2263 (1988).
[Crossref]

Burger, S.

J. Pomplun, S. Burger, F. Schmidt, A. Schliwa, D. Bimberg, A. Pietrzak, H. Wenzel, and G. Erbert, “Finite element simulation of the optical modes of semiconductor lasers,” Phys. Status Solidi B 247(4), 846–853 (2010).
[Crossref]

Byer, R. L.

T. Y. Fan and R. L. Byer, “Diode laser-pumped solid-state lasers,” IEEE J. Quantum Electron. 24(6), 895–912 (1988).
[Crossref]

Byvik, C. E.

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state laser rods,” IEEE J. Quantum Electron. 24(11), 2253–2263 (1988).
[Crossref]

Casperson, L. W.

Chaudhuri, S. K.

L. Xu, P. Huang, J. Chrostowski, and S. K. Chaudhuri, “Full-vectorial beam propagation method for anisotropic waveguides,” J. Lightw. Technol. 12(11), 1926–1931 (1994).
[Crossref]

Chen, D.

Chenais, S.

J. Didierjean, S. Forget, S. Chenais, F. Druon, F. Balembois, P. Georges, K. Altmann, and C. Pflaum, “High-resolution absolute temperature mapping of laser crystals in diode-end-pumped configuration,” Proc. SPIE 5707, 370–379 (2005).
[Crossref]

S. Chenais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and absolute temperature mapping and heat transfer measurements in diode-end-pumped Yb:YAG,” Appl. Phys. B 79(2), 221–224 (2004).
[Crossref]

Christ, A.

Chrostowski, J.

L. Xu, P. Huang, J. Chrostowski, and S. K. Chaudhuri, “Full-vectorial beam propagation method for anisotropic waveguides,” J. Lightw. Technol. 12(11), 1926–1931 (1994).
[Crossref]

Cook, W. R.

W. R. Cook, D. F. Nelson, and K. Vedam, “High frequency properties of dielectric crystals: piezooptic and electrooptic constants,” in Landolt-Börnstein - Group III Condensed Matter Vol. III/30A, D. F. Nelson, ed. (Springer, 1996).

Corless, R. M.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the LambertW function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

Cousins, A. K.

S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd: YAG lasers to high average powers,” IEEE J. Quantum Electron. 28(4), 997–1009 (1992).
[Crossref]

Dawes, J. M.

J. L. Blows, J. M. Dawes, and T. Omatsu, “Thermal lensing measurements in line-focus end-pumped neodymium yttrium aluminium garnet using holographic lateral shearing interferometry,” Appl. Phys. 83(6), 2901–2906 (1998).
[Crossref]

Didierjean, J.

J. Didierjean, S. Forget, S. Chenais, F. Druon, F. Balembois, P. Georges, K. Altmann, and C. Pflaum, “High-resolution absolute temperature mapping of laser crystals in diode-end-pumped configuration,” Proc. SPIE 5707, 370–379 (2005).
[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. 38(13), 5149–5153 (1967).
[Crossref]

Dong, S.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, “A novel approach for compensation of birefringence in cylindrical Nd:YAG rods,” Opt. Quant. Electron. 28(1), 57–69 (1996).
[Crossref]

Druon, F.

J. Didierjean, S. Forget, S. Chenais, F. Druon, F. Balembois, P. Georges, K. Altmann, and C. Pflaum, “High-resolution absolute temperature mapping of laser crystals in diode-end-pumped configuration,” Proc. SPIE 5707, 370–379 (2005).
[Crossref]

S. Chenais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and absolute temperature mapping and heat transfer measurements in diode-end-pumped Yb:YAG,” Appl. Phys. B 79(2), 221–224 (2004).
[Crossref]

Erbert, G.

J. Pomplun, S. Burger, F. Schmidt, A. Schliwa, D. Bimberg, A. Pietrzak, H. Wenzel, and G. Erbert, “Finite element simulation of the optical modes of semiconductor lasers,” Phys. Status Solidi B 247(4), 846–853 (2010).
[Crossref]

Fan, T. Y.

T. Y. Fan and R. L. Byer, “Diode laser-pumped solid-state lasers,” IEEE J. Quantum Electron. 24(6), 895–912 (1988).
[Crossref]

Farrukh, U. O.

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state laser rods,” IEEE J. Quantum Electron. 24(11), 2253–2263 (1988).
[Crossref]

Feit, M. D.

Fichtner, W.

Firth, W. J.

W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. 22(2), 226–230 (1977).
[Crossref]

Fleck, J. A.

Forget, S.

J. Didierjean, S. Forget, S. Chenais, F. Druon, F. Balembois, P. Georges, K. Altmann, and C. Pflaum, “High-resolution absolute temperature mapping of laser crystals in diode-end-pumped configuration,” Proc. SPIE 5707, 370–379 (2005).
[Crossref]

S. Chenais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and absolute temperature mapping and heat transfer measurements in diode-end-pumped Yb:YAG,” Appl. Phys. B 79(2), 221–224 (2004).
[Crossref]

Foster, J. D.

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

J. D. Foster and L. M. Osterink, “Index of refraction and expansion thermal coefficients of Nd:YAG,” Appl. Opt. 7(12), 2428–2429 (1968).
[Crossref] [PubMed]

Fox, A. G.

