Abstract

Knowledge about the temperature distribution inside solid-state laser crystals is essential for calculation of thermal phase shift, thermal lensing, thermally induced birefringence, and heat-induced crystal bending. Solutions for the temperature distribution for the case of steady-state heat loading have appeared in the literature only for simple cylindrical crystal shapes and are usually based on numerical techniques. For the first time, to our knowledge, a full analytical solution of the heat equation for an anisotropic cubic cross-section solid-state crystal is presented. The crystal is assumed to be longitudinally pumped by a Gaussian pump profile. The pump power attenuation along the crystal and the real cooling mechanisms, such as convection, are considered in detail. A comparison between our analytical solutions and its numerical counterparts shows excellent agreement when just a few terms are employed in the series solutions.

© 2008 Optical Society of America

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References

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  1. W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state laser,” J. Phys. D 34, 2381-2395 (2001).
  2. D. C. Brown, “Heat, fluorescence, and stimulated-emission power densities and fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560-572 (1998).
    [CrossRef]
  3. S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-inversion densities in Nd:YAG: upconversion and bleaching,” IEEE J. Quantum Electron. 34, 900-909 (1998).
    [CrossRef]
  4. C. Pfistner, 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 J. Quantum Electron. 30, 1605-1615 (1994).
  5. H. Nadgaran and M. Sabaian, “Pulsed pump: thermal effects in solid state lasers under super-Gaussian pulses,” Pramana J. Phys. 67, 1119-1128 (2006).
  6. 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 J. Phys. 66, 513-519 (2006).
  7. M. Sabaeian and H. Nadgaran, “Bessel-Gauss beams: investigations of thermal effects on their generation,” Opt. Commun. 281, 672-678 (2008).
  8. A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057-1069 (1992).
    [CrossRef]
  9. M. Schmid, Th. 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]
  10. 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]
  11. H. Zhang, J. Liu, J. Wang, Ch. Wang, L. Zhu, Z. Shao, X. Meng, X. Hu, and M. Jiang, “Characterization of the laser crystal Nd:GdVO4,” J. Opt. Soc. Am. B 19, 18-27 (2002).
    [CrossRef]
  12. P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser material: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253-258 (2002).
  13. R. A. Fielfs, M. Birnbaun, and C. N. Fincher, “High efficiency Nd:YVO4 diode-laser end-pumped laser,” Appl. Phys. Lett. 51, 1885-1886 (1987).
  14. 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]
  15. P. Shi, W. Chen, L. Li, and A. Gan, “Semianalytical thermal analysis on a Nd:YVO4 crystal,” Appl. Opt. 46, 4046-4051 (2007).
  16. P. Zeller and P. Peuser, “Efficient, multiwatt, continuous-wave laser operation on the 4F3/2-4I9/2 transitions of Nd:YVO4 and Nd:YAG,” Opt. Lett. 25, 34-36 (2000).
    [CrossRef]
  17. M. J. Damzen, M. Trew, E. Rosas, and G. J. Crofts, “Continuous-wave Nd:YVO4 grazing-incidence laser with 22.5 W output power and 64% conversion efficiency,” Opt. Commun. 196, 237-241 (2001).
    [CrossRef]
  18. Y. F. Chen and S. W. Tsai, “Diode-pumped Q-switched Nd:YVO4 yellow laser with intracavity sum-frequency mixing,” Opt. Lett. 27, 397-399 (2002).
    [CrossRef]
  19. Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979-986 (2003).
    [CrossRef]
  20. G. Arfken, Mathematical Methods for Physicists (Academic, 1988).
  21. 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).

2008 (1)

M. Sabaeian and H. Nadgaran, “Bessel-Gauss beams: investigations of thermal effects on their generation,” Opt. Commun. 281, 672-678 (2008).

2007 (2)

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]

P. Shi, W. Chen, L. Li, and A. Gan, “Semianalytical thermal analysis on a Nd:YVO4 crystal,” Appl. Opt. 46, 4046-4051 (2007).

