Abstract

Based on analytical theory of anisotropy, distributions of thermal distortion and temperature field within a diode-end-pumped rectangular Nd:YVO4 laser crystal are investigated. A thermal model that matches the actual working state of the laser crystal is established by analyzing the working characteristics of the Nd:YVO4 laser crystal. A novel method, to the best of our knowledge, is adopted to solve the heat conduction equation of the anisotropic medium. General solutions of the temperature field, thermal strain field, and thermal distortion field of the Nd:YVO4 crystal are obtained. The effect of anisotropic thermal parameters on the thermal strain field of the Nd:YVO4 laser crystal is also analyzed quantitatively. Research results show that a maximum temperature rise of 244.9°C and a maximum thermal distortion of 1.99  μm can be obtained in the center of the pump face when the Nd:YVO4 laser crystal doped with 0.5 at. % Nd3+ is diode end pumped in the center of the front end face with 15 W output power. This method can be applied to other thermal analyses of laser crystals and offers a theoretical basis to solve thermal problems effectively in the laser system.

© 2007 Optical Society of America

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References

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  1. Y. G. Wang, X. Y. Ma, J. Y. Peng, H. M. Tan, and L. S. Qian, "High-repetition rate Q-switched Nd:YVO4 laser with a composite semiconductor absorber," Appl. Opt. 45, 6616-6619 (2006).
    [CrossRef] [PubMed]
  2. B. A. Thompson, A. Minassian, and M. J. Damzen, "Unidirectional single-frequency operation of a Nd:YVO4 ring laser with and without a Faraday element," Appl. Opt. 43, 3174-3177 (2004).
    [CrossRef] [PubMed]
  3. S. A. Amarande and M. J. Damzen, "Measurement of the thermal lens of grazing-incidence diode-pumped Nd:YVO4 laser amplifier," Opt. Commun. 265, 306-313 (2006).
    [CrossRef]
  4. Y. Zhang, M. Y. Gao, and J. Q. Yao, "Study on thermal effect of anisotropic laser medium by LD end-pumped," J. Optoelectron. Laser 14, 1094-1098 (2003) (in Chinese).
  5. C. Li, J. Song, D. Shen, N. S. Kim, J. Lu, and K. Ueda, "Diode-pumped passively Q-switched Nd:YVO4 lasers operating at 1.06 μm wavelength," Appl. Phys. B 70, 471-474 (2000).
    [CrossRef]
  6. P. Shi, L. Li, X. F. Liu, A. S. Gan, and Z. W. Hua, "A influence of eccentricity on thermal effect of Nd:GdVO4 crystal with rectangle-section," J. Optoelectron. Laser 16, 1187-1192 (2005) (in Chinese).
  7. P. Shi, L. Li, A. S. Gan, Y. W. Ling, X. F. Liu, and J. T. Bai, "Thermal effect research of end-pumped rectangle Nd:GdVO4 crystals," Chin. J. Lasers 32, 923-928 (2005) (in Chinese).
  8. J. H. Liu, C. L. Du, Z. P. Wang, L. Zhu, H. J. Zhang, X. L. Meng, and J. Y. Wang, "Diode-pumped Q-switched Nd:YVO4/KTP green laser formed with a flat-flat resonator," Opt. Laser Technol. 33, 177-180 (2001).
    [CrossRef]
  9. J. H. Liu, Z. S. Shao, H. J. Zhang, X. L. Xian, L. Zhu, J. Y. Wang, Y. G. Liu, and M. H. Jiang, "Diode-laser array end-pumped intracavity frequency-doubled 3.6 W CW Nd:YVO4/KTP green laser," Opt. Commun. 173, 311-314 (2000).
    [CrossRef]
  10. M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, "Thermal modeling of continuous-wave end-pumped solid state lasers," Appl. Phys. Lett. 56, 1831-1833 (1990).
    [CrossRef]
  11. B. X. Wang, Engineering Heat and Mass Transfer(Science Press, 1982), pp. 35-37.
  12. X. Y. Peng, A. Asundi, Y. H. Chen, and Z. J. 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]

2006 (2)

S. A. Amarande and M. J. Damzen, "Measurement of the thermal lens of grazing-incidence diode-pumped Nd:YVO4 laser amplifier," Opt. Commun. 265, 306-313 (2006).
[CrossRef]

Y. G. Wang, X. Y. Ma, J. Y. Peng, H. M. Tan, and L. S. Qian, "High-repetition rate Q-switched Nd:YVO4 laser with a composite semiconductor absorber," Appl. Opt. 45, 6616-6619 (2006).
[CrossRef] [PubMed]

2005 (2)

P. Shi, L. Li, X. F. Liu, A. S. Gan, and Z. W. Hua, "A influence of eccentricity on thermal effect of Nd:GdVO4 crystal with rectangle-section," J. Optoelectron. Laser 16, 1187-1192 (2005) (in Chinese).