A. G. Fox and T. Li, “Effects of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron.  2(12), 774–783 (1966).
[Crossref]

A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51(1), 80–89 (1963).
[Crossref]

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Georges, P.

J. Didierjean, S. Forget, S. Chenais, F. Druon, F. Balembois, P. Georges, K. Altmann, and C. Pflaum, “High-resolution absolute temperature mapping of laser crystals in diode-end-pumped configuration,” Proc. SPIE 5707, 370–379 (2005).
[Crossref]

S. Chenais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and absolute temperature mapping and heat transfer measurements in diode-end-pumped Yb:YAG,” Appl. Phys. B 79(2), 221–224 (2004).
[Crossref]

Gonnet, G. H.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the LambertW function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

Graf, T.

Graf, Th.

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36(5), 620–626 (2000).
[Crossref]

Graupeter, T.

T. Graupeter, R. Hartmann, and C. Pflaum, “Calculations of eigenpolarization in Nd:YAG laser rods due to thermally induced birefringence,” IEEE J. Quantum Electron. 50(12), 1035–1043 (2014).
[Crossref]

Gruber, R.

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd : YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[Crossref]

Guo, W.-H.

Y.-Z. Huang, W.-H. Guo, and Q.-M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron. 37(1), 100–107 (2001).
[Crossref]

Hall, D. G.

Hare, D. E. G.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the LambertW function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

Hartmann, R.

T. Graupeter, R. Hartmann, and C. Pflaum, “Calculations of eigenpolarization in Nd:YAG laser rods due to thermally induced birefringence,” IEEE J. Quantum Electron. 50(12), 1035–1043 (2014).
[Crossref]

Hassani, M.

A. Hoorfar and M. Hassani, “Inequalities on the Lambert W function and hyperpower function,” J. Inequal. Pure and Appl. Math 9(2), 5–9 (2008).

Hodgson, N.

N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts, Applications (Springer Science, 2005).

Hoorfar, A.

A. Hoorfar and M. Hassani, “Inequalities on the Lambert W function and hyperpower function,” J. Inequal. Pure and Appl. Math 9(2), 5–9 (2008).

Huang, P.

L. Xu, P. Huang, J. Chrostowski, and S. K. Chaudhuri, “Full-vectorial beam propagation method for anisotropic waveguides,” J. Lightw. Technol. 12(11), 1926–1931 (1994).
[Crossref]

Huang, Y.-Z.

Y.-Z. Huang, W.-H. Guo, and Q.-M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron. 37(1), 100–107 (2001).
[Crossref]

Jeffrey, D. J.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the LambertW function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

Junghans, J.

Kaidan, M. V.

A. S. Andrushchak, Y. V. Bobitski, M. V. Kaidan, B. G. Mytsyk, A. V. Kityk, and W. Schranz, “Two-fold interferometric measurements of piezooptic constants: application to β-BaB2O4 crystals,” Opt. Laser Technol. 37(4), 319–328 (2005).
[Crossref]

Keller, M.

Kityk, A. V.

A. S. Andrushchak, Y. V. Bobitski, M. V. Kaidan, B. G. Mytsyk, A. V. Kityk, and W. Schranz, “Two-fold interferometric measurements of piezooptic constants: application to β-BaB2O4 crystals,” Opt. Laser Technol. 37(4), 319–328 (2005).
[Crossref]

Kliger, D. S.

D. S. Kliger and J. W. Lewis, Polarized Light in Optics and Spectroscopy (Elsevier, 1990).

Knuth, D. E.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the LambertW function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

Kobayashi, T.

Koechner, W.

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]

W. Koechner, “Thermal lensing in a Nd:YAG laser rod,” Appl. Opt. 9(11), 2548–2553 (1970).
[Crossref] [PubMed]

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, Solid-State Laser Engineering (Springer-Verlag, 1996).
[Crossref]

Kogelnik, H.

Kugler, N.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, “A novel approach for compensation of birefringence in cylindrical Nd:YAG rods,” Opt. Quant. Electron. 28(1), 57–69 (1996).
[Crossref]

Kuhn, M.

D. Asoubar, M. Kuhn, and F. Wyrowski, “Fully vectorial laser resonator modeling by vector extrapolation methods,” Proc. SPIE 9342, 934214 (2015).
[Crossref]

D. Asoubar, S. Zhang, F. Wyrowski, and M. Kuhn, “Laser resonator modeling by field tracing: a flexible approach for fully vectorial transversal eigenmode calculation,” J. Opt. Soc. Am. B 31(11), 2565–2573 (2014).
[Crossref]

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
[Crossref]

Kurtz, C. N.

C. N. Kurtz and W. Streifer, “Guided waves in inhomogeneous focusing media, part I: formulation, solution for quadratic inhomogeneity,” IEEE Trans. Microw. Theory Techn. 17(1), 11–15 (1969).
[Crossref]

Kuster, N.

Leger, J. R.

Lewis, J. W.

D. S. Kliger and J. W. Lewis, Polarized Light in Optics and Spectroscopy (Elsevier, 1990).

Li, T.

