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Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 51, Iss. 30 — Oct. 20, 2012
  • pp: 7150–7159

Analytical solutions for anisotropic time-dependent heat equations with Robin boundary condition for cubic-shaped solid-state laser crystals

Mohammad Sabaeian  »View Author Affiliations

Applied Optics, Vol. 51, Issue 30, pp. 7150-7159 (2012)

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The problem of finding analytical solutions for time-dependent or time-independent heat equations, especially for solid-state laser media, has required a lot of work in the field of thermal effects. However, to calculate the temperature distributions analytically, researchers often have to make some approximations or employ complex methods. In this work, we present full analytical solutions for anisotropic time-dependent heat equations in the Cartesian coordinates with various source terms corresponding to various pumping schemes. Moreover, the most general boundary condition of Robin (or impedance boundary condition), corresponding to the convection cooling mechanism, was applied. This general condition can be easily switched to constant temperature and thermal insulation as special cases. To this end, we first proposed a general approach to solving time-dependent heat equations with an arbitrary heat source. We then applied our approach to explore the temperature distribution for three cases: steady-state pumping or long transient, single-shot pumping or short transient, and repetitively pulsed pumping. Our results show the possibility of an easier and more accurate approach to analytical calculations of the thermal dispersion, thermal stresses (strains), thermal bending, thermal phase shift, and other thermal effects.

© 2012 Optical Society of America

OCIS Codes
(120.6780) Instrumentation, measurement, and metrology : Temperature
(120.6810) Instrumentation, measurement, and metrology : Thermal effects
(140.3480) Lasers and laser optics : Lasers, diode-pumped
(140.3580) Lasers and laser optics : Lasers, solid-state

ToC Category:
Lasers and Laser Optics

Original Manuscript: July 11, 2012
Revised Manuscript: September 5, 2012
Manuscript Accepted: September 7, 2012
Published: October 11, 2012

Mohammad Sabaeian, "Analytical solutions for anisotropic time-dependent heat equations with Robin boundary condition for cubic-shaped solid-state laser crystals," Appl. Opt. 51, 7150-7159 (2012)