P. Shi, L. Li, A. S. Gan, Y. W. Ling, X. F. Liu, and J. T. Bai, "Thermal effect research of end-pumped rectangle Nd:GdVO4 crystals," Chin. J. Lasers 32, 923-928 (2005) (in Chinese).

2004 (1)

2003 (1)

Y. Zhang, M. Y. Gao, and J. Q. Yao, "Study on thermal effect of anisotropic laser medium by LD end-pumped," J. Optoelectron. Laser 14, 1094-1098 (2003) (in Chinese).

2001 (2)

X. Y. Peng, A. Asundi, Y. H. Chen, and Z. J. 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]

J. H. Liu, C. L. Du, Z. P. Wang, L. Zhu, H. J. Zhang, X. L. Meng, and J. Y. Wang, "Diode-pumped Q-switched Nd:YVO4/KTP green laser formed with a flat-flat resonator," Opt. Laser Technol. 33, 177-180 (2001).
[CrossRef]

2000 (2)

J. H. Liu, Z. S. Shao, H. J. Zhang, X. L. Xian, L. Zhu, J. Y. Wang, Y. G. Liu, and M. H. Jiang, "Diode-laser array end-pumped intracavity frequency-doubled 3.6 W CW Nd:YVO4/KTP green laser," Opt. Commun. 173, 311-314 (2000).
[CrossRef]

C. Li, J. Song, D. Shen, N. S. Kim, J. Lu, and K. Ueda, "Diode-pumped passively Q-switched Nd:YVO4 lasers operating at 1.06 μm wavelength," Appl. Phys. B 70, 471-474 (2000).
[CrossRef]

1990 (1)

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, "Thermal modeling of continuous-wave end-pumped solid state lasers," Appl. Phys. Lett. 56, 1831-1833 (1990).
[CrossRef]

Amarande, S. A.

S. A. Amarande and M. J. Damzen, "Measurement of the thermal lens of grazing-incidence diode-pumped Nd:YVO4 laser amplifier," Opt. Commun. 265, 306-313 (2006).
[CrossRef]

Asundi, A.

Bai, J. T.

P. Shi, L. Li, A. S. Gan, Y. W. Ling, X. F. Liu, and J. T. Bai, "Thermal effect research of end-pumped rectangle Nd:GdVO4 crystals," Chin. J. Lasers 32, 923-928 (2005) (in Chinese).

Chen, Y. H.

Damzen, M. J.

S. A. Amarande and M. J. Damzen, "Measurement of the thermal lens of grazing-incidence diode-pumped Nd:YVO4 laser amplifier," Opt. Commun. 265, 306-313 (2006).
[CrossRef]

B. A. Thompson, A. Minassian, and M. J. Damzen, "Unidirectional single-frequency operation of a Nd:YVO4 ring laser with and without a Faraday element," Appl. Opt. 43, 3174-3177 (2004).
[CrossRef] [PubMed]

Du, C. L.

J. H. Liu, C. L. Du, Z. P. Wang, L. Zhu, H. J. Zhang, X. L. Meng, and J. Y. Wang, "Diode-pumped Q-switched Nd:YVO4/KTP green laser formed with a flat-flat resonator," Opt. Laser Technol. 33, 177-180 (2001).
[CrossRef]

Fields, R. A.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, "Thermal modeling of continuous-wave end-pumped solid state lasers," Appl. Phys. Lett. 56, 1831-1833 (1990).
[CrossRef]

Fincher, C. L.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, "Thermal modeling of continuous-wave end-pumped solid state lasers," Appl. Phys. Lett. 56, 1831-1833 (1990).
[CrossRef]

Gan, A. S.

P. Shi, L. Li, X. F. Liu, A. S. Gan, and Z. W. Hua, "A influence of eccentricity on thermal effect of Nd:GdVO4 crystal with rectangle-section," J. Optoelectron. Laser 16, 1187-1192 (2005) (in Chinese).