A. G. Fox and T. Li, “Effects of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron.  2(12), 774–783 (1966).
[Crossref]

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966).
[Crossref] [PubMed]

A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51(1), 80–89 (1963).
[Crossref]

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Lü, Q.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, “A novel approach for compensation of birefringence in cylindrical Nd:YAG rods,” Opt. Quant. Electron. 28(1), 57–69 (1996).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Merazzi, S.

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd : YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[Crossref]

Moulton, P. F.

P. F. Moulton, “An investigation of the Co:MgF2 laser system,” IEEE J. Quantum Electron. 21(10), 1582–1588 (1985).
[Crossref]

Mukai, A.

Müller, N.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, “A novel approach for compensation of birefringence in cylindrical Nd:YAG rods,” Opt. Quant. Electron. 28(1), 57–69 (1996).
[Crossref]

Mytsyk, B. G.

A. S. Andrushchak, Y. V. Bobitski, M. V. Kaidan, B. G. Mytsyk, A. V. Kityk, and W. Schranz, “Two-fold interferometric measurements of piezooptic constants: application to β-BaB2O4 crystals,” Opt. Laser Technol. 37(4), 319–328 (2005).
[Crossref]

Nelson, D. F.

W. R. Cook, D. F. Nelson, and K. Vedam, “High frequency properties of dielectric crystals: piezooptic and electrooptic constants,” in Landolt-Börnstein - Group III Condensed Matter Vol. III/30A, D. F. Nelson, ed. (Springer, 1996).

Neuenschwander, B.

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2), 245–254 (1999).
[Crossref]

Nozawa, Y.

Nyakas, P.

P. Nyakas, “Full-vectorial three-dimensional finite element optical simulation of vertical-cavity surface-emitting lasers,” J. Lightw. Technol. 25(9), 2427–2434 (2007).
[Crossref]

Omatsu, T.

J. L. Blows, J. M. Dawes, and T. Omatsu, “Thermal lensing measurements in line-focus end-pumped neodymium yttrium aluminium garnet using holographic lateral shearing interferometry,” Appl. Phys. 83(6), 2901–2906 (1998).
[Crossref]

Osterink, L. M.

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

J. D. Foster and L. M. Osterink, “Index of refraction and expansion thermal coefficients of Nd:YAG,” Appl. Opt. 7(12), 2428–2429 (1968).
[Crossref] [PubMed]

Pfeiffer, M.

M. Streiff, A. Witzig, M. Pfeiffer, P. Royo, and W. Fichtner, “A comprehensive VCSEL device simulator,” IEEE J. Sel. Topics Quantum Electron. 9(3), 879–891 (2003).
[Crossref]

Pfistner, C.

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd : YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[Crossref]

Pflaum, C.

T. Graupeter, R. Hartmann, and C. Pflaum, “Calculations of eigenpolarization in Nd:YAG laser rods due to thermally induced birefringence,” IEEE J. Quantum Electron. 50(12), 1035–1043 (2014).
[Crossref]

J. Didierjean, S. Forget, S. Chenais, F. Druon, F. Balembois, P. Georges, K. Altmann, and C. Pflaum, “High-resolution absolute temperature mapping of laser crystals in diode-end-pumped configuration,” Proc. SPIE 5707, 370–379 (2005).
[Crossref]

Pietrzak, A.

J. Pomplun, S. Burger, F. Schmidt, A. Schliwa, D. Bimberg, A. Pietrzak, H. Wenzel, and G. Erbert, “Finite element simulation of the optical modes of semiconductor lasers,” Phys. Status Solidi B 247(4), 846–853 (2010).
[Crossref]

Pomplun, J.

J. Pomplun, S. Burger, F. Schmidt, A. Schliwa, D. Bimberg, A. Pietrzak, H. Wenzel, and G. Erbert, “Finite element simulation of the optical modes of semiconductor lasers,” Phys. Status Solidi B 247(4), 846–853 (2010).
[Crossref]

Rice, D. K.

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]

Rice, R. R.

Rigrod, W. W.

W. W. Rigrod, “Saturation effects in high gain lasers,” Appl. Phys. 36(8), 2487–2490 (1965).
[Crossref]

Roos, M.

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36(5), 620–626 (2000).
[Crossref]

Royo, P.

M. Streiff, A. Witzig, M. Pfeiffer, P. Royo, and W. Fichtner, “A comprehensive VCSEL device simulator,” IEEE J. Sel. Topics Quantum Electron. 9(3), 879–891 (2003).
[Crossref]

Saijonmaa, J.

Saleh, B. A. E.

B. A. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley Interscience, 2007).

Sato, Y.

Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
[Crossref]

Schliwa, A.

J. Pomplun, S. Burger, F. Schmidt, A. Schliwa, D. Bimberg, A. Pietrzak, H. Wenzel, and G. Erbert, “Finite element simulation of the optical modes of semiconductor lasers,” Phys. Status Solidi B 247(4), 846–853 (2010).
[Crossref]

Schmid, M.

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36(5), 620–626 (2000).
[Crossref]

M. Schmid, T. Graf, and H. P. Weber, “Analytical model of the temperature distribution and the thermally induced birefringence in laser rods with cylindrically symmetric heating,” J. Opt. Soc. Am. B 17(8), 1398–1404 (2000).
[Crossref]

Schmidt, F.