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  1. X. Jiang, X. Yuan, H. Yu, M. Xu, D. Cao, and W. Duan, “Influence of the thermal effect on stability of the output in a heat capacity laser,” Chin. Opt. Lett. 5, S19–S20 (2007).
  2. C. Pfistrner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortion in end-pumped Nd:YAG, Nd:GSGG, and Nd:YLF rods,” IEEE Quantum Electron. 30, 1605–1615 (1994). [CrossRef]
  3. W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state laser,” J. Phys. D 34, 2381–2395 (2001). [CrossRef]
  4. I. B. Mukhin, O. V. Palashov, E. A. Khazanov, A. Ikesue, and Y. L. Aung, “Experimental study of thermally induced depolarization in Nd:YAG ceramics,” Opt. Express 13, 5983–5987 (2005). [CrossRef]
  5. J. Song, A. Liu, K. Okino, and K. I. Ueda, “Control of the thermal lensing effect with different pump light distributions,” Appl. Opt. 36, 8051–8055 (1997). [CrossRef]
  6. D. C. Brown, “Heat, fluorescence, and stimulated-emission power densities and fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560–572 (1998). [CrossRef]
  7. J. Zheng, Sh. Zhao, Q. Wang, X. Zhang, and L. Chen, “Influence of thermal effect on KTP type-II phase-matching second-harmonic generation,” Opt. Commun. 199, 207–214 (2001). [CrossRef]
  8. M. Sabaeian, L. Mousave, and H. Nadgaran, “Investigation of thermally-induced phase mismatching in continuous-wave second harmonic generation: A theoretical model,” Opt. Express 18, 18732–18743 (2010). [CrossRef]
  9. M. Sabaeian, H. Nadgaran, and L. Mousave, “Analytical solution of the heat equation in a longitudinally pumped cubic solid-state laser,” Appl. Opt. 47, 2317–2325 (2008). [CrossRef]
  10. R. Lausten and P. Baling, “Thermal lensing in pulsed laser amplifiers: an analytical model,” J. Opt. Soc. Am. B 20, 1479–1485 (2003). [CrossRef]
  11. P. Shi, W. Chen, L. Li, and A. Gan, “Semianalytical thermal analysis on a Nd:YVO4 crystal,” Appl. Opt. 46, 4046–4051 (2007). [CrossRef]
  12. J. K. Jabczynski, K. Kopczynski, and A. Szczesniak, “Thermal lensing and thermal aberration investigations in diode-pumped lasers,” Opt. Eng. 35, 3572–3578 (1996). [CrossRef]
  13. H. Nadgaran and P. Elahi, “The overall phase shift and lens effect calculation using Gaussian boundary conditions and paraxial ray approximation for an end-pumped solid state laser,” Pramana 66, 513–519 (2006). [CrossRef]
  14. A. K. Cousins, “Temperature and thermal stress scaling in finite-length, end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992). [CrossRef]
  15. 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, 1398–1404 (2000). [CrossRef]
  16. Z. Xiong, Zh. G. Li, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 laser,” IEEE J. Quantum Electron. 39, 979–986 (2003). [CrossRef]
  17. O. L. Antipov, E. A. Anashkina, and K. A. Fedorova, “Electronic and thermal lensing in diode end-pumped Yb:YAG laser rods and discs,” Quantum Electron. 39, 1131–1136 (2009). [CrossRef]
  18. L. L. Zhang, P. Shi, and L. Li, “Semianalytical thermal analysis of rectangle Nd:GGG in heat capacity laser,” Appl. Phys. B 101, 137–142 (2010). [CrossRef]
  19. N. P. Barnes, R. C. Eckhardt, D. J. Gettemy, and L. B. Edgett, “Absorption coefficient and the temperature variation of the refractive index difference of nonlinear optical crystal,” IEEE J. Quantum Electron. 15, 1074–1076 (1979). [CrossRef]
  20. M. Sabaeian and H. Nadgaran, “Bessel–Gauss beams: Investigations of thermal effects on their generation,” Opt. Commum. 281, 672–678 (2008). [CrossRef]
  21. H. Nadgaran, M. Servatkhah, and M. Sabaeian, “Mathieu–Gauss beams: A thermal consideration,” Opt. Commun. 283, 417–426 (2010). [CrossRef]
  22. H. Nadgaran and M. Servatkhah, “The effects of induced heat loads on the propagation of Ince–Gaussian beams,” Opt. Commun. 284, 5329–5337 (2011). [CrossRef]
  23. Sh. H. Li, H. B. He, Y. G. Shan, D. W. Li, Y. A. Zhao, and Zh. X. Fan, “Enhanced surface thermal lensing for absorption evaluation and defect identification of optical films,” Appl. Opt. 49, 2417–2421 (2010). [CrossRef]
  24. S. L. Prins, A. C. Barron, C. Herrmann, and J. R. McNeil, “Effect of stress on performance of dense wavelength division multiplexing filters: thermal properties,” Appl. Opt. 43, 633–637 (2004). [CrossRef]
  25. M. Shimosegawa, T. Omatsu, M. Tateda, I. Ogura, J. L. Blows, P. Wang, and J. M. Dawes, “Thermal conductivity of a self-frequency-doubling laser crystal measured by use of optical methods,” Appl. Opt. 40, 1372–1377 (2001). [CrossRef]
  26. F. Jürgensen and W. Schröer , “Studies on the diffraction image of a thermal lens,” Appl. Opt. 34, 41–50 (1995). [CrossRef]
  27. B. A. Usievich, V. A. Sychugov, F. Pigeon, and A. Tishchenko, “Analytical treatment of the thermal problem in axially pumped solid-state lasers,” IEEE J. Quantum Electron. 37, 1210–1214 (2001). [CrossRef]
  28. Y. Peng, Zh. Cheng, Y. Zhang, and J. Qiu, “Temperature distribution and thermal deformations of mirror substrates in laser resonator,” Appl. Opt. 40, 4824–4830 (2001). [CrossRef]
  29. P. Shi, W. Chen, L. Li, and A. Gan, “Semianalytical thermal analysis of thermal focal length on Nd:YAG rods,” Appl. Opt. 46, 6655–6661 (2007). [CrossRef]
  30. Z. Ma, D. Li, J. Gao, N. Wu, and K. Du, “Thermal effects of the diode end-pumped Nd:YVO4 slab,” Opt. Commun. 275, 179–185 (2007). [CrossRef]
  31. Zh. Li, X. Huai, Y. Tao, and Z. Guo, “Analysis of thermal effects in an orthotropic laser medium,” Appl. Opt. 48, 598–608(2009). [CrossRef]
  32. T. Liu, Z. M. Yang, and S. H. Xu, “Analytical investigation on transient thermal effects in pulse end-pumped short-length fiber laser,” Opt. Express 17, 12875–12890 (2009). [CrossRef]
  33. G. L. Bourdet and C. Gouedard, “Simple analytical derivations of thermal lensing in longitudinally Q-CW pumped Yb:YAG,” Appl. Opt. 49, 4160–4167 (2010). [CrossRef]
  34. T. T. L. Kristensen, W. Withayakumnankul, P. U. Jensen, and D. Abbott, “Modeling terahertz heating effects on water,” Opt. Express 18, 4727–4739 (2010). [CrossRef]
  35. H. Nadgaran and M. Sabaian, “Pulsed pump: Thermal effects in solid state lasers under super-Gaussian pulses,” Paramana 67, 1119–1128 (2006). [CrossRef]
  36. G. Arfken, Mathematical Methods for Physicists (Academic, 1988).
  37. X. Peng, A. Asundi, Y. Chen, and Zh. Xiong, “Study of the mechanical properties of Nd:YVO4 crystal by use of laser interferometry and finite-element analysis,” Appl. Opt. 40, 1396–1403 (2001). [CrossRef]

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