Abstract

Based on the theory of semianalytical thermal analysis, the temperature field and thermal lens effects within a diode-end-pumped Nd:YAG rods were investigated. A general expression of the temperature field within Nd:YAG rods was obtained through the analysis of a diode-end-pumped Nd:YAG rod, using what we believe to be a new method to solve a heat conduction equation of isotropic material. Calculating the thermal focal length within the diode-end-pumped Nd:YAG rods was done by an analysis of the additional optical path differences caused by heat, which was in good agreement with experimental results. These results show that the maximum temperature on the pump face of Nd:YAG rods is 51.9°C and the thermal focal length is 22 .4   cm when the output power of the diode laser is 10 W. Under the same conditions, the experimental value of the thermal focal length is 21 .0   cm. The difference between the theoretical analysis and the experimental result is only 6.7%. Results from this work can provide the theoretical basis for the optimized design of diode-end-pumped all-solid-state lasers.

© 2007 Optical Society of America

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  1. T. Y. Fan and R. L. Byer, "Diode laser pumped solid state lasers," IEEE J. Quantum Electron. 24, 895-912 (1988).
    [CrossRef]
  2. T. Y. Fan, "Heat generation in Nd:YAG and Nb:YAG," IEEE J. Quantum Electron. 29, 1457-1459 (1993).
    [CrossRef]
  3. A. K. Cousins, "Temperature and thermal stress scaling in finite-length end-pumped laser rods," IEEE J. Quantum Electron. 28, 1057-1069 (1992).
    [CrossRef]
  4. U. Brauch, "Temperature dependence of efficiency and thermal lensing of diode-laser pumped Nd:YAG laser," Appl. Phys. B 58, 397-402 (1994).
    [CrossRef]
  5. L. Li, P. Shi, and J. T. Bai, "Semi-analytical thermal analysis of single end-pumped laser crystal temperature distribution," J. Xi'an Jiaotong University 38, 369-372 (2004) (in Chinese).
  6. M. E. Innocenzi, H. T. Yura, C. L. Frucher, and R. A. Fields, "Thermal modeling of continuous-wave end-pumped solid state lasers," Appl. Phys. Lett. 56, 1831-1833 (1990).
    [CrossRef]
  7. Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
    [CrossRef]
  8. T. Graf, E. Wyss, M. Roth, and H. P. Weber, "Laser resonator with balanced thermal lenses," Opt. Commun. 190, 327-331 (2001).
    [CrossRef]
  9. B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, "Characterization of the heat loading of Nd-doped YAG, YOS, and GGG excited at diode pumping wavelengths," IEEE J. Quantum Electron. 29, 1457-1459 (1995).
  10. H. R. Yang and T. C. Zuo, "Investigation on thermal effects and output of all-solid-state miniature Yb:YAG laser," Acta Photonica Sin. 32, 907-910 (2003) (in Chinese).
  11. 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, 1605-1615 (1994).
    [CrossRef]
  12. Z. G. Li, Z. J. Xiong, W. L. Huang, T. Chen, N. Moore, and G. C. Lim, "Study of high power laser diode end-pumped composite crystal lasers," Chin. J. Lasers 32, 297-300 (2005) (in Chinese).
  13. X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, B. Ozygus, and H. Weber, "Study on thermal lens of Nd3+:YAG laser pumped by a laser diode," Chin. J. Lasers 27, 777-781 (2000) (in Chinese).
  14. Y. Yang, D. G. Xu, J. O. Yao, H. F. Li, Y. Y. Wang, and P. Wang, "Experimental research of an all-solid-state high power 1064 nm composite Nd:YAG laser," J. Optoelectron. Laser 17, 171-174 (2006) (in Chinese).

2006

Y. Yang, D. G. Xu, J. O. Yao, H. F. Li, Y. Y. Wang, and P. Wang, "Experimental research of an all-solid-state high power 1064 nm composite Nd:YAG laser," J. Optoelectron. Laser 17, 171-174 (2006) (in Chinese).

2005

Z. G. Li, Z. J. Xiong, W. L. Huang, T. Chen, N. Moore, and G. C. Lim, "Study of high power laser diode end-pumped composite crystal lasers," Chin. J. Lasers 32, 297-300 (2005) (in Chinese).

2004

L. Li, P. Shi, and J. T. Bai, "Semi-analytical thermal analysis of single end-pumped laser crystal temperature distribution," J. Xi'an Jiaotong University 38, 369-372 (2004) (in Chinese).