J. Pomplun, S. Burger, F. Schmidt, A. Schliwa, D. Bimberg, A. Pietrzak, H. Wenzel, and G. Erbert, “Finite element simulation of the optical modes of semiconductor lasers,” Phys. Status Solidi B 247(4), 846–853 (2010).
[Crossref]

Schranz, W.

A. S. Andrushchak, Y. V. Bobitski, M. V. Kaidan, B. G. Mytsyk, A. V. Kityk, and W. Schranz, “Two-fold interferometric measurements of piezooptic constants: application to β-BaB2O4 crystals,” Opt. Laser Technol. 37(4), 319–328 (2005).
[Crossref]

Seamans, J. F.

S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd: YAG lasers to high average powers,” IEEE J. Quantum Electron. 28(4), 997–1009 (1992).
[Crossref]

Siegman, A. E.

Smith, R. J.

Streifer, W.

C. N. Kurtz and W. Streifer, “Guided waves in inhomogeneous focusing media, part I: formulation, solution for quadratic inhomogeneity,” IEEE Trans. Microw. Theory Techn. 17(1), 11–15 (1969).
[Crossref]

Streiff, M.

Svelto, O.

O. Svelto, Principles of Lasers (Springer, 2010).
[Crossref]

Sziklas, E. A.

Taira, T.

Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
[Crossref]

T. Taira, A. Mukai, Y. Nozawa, and T. Kobayashi, “Single-mode oscillation of laser-diode-pumped Nd:YVO4 microchip lasers,” Opt. Lett. 16(24), 1955–1957 (1991).
[Crossref] [PubMed]

Teich, M. C.

B. A. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley Interscience, 2007).

Thylen, L.

Tidwell, S. C.

S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd: YAG lasers to high average powers,” IEEE J. Quantum Electron. 28(4), 997–1009 (1992).
[Crossref]

Vedam, K.

W. R. Cook, D. F. Nelson, and K. Vedam, “High frequency properties of dielectric crystals: piezooptic and electrooptic constants,” in Landolt-Börnstein - Group III Condensed Matter Vol. III/30A, D. F. Nelson, ed. (Springer, 1996).

Wang, Q.-M.

Y.-Z. Huang, W.-H. Guo, and Q.-M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron. 37(1), 100–107 (2001).
[Crossref]

Wang, Z.

Weber, H.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, “A novel approach for compensation of birefringence in cylindrical Nd:YAG rods,” Opt. Quant. Electron. 28(1), 57–69 (1996).
[Crossref]

J. Junghans, M. Keller, and H. Weber, “Laser resonators with polarizing elements - eigenstates and eigenvalues of polarization,” Appl. Opt. 13(12), 2793–2798 (1974).
[Crossref] [PubMed]

N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts, Applications (Springer Science, 2005).

Weber, H. P.

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36(5), 620–626 (2000).
[Crossref]

M. Schmid, T. Graf, and H. P. Weber, “Analytical model of the temperature distribution and the thermally induced birefringence in laser rods with cylindrically symmetric heating,” J. Opt. Soc. Am. B 17(8), 1398–1404 (2000).
[Crossref]

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2), 245–254 (1999).
[Crossref]

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd : YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[Crossref]

Weber, R.

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36(5), 620–626 (2000).
[Crossref]

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2), 245–254 (1999).
[Crossref]

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd : YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[Crossref]

Wenzel, H.

J. Pomplun, S. Burger, F. Schmidt, A. Schliwa, D. Bimberg, A. Pietrzak, H. Wenzel, and G. Erbert, “Finite element simulation of the optical modes of semiconductor lasers,” Phys. Status Solidi B 247(4), 846–853 (2010).
[Crossref]

Wittrock, U.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, “A novel approach for compensation of birefringence in cylindrical Nd:YAG rods,” Opt. Quant. Electron. 28(1), 57–69 (1996).
[Crossref]

Witzig, A.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Wyrowski, F.

D. Asoubar, M. Kuhn, and F. Wyrowski, “Fully vectorial laser resonator modeling by vector extrapolation methods,” Proc. SPIE 9342, 934214 (2015).
[Crossref]

D. Asoubar, S. Zhang, F. Wyrowski, and M. Kuhn, “Laser resonator modeling by field tracing: a flexible approach for fully vectorial transversal eigenmode calculation,” J. Opt. Soc. Am. B 31(11), 2565–2573 (2014).
[Crossref]

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
[Crossref]

Xu, L.

L. Xu, P. Huang, J. Chrostowski, and S. K. Chaudhuri, “Full-vectorial beam propagation method for anisotropic waveguides,” J. Lightw. Technol. 12(11), 1926–1931 (1994).
[Crossref]

Yevick, D.

Zhang, S.