2006 (2)

H. Nadgaran and M. Sabaian, “Pulsed pump: thermal effects in solid state lasers under super-Gaussian pulses,” Pramana J. Phys. 67, 1119-1128 (2006).

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 J. Phys. 66, 513-519 (2006).

2003 (1)

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979-986 (2003).
[CrossRef]

2002 (3)

Y. F. Chen and S. W. Tsai, “Diode-pumped Q-switched Nd:YVO4 yellow laser with intracavity sum-frequency mixing,” Opt. Lett. 27, 397-399 (2002).
[CrossRef]

H. Zhang, J. Liu, J. Wang, Ch. Wang, L. Zhu, Z. Shao, X. Meng, X. Hu, and M. Jiang, “Characterization of the laser crystal Nd:GdVO4,” J. Opt. Soc. Am. B 19, 18-27 (2002).
[CrossRef]

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser material: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253-258 (2002).

2001 (4)

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]

M. J. Damzen, M. Trew, E. Rosas, and G. J. Crofts, “Continuous-wave Nd:YVO4 grazing-incidence laser with 22.5 W output power and 64% conversion efficiency,” Opt. Commun. 196, 237-241 (2001).
[CrossRef]

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state laser,” J. Phys. D 34, 2381-2395 (2001).

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).

2000 (2)

1998 (2)

D. C. Brown, “Heat, fluorescence, and stimulated-emission power densities and fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560-572 (1998).
[CrossRef]

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-inversion densities in Nd:YAG: upconversion and bleaching,” IEEE J. Quantum Electron. 34, 900-909 (1998).
[CrossRef]

1994 (1)

C. Pfistner, 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 J. Quantum Electron. 30, 1605-1615 (1994).

1992 (1)

A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057-1069 (1992).
[CrossRef]

1988 (1)

G. Arfken, Mathematical Methods for Physicists (Academic, 1988).

1987 (1)

R. A. Fielfs, M. Birnbaun, and C. N. Fincher, “High efficiency Nd:YVO4 diode-laser end-pumped laser,” Appl. Phys. Lett. 51, 1885-1886 (1987).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, 1988).

Asundi, A.

Birnbaun, M.

R. A. Fielfs, M. Birnbaun, and C. N. Fincher, “High efficiency Nd:YVO4 diode-laser end-pumped laser,” Appl. Phys. Lett. 51, 1885-1886 (1987).

Bonner, C. L.

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-inversion densities in Nd:YAG: upconversion and bleaching,” IEEE J. Quantum Electron. 34, 900-909 (1998).
[CrossRef]

Brown, D. C.

D. C. Brown, “Heat, fluorescence, and stimulated-emission power densities and fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560-572 (1998).
[CrossRef]

Chen, W.

Chen, Y.

Chen, Y. F.

Clarkson, W. A.

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state laser,” J. Phys. D 34, 2381-2395 (2001).

Cousins, A. K.

A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057-1069 (1992).
[CrossRef]

Crofts, G. J.

M. J. Damzen, M. Trew, E. Rosas, and G. J. Crofts, “Continuous-wave Nd:YVO4 grazing-incidence laser with 22.5 W output power and 64% conversion efficiency,” Opt. Commun. 196, 237-241 (2001).
[CrossRef]

Damzen, M. J.

M. J. Damzen, M. Trew, E. Rosas, and G. J. Crofts, “Continuous-wave Nd:YVO4 grazing-incidence laser with 22.5 W output power and 64% conversion efficiency,” Opt. Commun. 196, 237-241 (2001).
[CrossRef]

Du, K.

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]

Elahi, P.

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 J. Phys. 66, 513-519 (2006).

Ferrand, B.

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-inversion densities in Nd:YAG: upconversion and bleaching,” IEEE J. Quantum Electron. 34, 900-909 (1998).
[CrossRef]

Fielfs, R. A.

R. A. Fielfs, M. Birnbaun, and C. N. Fincher, “High efficiency Nd:YVO4 diode-laser end-pumped laser,” Appl. Phys. Lett. 51, 1885-1886 (1987).