P. Shi, L. Li, A. S. Gan, Y. W. Ling, X. F. Liu, and J. T. Bai, "Thermal effect research of end-pumped rectangle Nd:GdVO4 crystals," Chin. J. Lasers 32, 923-928 (2005) (in Chinese).

Gao, M. Y.

Y. Zhang, M. Y. Gao, and J. Q. Yao, "Study on thermal effect of anisotropic laser medium by LD end-pumped," J. Optoelectron. Laser 14, 1094-1098 (2003) (in Chinese).

Hua, Z. W.

P. Shi, L. Li, X. F. Liu, A. S. Gan, and Z. W. Hua, "A influence of eccentricity on thermal effect of Nd:GdVO4 crystal with rectangle-section," J. Optoelectron. Laser 16, 1187-1192 (2005) (in Chinese).

Innocenzi, M. E.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, "Thermal modeling of continuous-wave end-pumped solid state lasers," Appl. Phys. Lett. 56, 1831-1833 (1990).
[CrossRef]

Jiang, M. H.

J. H. Liu, Z. S. Shao, H. J. Zhang, X. L. Xian, L. Zhu, J. Y. Wang, Y. G. Liu, and M. H. Jiang, "Diode-laser array end-pumped intracavity frequency-doubled 3.6 W CW Nd:YVO4/KTP green laser," Opt. Commun. 173, 311-314 (2000).
[CrossRef]

Kim, N. S.

C. Li, J. Song, D. Shen, N. S. Kim, J. Lu, and K. Ueda, "Diode-pumped passively Q-switched Nd:YVO4 lasers operating at 1.06 μm wavelength," Appl. Phys. B 70, 471-474 (2000).
[CrossRef]

Li, C.

C. Li, J. Song, D. Shen, N. S. Kim, J. Lu, and K. Ueda, "Diode-pumped passively Q-switched Nd:YVO4 lasers operating at 1.06 μm wavelength," Appl. Phys. B 70, 471-474 (2000).
[CrossRef]

Li, L.

P. Shi, L. Li, A. S. Gan, Y. W. Ling, X. F. Liu, and J. T. Bai, "Thermal effect research of end-pumped rectangle Nd:GdVO4 crystals," Chin. J. Lasers 32, 923-928 (2005) (in Chinese).

P. Shi, L. Li, X. F. Liu, A. S. Gan, and Z. W. Hua, "A influence of eccentricity on thermal effect of Nd:GdVO4 crystal with rectangle-section," J. Optoelectron. Laser 16, 1187-1192 (2005) (in Chinese).

Ling, Y. W.

P. Shi, L. Li, A. S. Gan, Y. W. Ling, X. F. Liu, and J. T. Bai, "Thermal effect research of end-pumped rectangle Nd:GdVO4 crystals," Chin. J. Lasers 32, 923-928 (2005) (in Chinese).

Liu, J. H.

J. H. Liu, C. L. Du, Z. P. Wang, L. Zhu, H. J. Zhang, X. L. Meng, and J. Y. Wang, "Diode-pumped Q-switched Nd:YVO4/KTP green laser formed with a flat-flat resonator," Opt. Laser Technol. 33, 177-180 (2001).
[CrossRef]

J. H. Liu, Z. S. Shao, H. J. Zhang, X. L. Xian, L. Zhu, J. Y. Wang, Y. G. Liu, and M. H. Jiang, "Diode-laser array end-pumped intracavity frequency-doubled 3.6 W CW Nd:YVO4/KTP green laser," Opt. Commun. 173, 311-314 (2000).
[CrossRef]

Liu, X. F.

P. Shi, L. Li, A. S. Gan, Y. W. Ling, X. F. Liu, and J. T. Bai, "Thermal effect research of end-pumped rectangle Nd:GdVO4 crystals," Chin. J. Lasers 32, 923-928 (2005) (in Chinese).

P. Shi, L. Li, X. F. Liu, A. S. Gan, and Z. W. Hua, "A influence of eccentricity on thermal effect of Nd:GdVO4 crystal with rectangle-section," J. Optoelectron. Laser 16, 1187-1192 (2005) (in Chinese).

Liu, Y. G.

J. H. Liu, Z. S. Shao, H. J. Zhang, X. L. Xian, L. Zhu, J. Y. Wang, Y. G. Liu, and M. H. Jiang, "Diode-laser array end-pumped intracavity frequency-doubled 3.6 W CW Nd:YVO4/KTP green laser," Opt. Commun. 173, 311-314 (2000).
[CrossRef]

Lu, J.