Adv. Comput. Math. (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the LambertW function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

Appl. Opt. (11)

J. D. Foster and L. M. Osterink, “Index of refraction and expansion thermal coefficients of Nd:YAG,” Appl. Opt. 7(12), 2428–2429 (1968).
[Crossref] [PubMed]

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]

W. Koechner, “Thermal lensing in a Nd:YAG laser rod,” Appl. Opt. 9(11), 2548–2553 (1970).
[Crossref] [PubMed]

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966).
[Crossref] [PubMed]

H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain,” Appl. Opt. 4(12), 1562–1569 (1965).
[Crossref]

D. G. Hall, R. J. Smith, and R. R. Rice, “Pump-size effects in Nd:YAG lasers,” Appl. Opt. 19(18), 3041–3043 (1980).
[Crossref] [PubMed]

L. W. Casperson, “Laser power calculations: sources of error,” Appl. Opt. 19(3), 422–431 (1980).
[Crossref] [PubMed]

J. Junghans, M. Keller, and H. Weber, “Laser resonators with polarizing elements - eigenstates and eigenvalues of polarization,” Appl. Opt. 13(12), 2793–2798 (1974).
[Crossref] [PubMed]

L. Thylen and D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. 21(15), 2751–2754 (1982).
[Crossref] [PubMed]

A. E. Siegman and E. A. Sziklas, “Mode calculations in unstable resonators with flowing saturable gain. 1: Hermite-Gaussian expansion,” Appl. Opt. 13(12), 2775–2792 (1974).
[Crossref] [PubMed]

E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: fast Fourier transform method,” Appl. Opt. 14(8), 1874–1889 (1975).
[Crossref] [PubMed]

Appl. Phys. (4)

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

W. W. Rigrod, “Saturation effects in high gain lasers,” Appl. Phys. 36(8), 2487–2490 (1965).
[Crossref]

R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” Appl. Phys. 38(13), 5149–5153 (1967).
[Crossref]

J. L. Blows, J. M. Dawes, and T. Omatsu, “Thermal lensing measurements in line-focus end-pumped neodymium yttrium aluminium garnet using holographic lateral shearing interferometry,” Appl. Phys. 83(6), 2901–2906 (1998).
[Crossref]

Appl. Phys. B (1)

S. Chenais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and absolute temperature mapping and heat transfer measurements in diode-end-pumped Yb:YAG,” Appl. Phys. B 79(2), 221–224 (2004).
[Crossref]

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

IEEE J. Quantum Electron (1)

A. G. Fox and T. Li, “Effects of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron.  2(12), 774–783 (1966).
[Crossref]

IEEE J. Quantum Electron. (10)

Y.-Z. Huang, W.-H. Guo, and Q.-M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron. 37(1), 100–107 (2001).
[Crossref]

T. Graupeter, R. Hartmann, and C. Pflaum, “Calculations of eigenpolarization in Nd:YAG laser rods due to thermally induced birefringence,” IEEE J. Quantum Electron. 50(12), 1035–1043 (2014).
[Crossref]

P. F. Moulton, “An investigation of the Co:MgF2 laser system,” IEEE J. Quantum Electron. 21(10), 1582–1588 (1985).
[Crossref]

T. Y. Fan and R. L. Byer, “Diode laser-pumped solid-state lasers,” IEEE J. Quantum Electron. 24(6), 895–912 (1988).
[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]

Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
[Crossref]

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd : YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[Crossref]

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state laser rods,” IEEE J. Quantum Electron. 24(11), 2253–2263 (1988).
[Crossref]

S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd: YAG lasers to high average powers,” IEEE J. Quantum Electron. 28(4), 997–1009 (1992).
[Crossref]

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36(5), 620–626 (2000).
[Crossref]

IEEE J. Sel. Topics Quantum Electron. (1)

M. Streiff, A. Witzig, M. Pfeiffer, P. Royo, and W. Fichtner, “A comprehensive VCSEL device simulator,” IEEE J. Sel. Topics Quantum Electron. 9(3), 879–891 (2003).
[Crossref]

IEEE Trans. Microw. Theory Techn. (1)

C. N. Kurtz and W. Streifer, “Guided waves in inhomogeneous focusing media, part I: formulation, solution for quadratic inhomogeneity,” IEEE Trans. Microw. Theory Techn. 17(1), 11–15 (1969).
[Crossref]

J. Inequal. Pure and Appl. Math (1)

A. Hoorfar and M. Hassani, “Inequalities on the Lambert W function and hyperpower function,” J. Inequal. Pure and Appl. Math 9(2), 5–9 (2008).

J. Lightw. Technol. (2)

P. Nyakas, “Full-vectorial three-dimensional finite element optical simulation of vertical-cavity surface-emitting lasers,” J. Lightw. Technol. 25(9), 2427–2434 (2007).
[Crossref]

L. Xu, P. Huang, J. Chrostowski, and S. K. Chaudhuri, “Full-vectorial beam propagation method for anisotropic waveguides,” J. Lightw. Technol. 12(11), 1926–1931 (1994).
[Crossref]

J. Mod. Opt. (1)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. 22(2), 226–230 (1977).
[Crossref]

Opt. Laser Technol. (1)

A. S. Andrushchak, Y. V. Bobitski, M. V. Kaidan, B. G. Mytsyk, A. V. Kityk, and W. Schranz, “Two-fold interferometric measurements of piezooptic constants: application to β-BaB2O4 crystals,” Opt. Laser Technol. 37(4), 319–328 (2005).
[Crossref]