2003

H. R. Yang and T. C. Zuo, "Investigation on thermal effects and output of all-solid-state miniature Yb:YAG laser," Acta Photonica Sin. 32, 907-910 (2003) (in Chinese).

2001

T. Graf, E. Wyss, M. Roth, and H. P. Weber, "Laser resonator with balanced thermal lenses," Opt. Commun. 190, 327-331 (2001).
[CrossRef]

2000

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, B. Ozygus, and H. Weber, "Study on thermal lens of Nd3+:YAG laser pumped by a laser diode," Chin. J. Lasers 27, 777-781 (2000) (in Chinese).

1997

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[CrossRef]

1995

B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, "Characterization of the heat loading of Nd-doped YAG, YOS, and GGG excited at diode pumping wavelengths," IEEE J. Quantum Electron. 29, 1457-1459 (1995).

1994

U. Brauch, "Temperature dependence of efficiency and thermal lensing of diode-laser pumped Nd:YAG laser," Appl. Phys. B 58, 397-402 (1994).
[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, 1605-1615 (1994).
[CrossRef]

1993

T. Y. Fan, "Heat generation in Nd:YAG and Nb:YAG," IEEE J. Quantum Electron. 29, 1457-1459 (1993).
[CrossRef]

1992

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

1990

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

1988

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

Albrecht, G. F.

B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, "Characterization of the heat loading of Nd-doped YAG, YOS, and GGG excited at diode pumping wavelengths," IEEE J. Quantum Electron. 29, 1457-1459 (1995).

Bai, J. T.

L. Li, P. Shi, and J. T. Bai, "Semi-analytical thermal analysis of single end-pumped laser crystal temperature distribution," J. Xi'an Jiaotong University 38, 369-372 (2004) (in Chinese).

Beach, R. J.

B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, "Characterization of the heat loading of Nd-doped YAG, YOS, and GGG excited at diode pumping wavelengths," IEEE J. Quantum Electron. 29, 1457-1459 (1995).

Brauch, U.

U. Brauch, "Temperature dependence of efficiency and thermal lensing of diode-laser pumped Nd:YAG laser," Appl. Phys. B 58, 397-402 (1994).
[CrossRef]

Byer, R. L.

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

Chen, T.

Z. G. Li, Z. J. Xiong, W. L. Huang, T. Chen, N. Moore, and G. C. Lim, "Study of high power laser diode end-pumped composite crystal lasers," Chin. J. Lasers 32, 297-300 (2005) (in Chinese).

Chen, Y. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[CrossRef]

Comaskey, B.

B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, "Characterization of the heat loading of Nd-doped YAG, YOS, and GGG excited at diode pumping wavelengths," IEEE J. Quantum Electron. 29, 1457-1459 (1995).

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]

Fan, T. Y.

T. Y. Fan, "Heat generation in Nd:YAG and Nb:YAG," IEEE J. Quantum Electron. 29, 1457-1459 (1993).
[CrossRef]

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

Fields, R. A.

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

Frucher, C. L.

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

Graf, T.

T. Graf, E. Wyss, M. Roth, and H. P. Weber, "Laser resonator with balanced thermal lenses," Opt. Commun. 190, 327-331 (2001).
[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, 1605-1615 (1994).
[CrossRef]

Huang, T. M.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[CrossRef]

Huang, W. L.

Z. G. Li, Z. J. Xiong, W. L. Huang, T. Chen, N. Moore, and G. C. Lim, "Study of high power laser diode end-pumped composite crystal lasers," Chin. J. Lasers 32, 297-300 (2005) (in Chinese).

Innocenzi, M. E.

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

Kao, C. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[CrossRef]

Li, H. F.

Y. Yang, D. G. Xu, J. O. Yao, H. F. Li, Y. Y. Wang, and P. Wang, "Experimental research of an all-solid-state high power 1064 nm composite Nd:YAG laser," J. Optoelectron. Laser 17, 171-174 (2006) (in Chinese).

Li, L.

L. Li, P. Shi, and J. T. Bai, "Semi-analytical thermal analysis of single end-pumped laser crystal temperature distribution," J. Xi'an Jiaotong University 38, 369-372 (2004) (in Chinese).

Li, Z. G.

Z. G. Li, Z. J. Xiong, W. L. Huang, T. Chen, N. Moore, and G. C. Lim, "Study of high power laser diode end-pumped composite crystal lasers," Chin. J. Lasers 32, 297-300 (2005) (in Chinese).

Lim, G. C.