Fincher, C. N.

R. A. Fielfs, M. Birnbaun, and C. N. Fincher, “High efficiency Nd:YVO4 diode-laser end-pumped laser,” Appl. Phys. Lett. 51, 1885-1886 (1987).

Gan, A.

Gao, J.

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]

George, J.

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser material: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253-258 (2002).

Graf, Th.

Gruber, R.

C. Pfistner, 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 J. Quantum Electron. 30, 1605-1615 (1994).

Guy, S.

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-inversion densities in Nd:YAG: upconversion and bleaching,” IEEE J. Quantum Electron. 34, 900-909 (1998).
[CrossRef]

Hanna, D. C.

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-inversion densities in Nd:YAG: upconversion and bleaching,” IEEE J. Quantum Electron. 34, 900-909 (1998).
[CrossRef]

Hu, X.

Huang, W. L.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979-986 (2003).
[CrossRef]

Jiang, M.

Li, D.

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]

Li, L.

Li, Z. G.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979-986 (2003).
[CrossRef]

Lim, G. C.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979-986 (2003).
[CrossRef]

Liu, J.

Ma, Z.

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]

Meng, X.

Merazzi, S.

C. Pfistner, 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 J. Quantum Electron. 30, 1605-1615 (1994).

Moore, N.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979-986 (2003).
[CrossRef]

Mukhopadhyay, P. K.

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser material: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253-258 (2002).

Nadgaran, H.

M. Sabaeian and H. Nadgaran, “Bessel-Gauss beams: investigations of thermal effects on their generation,” Opt. Commun. 281, 672-678 (2008).

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 J. Phys. 66, 513-519 (2006).

H. Nadgaran and M. Sabaian, “Pulsed pump: thermal effects in solid state lasers under super-Gaussian pulses,” Pramana J. Phys. 67, 1119-1128 (2006).

Nathan, T. P. S.

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser material: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253-258 (2002).

Peng, X.

Peuser, P.

Pfistner, C.

C. Pfistner, 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 J. Quantum Electron. 30, 1605-1615 (1994).

Pigeon, F.

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]

Ranganathan, K.

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser material: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253-258 (2002).

Rosas, E.

M. J. Damzen, M. Trew, E. Rosas, and G. J. Crofts, “Continuous-wave Nd:YVO4 grazing-incidence laser with 22.5 W output power and 64% conversion efficiency,” Opt. Commun. 196, 237-241 (2001).
[CrossRef]

Sabaeian, M.

M. Sabaeian and H. Nadgaran, “Bessel-Gauss beams: investigations of thermal effects on their generation,” Opt. Commun. 281, 672-678 (2008).

Sabaian, M.

H. Nadgaran and M. Sabaian, “Pulsed pump: thermal effects in solid state lasers under super-Gaussian pulses,” Pramana J. Phys. 67, 1119-1128 (2006).

Schmid, M.

Shao, Z.

Sharma, S. K.

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser material: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253-258 (2002).

Shepherd, D. P.

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-inversion densities in Nd:YAG: upconversion and bleaching,” IEEE J. Quantum Electron. 34, 900-909 (1998).
[CrossRef]

Shi, P.

Sychugov, V. A.

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]

Tishchenko, A.

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]

Trew, M.

M. J. Damzen, M. Trew, E. Rosas, and G. J. Crofts, “Continuous-wave Nd:YVO4 grazing-incidence laser with 22.5 W output power and 64% conversion efficiency,” Opt. Commun. 196, 237-241 (2001).
[CrossRef]

Tropper, A. C.

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-inversion densities in Nd:YAG: upconversion and bleaching,” IEEE J. Quantum Electron. 34, 900-909 (1998).
[CrossRef]

Tsai, S. W.

Usievich, B. A.

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]

Wang, Ch.

Wang, J.

Weber, H. P.

M. Schmid, Th. 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]

C. Pfistner, 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 J. Quantum Electron. 30, 1605-1615 (1994).

Weber, R.