Opt. Lett. (2)

Opt. Mater. (1)

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2), 245–254 (1999).
[Crossref]

Opt. Quant. Electron. (1)

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, “A novel approach for compensation of birefringence in cylindrical Nd:YAG rods,” Opt. Quant. Electron. 28(1), 57–69 (1996).
[Crossref]

Phys. Status Solidi B (1)

J. Pomplun, S. Burger, F. Schmidt, A. Schliwa, D. Bimberg, A. Pietrzak, H. Wenzel, and G. Erbert, “Finite element simulation of the optical modes of semiconductor lasers,” Phys. Status Solidi B 247(4), 846–853 (2010).
[Crossref]

Proc. IEEE (1)

A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51(1), 80–89 (1963).
[Crossref]

Proc. SPIE (2)

D. Asoubar, M. Kuhn, and F. Wyrowski, “Fully vectorial laser resonator modeling by vector extrapolation methods,” Proc. SPIE 9342, 934214 (2015).
[Crossref]

J. Didierjean, S. Forget, S. Chenais, F. Druon, F. Balembois, P. Georges, K. Altmann, and C. Pflaum, “High-resolution absolute temperature mapping of laser crystals in diode-end-pumped configuration,” Proc. SPIE 5707, 370–379 (2005).
[Crossref]

Other (10)

N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts, Applications (Springer Science, 2005).

Meta Numerics, Library for advanced scientific computation in the .NET Framework, www.meta-numerics.net , accessed (2/18/2015).

W. Koechner, Solid-State Laser Engineering (Springer-Verlag, 1996).
[Crossref]

LightTrans GmbH, LightTrans VirtualLab Advanced, www.lighttrans.com , accessed (4/26/2015).

W. R. Cook, D. F. Nelson, and K. Vedam, “High frequency properties of dielectric crystals: piezooptic and electrooptic constants,” in Landolt-Börnstein - Group III Condensed Matter Vol. III/30A, D. F. Nelson, ed. (Springer, 1996).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

B. A. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley Interscience, 2007).

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

D. S. Kliger and J. W. Lewis, Polarized Light in Optics and Spectroscopy (Elsevier, 1990).

O. Svelto, Principles of Lasers (Springer, 2010).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Exemplary resonator setup consisting of several different active and passive optical components. The transversal resonator mode can be calculated in a arbitrary plane using a sequence of diverse component operators C and free space propagation operators P describing a full resonator round trip. Each operator solves Maxwell’s equations in a rigorous or approximated manner for a single subdomain of the resonator.
Fig. 2
Fig. 2 Schematic illustration of the symmetrized vBPM for light propagation through an active intracavity component: the component is subdivided into segments of width Δz. Within a segment only linear effects, like diffraction or linear absorption are included in the angular spectrum of plane waves operators P s SPW. The effect of nonlinearity is introduced by the nonlinear operators C s NL in the center of the segments (represented by dashed lines).
Fig. 3
Fig. 3 Physical interpretation of a single nonlinear vBPM operation C s NL within a single slab with thickness Δz: the result of the operation is obtained by mapping the complex optical path length (OPL) introduced by the complex refractive index modulation within the slab onto the initial field distribution (V1,V2)T, which is sampled on a transversal grid ρn. Due to the small slab thickness and a paraxial initial field, the OPL can be evaluated along rays propagating parallel to the optical axis. Each transversal grid point is the origin of a ray/local plane wave with a certain polarization vector V in the transversal x-y-plane. Due to the stress-induced birefringence, the OPL for each ray/local plane wave has to be evaluated for the two orthogonal polarization components parallel to the principal axes α′ and β′. Due to the position-dependent nature of the electric permittivity tensor of the active medium, for each ray another orientation and shape of the indicatrix must be considered in the OPL analysis. In the left part of the figure two example indicatrices for different rays with different polarization vectors V are given.
Fig. 4
Fig. 4 Initial setup 1: solid state laser, consisting of two spherical mirrors M1 and M2, a linear polarizer and a flash-pumped Nd:YAG crystal. To suppress higher order modes, an aperture is placed in the plane of the outcoupling mirror M1. The parameters of the Nd:YAG crystal are given in Table 1.
Fig. 5
Fig. 5 Dominant transversal eigenmode of the laser resonator setup given in Fig. 4. The upper row shows the V1 (a) and V2 (b) field components of the mode in the plane of the outcoupling mirror M1. The lower row shows the V1 (c) and V2 (d) field components of the mode in the plane between the active medium and the linear polarizer. In both rows the propagation direction of the mode is towards the outcoupling mirror and the fundamental transversal eigenmode has a non-Gaussian shape due to birefringence, diffraction and thermal lensing effects.
Fig. 6
Fig. 6 Resonator setup 2 with thermal lens compensation: a stable Fourier transform resonator geometry is used to improve the beam quality and output power. To ensure that the thermal lens in the flash-pumped Nd:YAG laser is compensated an aspherical mirror was designed. The parameters of the Nd:YAG crystal are the same as in the initial setup.
Fig. 7
Fig. 7 Dominant transversal eigenmode of the laser resonator setup 2 given in Fig. 6. The upper row shows the V1 (a) and V2 (b) field components of the mode in the plane of the outcoupling mirror M1. The lower row shows the V1 (c) and V2 (d) field components of the mode in the plane behind the active medium, next to the linear polarizer. In both rows the propagation direction of the mode is towards the outcoupling mirror.
Fig. 8
Fig. 8 Resonator setup 3 with thermal lens and birefringence compensation: a stable Fourier transform resonator geometry is used to improve the beam quality and output power. To ensure that the thermal lens in the flash-pumped Nd:YAG laser is compensated an aspherical mirror was designed. The initial Nd:YAG crystal is split into two rods with equal length and optical pump. Stress-induced birefringence is compensated by placing a 90° quartz polarization rotator and a thin lens between the two Nd:YAG crystals.
Fig. 9
Fig. 9 Dominant transversal eigenmode of the laser resonator setup 3 given in Fig. 8. The upper row shows the V1 (a) and V2 (b) field components of the mode in the plane of the outcoupling mirror M1. The lower row shows the V1 (c) and V2 (d) field components of the mode in the plane behind the active medium 1 next to the linear polarizer. In both rows the propagation direction of the mode is towards the outcoupling mirror.