C. Li, J. Song, D. Shen, N. S. Kim, J. Lu, and K. Ueda, "Diode-pumped passively Q-switched Nd:YVO4 lasers operating at 1.06 μm wavelength," Appl. Phys. B 70, 471-474 (2000).
[CrossRef]

Ma, X. Y.

Meng, X. L.

J. H. Liu, C. L. Du, Z. P. Wang, L. Zhu, H. J. Zhang, X. L. Meng, and J. Y. Wang, "Diode-pumped Q-switched Nd:YVO4/KTP green laser formed with a flat-flat resonator," Opt. Laser Technol. 33, 177-180 (2001).
[CrossRef]

Minassian, A.

Peng, J. Y.

Peng, X. Y.

Qian, L. S.

Shao, Z. S.

J. H. Liu, Z. S. Shao, H. J. Zhang, X. L. Xian, L. Zhu, J. Y. Wang, Y. G. Liu, and M. H. Jiang, "Diode-laser array end-pumped intracavity frequency-doubled 3.6 W CW Nd:YVO4/KTP green laser," Opt. Commun. 173, 311-314 (2000).
[CrossRef]

Shen, D.

C. Li, J. Song, D. Shen, N. S. Kim, J. Lu, and K. Ueda, "Diode-pumped passively Q-switched Nd:YVO4 lasers operating at 1.06 μm wavelength," Appl. Phys. B 70, 471-474 (2000).
[CrossRef]

Shi, P.

P. Shi, L. Li, A. S. Gan, Y. W. Ling, X. F. Liu, and J. T. Bai, "Thermal effect research of end-pumped rectangle Nd:GdVO4 crystals," Chin. J. Lasers 32, 923-928 (2005) (in Chinese).

P. Shi, L. Li, X. F. Liu, A. S. Gan, and Z. W. Hua, "A influence of eccentricity on thermal effect of Nd:GdVO4 crystal with rectangle-section," J. Optoelectron. Laser 16, 1187-1192 (2005) (in Chinese).

Song, J.

C. Li, J. Song, D. Shen, N. S. Kim, J. Lu, and K. Ueda, "Diode-pumped passively Q-switched Nd:YVO4 lasers operating at 1.06 μm wavelength," Appl. Phys. B 70, 471-474 (2000).
[CrossRef]

Tan, H. M.

Thompson, B. A.

Ueda, K.

C. Li, J. Song, D. Shen, N. S. Kim, J. Lu, and K. Ueda, "Diode-pumped passively Q-switched Nd:YVO4 lasers operating at 1.06 μm wavelength," Appl. Phys. B 70, 471-474 (2000).
[CrossRef]

Wang, B. X.

B. X. Wang, Engineering Heat and Mass Transfer(Science Press, 1982), pp. 35-37.

Wang, J. Y.

J. H. Liu, C. L. Du, Z. P. Wang, L. Zhu, H. J. Zhang, X. L. Meng, and J. Y. Wang, "Diode-pumped Q-switched Nd:YVO4/KTP green laser formed with a flat-flat resonator," Opt. Laser Technol. 33, 177-180 (2001).
[CrossRef]

J. H. Liu, Z. S. Shao, H. J. Zhang, X. L. Xian, L. Zhu, J. Y. Wang, Y. G. Liu, and M. H. Jiang, "Diode-laser array end-pumped intracavity frequency-doubled 3.6 W CW Nd:YVO4/KTP green laser," Opt. Commun. 173, 311-314 (2000).
[CrossRef]

Wang, Y. G.

Wang, Z. P.

J. H. Liu, C. L. Du, Z. P. Wang, L. Zhu, H. J. Zhang, X. L. Meng, and J. Y. Wang, "Diode-pumped Q-switched Nd:YVO4/KTP green laser formed with a flat-flat resonator," Opt. Laser Technol. 33, 177-180 (2001).
[CrossRef]

Xian, X. L.

J. H. Liu, Z. S. Shao, H. J. Zhang, X. L. Xian, L. Zhu, J. Y. Wang, Y. G. Liu, and M. H. Jiang, "Diode-laser array end-pumped intracavity frequency-doubled 3.6 W CW Nd:YVO4/KTP green laser," Opt. Commun. 173, 311-314 (2000).
[CrossRef]

Xiong, Z. J.

Yao, J. Q.