C. Pfistner, 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 J. Quantum Electron. 30, 1605-1615 (1994).

Wu, N.

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]

Xiong, Z.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979-986 (2003).
[CrossRef]

Xiong, Zh.

Zeller, P.

Zhang, H.

Zhu, L.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

R. A. Fielfs, M. Birnbaun, and C. N. Fincher, “High efficiency Nd:YVO4 diode-laser end-pumped laser,” Appl. Phys. Lett. 51, 1885-1886 (1987).

IEEE J. Quantum Electron. (6)

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]

D. C. Brown, “Heat, fluorescence, and stimulated-emission power densities and fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560-572 (1998).
[CrossRef]

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-inversion densities in Nd:YAG: upconversion and bleaching,” IEEE J. Quantum Electron. 34, 900-909 (1998).
[CrossRef]

C. Pfistner, 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 J. Quantum Electron. 30, 1605-1615 (1994).

A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057-1069 (1992).
[CrossRef]

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979-986 (2003).
[CrossRef]

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

J. Phys. D (1)

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state laser,” J. Phys. D 34, 2381-2395 (2001).

Opt. Commun. (3)

M. Sabaeian and H. Nadgaran, “Bessel-Gauss beams: investigations of thermal effects on their generation,” Opt. Commun. 281, 672-678 (2008).

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]

M. J. Damzen, M. Trew, E. Rosas, and G. J. Crofts, “Continuous-wave Nd:YVO4 grazing-incidence laser with 22.5 W output power and 64% conversion efficiency,” Opt. Commun. 196, 237-241 (2001).
[CrossRef]

Opt. Laser Technol. (1)

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser material: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253-258 (2002).

Opt. Lett. (2)

Pramana J. Phys. (2)

H. Nadgaran and M. Sabaian, “Pulsed pump: thermal effects in solid state lasers under super-Gaussian pulses,” Pramana J. Phys. 67, 1119-1128 (2006).

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 J. Phys. 66, 513-519 (2006).

Other (1)

G. Arfken, Mathematical Methods for Physicists (Academic, 1988).

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Figures (7)

Fig. 1
Fig. 1

Schematic of cubic geometry crystal.

Fig. 2
Fig. 2

Three-dimensional temperature distribution computed by Eq. (8) in the X Y plane at Z = 0 with m , n = 1 , 3 , 5 , 7 , 9 .

Fig. 3
Fig. 3

Temperature distribution at Z = 0 and Z = c / 2 planes along X axis for P = 15 W absorbed pump power. Solid curves show our analytical solution with m , n = 1 , 3 , 5 , 7 , 9 ; dotted curves show the numerical solution.

Fig. 4
Fig. 4

Temperature distribution along the crystal axis at X = a / 2 and Y = b / 2 . The solid curve shows the analytical solution with m , n = 1 , 3 , 5 , 7 , 9 ; the dotted curve shows the numerical calculation.

Fig. 5
Fig. 5

Temperature distribution along the x direction in the planes z = 0 and z = c / 2 at y = b / 2 , calculated by Eq. (19).

Fig. 6
Fig. 6

Three-dimensional temperature distribution of Eq. (30) in the X Z plane at y = b / 2 with convection cooling boundary conditions for all six faces.

Fig. 7
Fig. 7

Comparison between the analytical (solid) and numerical (dotted) temperature along the X direction in the X Y planes z = 0 and z = c / 2 at y = b / 2 , obtained by Eq. (30).