Tables (2)

Tables Icon

Table 1 Parameters of the Nd:YAG active medium. If no extra citation is given for the parameter, its value was taken from Koechner [5, 57].

Tables Icon

Table 2 Output power and beam quality of laser resonator setups 1-3. The depolarization loss and the diffraction loss are defined by the ratio of the beam power before and after the polarizer and mirror aperture, respectively.

Equations (52)

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( γ 1 0 0 γ 2 ) ( V 1 V 2 ) = ( V 1 V 2 ) = ( 11 12 21 22 ) ( V 1 V 2 ) .
= m = 1 n ( C m P m , m 1 ) .
C m = ( C m , 11 0 0 C m , 22 )
C m = ( C m , 11 C m , 12 C m , 21 C m , 22 )
ε ¯ ( r , V ) = ( ε 11 ( r , V ) ε 12 ( r , V ) ε 13 ( r , V ) ε 21 ( r , V ) ε 22 ( r , V ) ε 23 ( r , V ) ε 31 ( r , V ) ε 32 ( r , V ) ε 33 ( r , V ) )
( I ¯ 2 z 2 M ¯ 2 ) ( V 1 V 2 ) = 0
M ¯ 2 = ( 2 + k 0 2 ε 11 ( r , V ) k 0 2 ε 12 ( r , V ) k 0 2 ε 21 ( r , V ) 2 + k 0 2 ε 22 ( r , V ) ) .
ε α β ( r , V ) = ( n 0 + n ˜ α β ( r , V ) ) 2 n 0 2 + 2 n 0 n ˜ α β ( r , V )
ñ α β ( r , V ) = Δ n α β therm ( r ) + Δ n α β birefring ( r , V ) + Δ n α β gain ( r , V ) .
( V 1 ( z 0 + L ) V 2 ( z 0 + L ) ) = s = 1 S [ P s SPW ( Δ z 2 ) C s NL ( z 0 + ( s 1 2 ) Δ z ) P s SPW ( Δ z 2 ) ] ( V 1 ( z 0 ) V 2 ( z 0 ) ) exp ( i k 0 n 0 L )
P s SPW ( Δ z 2 ) = ( P SPW 0 0 P SPW )
P SPW V ( ρ , z 0 ) = 1 { [ V ( ρ , z 0 ) ] exp [ i k z Δ z 2 ] } .
C s NL ( z 0 + ( s 1 2 ) Δ z ) = ( C 11 , s NL C 12 , s NL C 21 , s NL C 22 , s NL )
C α β , s NL = exp ( i 2 π λ z 0 + ( s 1 ) Δ z z 0 + s Δ z n ˜ α β ( z , V ) d z )
C α β , s NL = C α β , s gain C α β , s t + b
C α β , s gain = exp ( i 2 π λ z 0 + ( s 1 ) Δ z z 0 + s Δ z Δ n α β gain ( z , V ) d z )
C α β , s t + b = exp [ i 2 π λ z 0 + ( s 1 ) Δ z z 0 + s Δ z ( Δ n α β therm ( z ) + Δ n α β birefring ( z , V ) ) d z ] .
C α β , s t + b exp { i 2 Δ z π λ [ Δ n α β therm ( z 0 + ( s 1 2 ) Δ z ) + Δ n α β birefring ( z 0 + ( s 1 2 ) Δ z , V ) ] } .
d I d z = g I
I | S | Re ( n 0 ) ε 0 2 c ( | V 1 | 2 + | V 2 | 2 ) .
I ( z 0 + Δ z ) = I ( z 0 ) exp ( z 0 z 0 + Δ z g ( z ) d z )
i 2 π λ 2 Δ n α β gain ( r , V ) = g .
Δ n α β gain ( r , V ) λ i 4 π g 0 1 + I I s .
d V d z = 1 2 g 0 1 + ( | V 1 | 2 + | V 2 | 2 ) / V sat 2 V , = 1 , 2
V sat 2 = I s Re ( n 0 ) ε 0 2 c
V ( z 0 + Δ z ) = [ V sat 2 | V 1 ( z 0 ) | 2 + | V 2 ( z 0 ) | 2 W ( 1 V sat 2 exp ( g 0 Δ z + c 1 ) ) ] ( 1 / 2 ) V ( z 0 ) , = 1 , 2