Y. Zhang, M. Y. Gao, and J. Q. Yao, "Study on thermal effect of anisotropic laser medium by LD end-pumped," J. Optoelectron. Laser 14, 1094-1098 (2003) (in Chinese).

Yura, H. T.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, "Thermal modeling of continuous-wave end-pumped solid state lasers," Appl. Phys. Lett. 56, 1831-1833 (1990).
[CrossRef]

Zhang, H. J.

J. H. Liu, C. L. Du, Z. P. Wang, L. Zhu, H. J. Zhang, X. L. Meng, and J. Y. Wang, "Diode-pumped Q-switched Nd:YVO4/KTP green laser formed with a flat-flat resonator," Opt. Laser Technol. 33, 177-180 (2001).
[CrossRef]

J. H. Liu, Z. S. Shao, H. J. Zhang, X. L. Xian, L. Zhu, J. Y. Wang, Y. G. Liu, and M. H. Jiang, "Diode-laser array end-pumped intracavity frequency-doubled 3.6 W CW Nd:YVO4/KTP green laser," Opt. Commun. 173, 311-314 (2000).
[CrossRef]

Zhang, Y.

Y. Zhang, M. Y. Gao, and J. Q. Yao, "Study on thermal effect of anisotropic laser medium by LD end-pumped," J. Optoelectron. Laser 14, 1094-1098 (2003) (in Chinese).

Zhu, L.

J. H. Liu, C. L. Du, Z. P. Wang, L. Zhu, H. J. Zhang, X. L. Meng, and J. Y. Wang, "Diode-pumped Q-switched Nd:YVO4/KTP green laser formed with a flat-flat resonator," Opt. Laser Technol. 33, 177-180 (2001).
[CrossRef]

J. H. Liu, Z. S. Shao, H. J. Zhang, X. L. Xian, L. Zhu, J. Y. Wang, Y. G. Liu, and M. H. Jiang, "Diode-laser array end-pumped intracavity frequency-doubled 3.6 W CW Nd:YVO4/KTP green laser," Opt. Commun. 173, 311-314 (2000).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (1)

C. Li, J. Song, D. Shen, N. S. Kim, J. Lu, and K. Ueda, "Diode-pumped passively Q-switched Nd:YVO4 lasers operating at 1.06 μm wavelength," Appl. Phys. B 70, 471-474 (2000).
[CrossRef]

Appl. Phys. Lett. (1)

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, "Thermal modeling of continuous-wave end-pumped solid state lasers," Appl. Phys. Lett. 56, 1831-1833 (1990).
[CrossRef]

Chin. J. Lasers (1)

P. Shi, L. Li, A. S. Gan, Y. W. Ling, X. F. Liu, and J. T. Bai, "Thermal effect research of end-pumped rectangle Nd:GdVO4 crystals," Chin. J. Lasers 32, 923-928 (2005) (in Chinese).

J. Optoelectron. Laser (2)

Y. Zhang, M. Y. Gao, and J. Q. Yao, "Study on thermal effect of anisotropic laser medium by LD end-pumped," J. Optoelectron. Laser 14, 1094-1098 (2003) (in Chinese).

P. Shi, L. Li, X. F. Liu, A. S. Gan, and Z. W. Hua, "A influence of eccentricity on thermal effect of Nd:GdVO4 crystal with rectangle-section," J. Optoelectron. Laser 16, 1187-1192 (2005) (in Chinese).

Opt. Commun. (2)

S. A. Amarande and M. J. Damzen, "Measurement of the thermal lens of grazing-incidence diode-pumped Nd:YVO4 laser amplifier," Opt. Commun. 265, 306-313 (2006).
[CrossRef]

J. H. Liu, Z. S. Shao, H. J. Zhang, X. L. Xian, L. Zhu, J. Y. Wang, Y. G. Liu, and M. H. Jiang, "Diode-laser array end-pumped intracavity frequency-doubled 3.6 W CW Nd:YVO4/KTP green laser," Opt. Commun. 173, 311-314 (2000).
[CrossRef]

Opt. Laser Technol. (1)

J. H. Liu, C. L. Du, Z. P. Wang, L. Zhu, H. J. Zhang, X. L. Meng, and J. Y. Wang, "Diode-pumped Q-switched Nd:YVO4/KTP green laser formed with a flat-flat resonator," Opt. Laser Technol. 33, 177-180 (2001).
[CrossRef]

Other (1)

B. X. Wang, Engineering Heat and Mass Transfer(Science Press, 1982), pp. 35-37.

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Figures (7)

Fig. 1
Fig. 1

Schematic of the diode laser end-pumped Nd:YVO4∕KTP laser.