Equations (52)

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

K x 2 T ( x , y , z ) x 2 + K y 2 T ( x , y , z ) y 2 + K z 2 T ( x , y , z ) z 2 = S ( x , y , z ) ,
S ( x , y , z ) = Q 0 exp { 2 [ ( x a 2 ) 2 + ( y b 2 ) 2 ] / ω p 2 } exp ( γ z )
Q 0 = 2 η P π ω p 2 ( 1 e γ c ) e r f ( a 2 2 ω p ) e r f ( b 2 2 ω p ) ,
K z T ( x , y , z ) z | z = 0 = h a [ T ( x , y , z = 0 ) T 0 ]
K z T ( x , y , z ) z | z = c = h a [ T ( x , y , z = c ) T 0 ]
T ( x = 0 , y , z ) T 0 = T ( x = a , y , z ) T 0 = 0 ,
T ( x , y = 0 , z ) T 0 = T ( x , y = b , z ) T 0 = 0.
T ( x , y , z ) = m , n = 1 T m n ( z ) sin ( m π x a ) sin ( n π y b ) + T 0 ,
d 2 T m n ( z ) d z 2 Γ m n 2 T m n ( z ) = 4 Q 0 a b K z p m q n e γ z ,
Γ m n 2 = K x K z ( m π a ) 2 + K y K z ( n π b ) 2 ,
p m = 0 a e 2 ( x a / 2 ) 2 / ω p 2 sin ( m π x / a ) d x = 2 π ω p 2 e m 2 π 2 ω p 2 / 8 a 2 sin ( m π 2 ) Re { erf ( 2 a 2 ω p + i 2 m π ω p 4 a ) } ,
q n = 0 b e 2 ( y b / 2 ) 2 / ω p 2 sin ( n π y / b ) d y = 2 π ω p 2 e n 2 π 2 ω p 2 / 8 b 2 sin ( n π 2 ) Re { erf ( 2 b 2 ω p + i 2 n π ω p 4 b ) } .
T m n ( z ) = A m n + e Γ m n z + A m n e Γ m n z B m n e γ z ,
B m n = 4 Q 0 p m q n a b K z ( γ 2 Γ m n 2 ) .
K z d T m n d z | z = 0 h a T m n ( z = 0 ) = 0 ,
K z d T m n d z | z = c + h a T m n ( z = c ) = 0.
T ( x = 0 , y , z ) T 1 = T ( x = a , y , z ) T 1 = 0 ,
T ( x , y = 0 , z ) T 1 = T ( x , y = b , z ) T 1 = 0 ,
T ( x , y , z ) = m , n = 1 T m n ( z ) sin ( m π x a ) sin ( n π y b ) + T 1 .
m , n = 1 { sin ( m π x a ) sin ( n π y b ) ( K z d T m n d z h a T m n ) | z = 0 } = h a Δ T ,
m , n = 1 { sin ( m π x a ) sin ( n π y b ) ( K d T m n d z h T m n ) | z = c } = h a Δ T ,
K z d T m n d z | z = 0 h a T m n ( z = 0 ) = { 16 h a m n π 2 Δ T n , m = odd 0 others ,
K z d T m n d z | z = c + h a T m n ( z = c ) = { 16 h a m n π 2 Δ T n , m = odd 0 others .
K x T x | x = 0 = h s [ T ( x , y , z ) T 0 ] x = 0 ,
- K x T x | x = a = h s [ T ( x , y , z ) T 0 ] x = a ,
K y T y | y = 0 = h s [ T ( x , y , z ) T 0 ] y = 0 ,
- K y T y | y = b = h s [ T ( x , y , z ) T 0 ] y = b ,
K z T z | z = 0 = h a [ T ( x , y , z ) T 0 ] z = 0 ,
- K z T z | z = c = h a [ T ( x , y , z ) T 0 ] z = c .
T ( x , y , z ) = m , n = 1 T m n ( z ) sin ( α m x a + β m ) sin ( δ n y b + θ n ) + T 0 ,
tan ( β m ) = K x α m h s a ,
tan ( α m + β m ) = K x α m h s a .
2 cot ( α m ) = K x α m h s a h s a K x α m .