c 1 = ln ( | V 1 ( z 0 ) | 2 + | V 2 ( z 0 ) | 2 ) + ( | V 1 ( z 0 ) | 2 + | V 2 ( z 0 ) | 2 V sat 2 ) .
W ( a ) ln ( a ) ln ln ( a ) + ln ln ( a ) ln ( a ) .
I = I + + I .
Δ n α β therm ( r ) = { d n d T ( T T 0 ) + 1 2 d 2 n d T 2 ( T T 0 ) 2 + for α = β 0 else
( B x x 0 + Δ B x x ) x 2 + ( B y y 0 + Δ B y y ) y 2 + ( B z z 0 + Δ B z z ) z 2 + 2 Δ B x y x y + 2 Δ B x z x z + 2 Δ B y z y z = 1
Δ B i j = k l p i j k l ε ^ k l
p i j k l = m n q i j m n C m n k l
( B x x 0 + Δ B x x Δ B x y Δ B y x B y y 0 + Δ B y y ) ( V 1 ( r ) V 2 ( r ) ) = 1 n 2 ( V 1 ( r ) V 2 ( r ) )
C s NL = ( C s Proj ) 1 ( C 11 , s NL 0 0 C 22 , s NL ) C s Proj
C α α , s NL = exp ( i 2 π λ z 0 + ( s 1 ) Δ z z 0 + s Δ z n ˜ α α ( z , V ) d z ) ,
C s Proj ( V 1 ( ρ ) V 2 ( ρ ) ) = ( V 1 ( ρ ) V 2 ( ρ ) ) V x ( ρ ) | V x ( ρ ) | V x ( ρ ) + ( V 1 ( ρ ) V 2 ( ρ ) ) V y ( ρ ) | V y ( ρ ) | V y ( ρ )
n ˜ α β ( z , V ) = Δ n α β therm ( r ) + Δ n α β birefring ( r , V ) + Δ n α β gain ( r , V ) .
1 / n x 2 = B 0 + 1 2 ( Δ B x x + Δ B y y ) + 1 2 [ ( Δ B x x Δ B y y ) 2 + 4 Δ B x y 2 ] 1 / 2
1 / n y 2 = B 0 + 1 2 ( Δ B x x + Δ B y y ) 1 2 [ ( Δ B x x Δ B y y ) 2 + 4 Δ B x y 2 ] 1 / 2 .
Δ n α β birefring ( r , V ) = { n 0 3 2 ( 1 n x 2 B 0 ) for α = β = 1 n 0 3 2 ( 1 n y 2 B 0 ) for α = β = 2 0 else
T ( ρ ) = T ( 0 ) | ρ | 2 ( 0.5 d ) 2 [ T ( 0 ) T ( 0.5 d ) ]
Δ n α α therm ( ρ ) = 41.277 m 2 × | ρ | 2 .
C s NL = ( cos θ sin θ sin θ cos θ ) ( C 11 , s NL 0 0 C 22 , s NL ) ( cos θ sin θ sin θ cos θ )
Δ n 11 birefring ( ρ , V ) = α ν n Nd:YAG 3 [ T ( 0 ) T ( 0.5 d ) ] 6 ( 1 ν ) ( 0.5 d ) 2 ( p 1111 p 1122 + 4 p 3232 ) | ρ | 2
g 0 η pe P e τ σ 12 π ( 0.5 d ) 2 L h ν 20 .
d | V | 2 d z = g 0 1 + ( | V 1 | 2 + | V 2 | 2 ) / V sat 2 | V | 2 , = 1 , 2.
d ( | V 1 | 2 + | V 2 | 2 ) d z = g 0 ( | V 1 | 2 + | V 2 | 2 ) 1 + ( | V 1 | 2 + | V 2 | 2 ) / V sat 2
| V 1 ( z 0 + Δ z ) | 2 + | V 2 ( z 0 + Δ z ) | 2 = V sat 2 W ( exp ( g 0 Δ z + c 1 ) V sat 2 ) .
c 1 = ln ( | V 1 ( z 0 ) | 2 + | V 2 ( z 0 ) | 2 ) + ( | V 1 ( z 0 ) | 2 + | V 2 ( z 0 ) | 2 V sat 2 )
| V ( z 0 + Δ z ) ) | 2 = | V ( z 0 ) | 2 | V 1 ( z 0 ) | 2 + | V 2 ( z 0 ) | 2 V sat 2 W ( exp ( g 0 Δ z + c 1 ) V sat 2 ) , = 1 , 2.
V ( z 0 + Δ z ) = V ( z 0 ) ( | V 1 ( z 0 ) | 2 + | V 2 ( z 0 ) | 2 ) 1 / 2 [ V sat 2 W ( exp ( g 0 Δ z + c 1 ) V sat 2 ) ] ( 1 / 2 ) , = 1 , 2.

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