Fig. 2
Fig. 2

Schematic of the laser crystal thermal model.

Fig. 3
Fig. 3

(Color online) Three-dimensional temperature field distribution diagram of the Nd:YVO4 crystal by the diode laser.

Fig. 4
Fig. 4

Temperature distribution diagram of the Nd:YVO4 crystal interior x = a / 2 , y = b / 2 .

Fig. 5
Fig. 5

(Color online) Thermal distortion field distribution diagram of the Nd:YVO4 crystal pumped face ( z = 0 , y = b / 2 ) .

Fig. 6
Fig. 6

(Color online) Three-dimensional thermal distortion field distribution diagram of the Nd:YVO4 crystal pumped face.

Fig. 7
Fig. 7

Thermal distortion strain distribution diagram of the Nd:YVO4 crystal ( z = 0 , y = b / 2 ) .

Equations (29)

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

I ( x , y , 0 ) = I 0 exp [ 2 ( x a 2 ) 2 + ( y b 2 ) 2 w 2 ] ,
I ( x , y , z ) = I ( x , y , 0 ) exp ( β z ) .
q v ( x , y , z ) = η I ( x , y , z ) = I 0 η exp ( 2 ( x a 2 ) 2 + ( y b 2 ) 2 w 2 ) × exp ( β z ) ,
η = 1 λ P / λ L ,
u ( 0 , y , z ) = 0 , u ( a , y , z ) = 0 ,
u ( x , 0 , z ) = 0 , u ( x , b , z ) = 0 ,
u z ( x , y , 0 ) = 0 , u z ( x , y , c ) = 0.
λ x 2 u ( x , y , z ) x 2 + λ y 2 u ( x , y , z ) y 2 + λ z 2 u ( x , y , z ) z 2 = q v ( x , y , z ) ,
u ( x , y , z ) = n = 1 m = 1 l = 0 A n m l sin n π a x sin m π b y cos l π c z ,
n = 1 m = 1 l = 0 B n m l sin n π a x sin m π b y cos l π c z = q v ,
B n m l = A n m l [ λ x ( n π a ) 2 + λ y ( m π b ) 2 + λ z ( l π c ) 2 ] .
0 a sin n π a x sin k π a x d x = a 2 δ n k .
m = 1 l = 0 B n m l sin m π b y cos l π c z = 2 a 0 a q v sin n π a x d x .
0 b sin m π b y sin k π b y d y = b 2 δ m k .
l = 0 B n m l cos l π c z = 4 a b 0 b 0 a q v sin n π a x sin m π b y d x d y .
0 c cos l π c z cos k π c z d z = c 2 δ l k .
B n m l = 8 a b c 0 c 0 b 0 a q v sin n π a x sin m π b y × cos l π c z d x d y d z .
q v ( x , y , z ) = I 0 η exp [ 2 ( x a 2 ) 2 + ( y b 2 ) 2 w 2 ] × exp ( β z )
B n m l = 8 I 0 β η a b c × 0 b 0 a exp ( 2 ( x a 2 ) 2 + ( y b 2 ) 2 w 2 ) × sin n π a x sin m π b y d x d y × 0 c exp ( β z ) cos l π c z d z .
0 c exp ( β z ) cos l π c z d z = β c 2 [ 1 exp ( β c ) cos l π ] β 2 c 2 + l 2 π 2 ,
A n m l = 8 I 0 β η c [ 1 exp ( β c ) cos l π ] a b π 2 ( β 2 c 2 + l 2 π 2 ) ( λ x n 2 a 2 + λ y m 2 b 2 + λ z l 2 c 2 ) × 0 b 0 a exp ( 2 ( x a 2 ) 2 + ( y b 2 ) 2 w 2 ) × sin n π a x sin m π b y d x d y .
I m = 0 I 0 exp ( 2 r 2 w 2 ) 2 π r d r .
I 0 = I m 2 π 0 exp ( 2 r 2 w 2 ) r d r .
d x = α x u ( x , y , z ) d x ,
d y = α y u ( x , y , z ) d y ,
d z = α z u ( x , y , z ) d z ,
x = 0 a d x = α x 0 a u ( x , y , z ) d x ,
y = 0 b d y = α y 0 b u ( x , y , z ) d y ,
z = 0 c d z = α z 0 c u ( x , y , z ) d z .

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