β m = tan 1 ( K x α m h s a ) .
tan ( δ n ) = K y δ n h s b ,
tan ( δ n + θ n ) = K y δ n h s b .
d 2 T m n ( z ) d 2 z Γ m n 2 T m n ( z ) = 16 Q 0 K z a b s m t n e γ z ,
Γ m n 2 = K x K z ( α m a ) 2 + K y K z ( δ n b ) 2 ,
s m = 0 a exp [ 2 ( x a / 2 ) 2 / ω p 2 ) ] sin [ ( α m x / a ) + β m ] d x 2 [ sin ( 2 α m + 2 β m ) sin ( 2 β m ) ] / α m = π / 2 ω p e α m 2 ω p 2 / 8 a 2 sin ( β m + α m / 2 ) Re [ erf ( 2 a 2 ω p + i 2 α m ω p 4 a ) ] 2 [ sin ( 2 α m + 2 β m ) sin ( 2 β m ) ] / α m ,
t n = 0 a exp [ 2 ( y b / 2 ) 2 / ω p 2 ) ] sin [ ( δ n y / b ) + θ n ] d y 2 [ sin ( 2 δ n + 2 θ n ) sin ( 2 θ n ) ] / δ n = π / 2 ω p e δ n 2 ω p 2 / 8 b 2 sin ( θ n + δ n / 2 ) Re [ erf ( 2 b 2 ω p + i 2 δ n ω p 4 b ) ] 2 [ sin ( 2 δ n + 2 θ n ) sin ( 2 θ n ) ] / δ n .
T m n ( z ) = A m n + e Γ m n z + A m n e Γ m n z B m n e γ z ,
B m n = 16 Q 0 s m t n a b K z ( γ 2 Γ m n 2 ) .
A m n ± = B m n ( K z Γ m n ± h a ) ( K z γ h a ) e γ c ( K z Γ m n h a ) ( K z γ + h a ) e Γ m n c ( K z Γ m n h a ) 2 e Γ m n c ( K z Γ m n + h a ) 2 e Γ m n c .
A m n ± = B m n ( K z Γ m n ± h a ) ( K z γ h a ) e γ c ( K z Γ m n h a ) ( K z γ + h a ) e Γ m n c ( K z Γ m n h a ) 2 e Γ m n c ( K z Γ m n + h a ) 2 e Γ m n c + 16 h Δ T m n π 2 ( K z Γ m n ± h a ) + ( K z Γ m n h a ) e Γ m n c ( K z Γ m n h a ) 2 e Γ m n c ( K z Γ m n + h a ) 2 e Γ m n c .
A m n ± = B m n ( K z Γ m n ± h a ) ( K z γ h a ) e γ c ( K z Γ m n h a ) ( K z γ + h a ) e Γ m n c ( K z Γ m n h a ) 2 e Γ m n c ( K z Γ m n + h a ) 2 e Γ m n c .
m , n = 1 [ d 2 T m n ( z ) d z 2 Γ m n 2 T m n ( z ) ] sin ( α m x a + β m ) sin ( δ n y b + θ n ) = S ( x , y , z ) ,
Γ m n 2 = K x K z ( α m a ) 2 + K y K z ( δ n b ) 2
m , n = 1 [ d 2 T m n ( z ) d z 2 Γ m n 2 T m n ( z ) ] [ 0 a sin ( α m x a + β m ) sin ( α m x a + β m ) d x ] [ 0 a sin ( δ n y b + θ n ) sin ( δ n y b + θ n ) d y ] = 16 β K z a b s m t n e γ z .
0 a sin ( α m x a + β m ) sin ( α m x a + β m ) d x = a 2 a 4 α [ sin ( 2 α m + 2 β m ) sin ( 2 β m ) ]
0 a sin ( α m x a + β m ) sin ( α m x a + β m ) d x = a 2 ( α m α m ) { sin [ ( α m + β m ) ( α m + β m ) ] sin ( β m + β m ) } = a 2 ( α m α m ) [ sin ( α m + β m ) cos ( α m + β m ) cos ( α m + β m ) sin ( α m + β m ) sin ( β m ) cos ( β m ) cos ( β m ) sin ( β m ) ]
sin ( β m ) = K x α m h s a cos ( β m ) ,
sin ( α m + β m ) = K x α m h s a cos ( α m + β m )

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