Z. G. Li, Z. J. Xiong, W. L. Huang, T. Chen, N. Moore, and G. C. Lim, "Study of high power laser diode end-pumped composite crystal lasers," Chin. J. Lasers 32, 297-300 (2005) (in Chinese).

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, 1605-1615 (1994).
[CrossRef]

Moore, N.

Z. G. Li, Z. J. Xiong, W. L. Huang, T. Chen, N. Moore, and G. C. Lim, "Study of high power laser diode end-pumped composite crystal lasers," Chin. J. Lasers 32, 297-300 (2005) (in Chinese).

Moran, B. D.

B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, "Characterization of the heat loading of Nd-doped YAG, YOS, and GGG excited at diode pumping wavelengths," IEEE J. Quantum Electron. 29, 1457-1459 (1995).

Ozygus, B.

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, B. Ozygus, and H. Weber, "Study on thermal lens of Nd3+:YAG laser pumped by a laser diode," Chin. J. Lasers 27, 777-781 (2000) (in Chinese).

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, 1605-1615 (1994).
[CrossRef]

Roth, M.

T. Graf, E. Wyss, M. Roth, and H. P. Weber, "Laser resonator with balanced thermal lenses," Opt. Commun. 190, 327-331 (2001).
[CrossRef]

Shi, P.

L. Li, P. Shi, and J. T. Bai, "Semi-analytical thermal analysis of single end-pumped laser crystal temperature distribution," J. Xi'an Jiaotong University 38, 369-372 (2004) (in Chinese).

Wang, C. L.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[CrossRef]

Wang, P.

Y. Yang, D. G. Xu, J. O. Yao, H. F. Li, Y. Y. Wang, and P. Wang, "Experimental research of an all-solid-state high power 1064 nm composite Nd:YAG laser," J. Optoelectron. Laser 17, 171-174 (2006) (in Chinese).

Wang, Q. P.

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, B. Ozygus, and H. Weber, "Study on thermal lens of Nd3+:YAG laser pumped by a laser diode," Chin. J. Lasers 27, 777-781 (2000) (in Chinese).

Wang, S. C.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[CrossRef]

Wang, Y. Y.

Y. Yang, D. G. Xu, J. O. Yao, H. F. Li, Y. Y. Wang, and P. Wang, "Experimental research of an all-solid-state high power 1064 nm composite Nd:YAG laser," J. Optoelectron. Laser 17, 171-174 (2006) (in Chinese).

Weber, H.

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, B. Ozygus, and H. Weber, "Study on thermal lens of Nd3+:YAG laser pumped by a laser diode," Chin. J. Lasers 27, 777-781 (2000) (in Chinese).

Weber, H. P.

T. Graf, E. Wyss, M. Roth, and H. P. Weber, "Laser resonator with balanced thermal lenses," Opt. Commun. 190, 327-331 (2001).
[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, 1605-1615 (1994).
[CrossRef]

Weber, 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, 1605-1615 (1994).
[CrossRef]

Wyss, E.

T. Graf, E. Wyss, M. Roth, and H. P. Weber, "Laser resonator with balanced thermal lenses," Opt. Commun. 190, 327-331 (2001).
[CrossRef]

Xiong, Z. J.

Z. G. Li, Z. J. Xiong, W. L. Huang, T. Chen, N. Moore, and G. C. Lim, "Study of high power laser diode end-pumped composite crystal lasers," Chin. J. Lasers 32, 297-300 (2005) (in Chinese).

Xu, D. G.

Y. Yang, D. G. Xu, J. O. Yao, H. F. Li, Y. Y. Wang, and P. Wang, "Experimental research of an all-solid-state high power 1064 nm composite Nd:YAG laser," J. Optoelectron. Laser 17, 171-174 (2006) (in Chinese).

Yang, H. R.

H. R. Yang and T. C. Zuo, "Investigation on thermal effects and output of all-solid-state miniature Yb:YAG laser," Acta Photonica Sin. 32, 907-910 (2003) (in Chinese).

Yang, Y.

Y. Yang, D. G. Xu, J. O. Yao, H. F. Li, Y. Y. Wang, and P. Wang, "Experimental research of an all-solid-state high power 1064 nm composite Nd:YAG laser," J. Optoelectron. Laser 17, 171-174 (2006) (in Chinese).

Yao, J. O.

Y. Yang, D. G. Xu, J. O. Yao, H. F. Li, Y. Y. Wang, and P. Wang, "Experimental research of an all-solid-state high power 1064 nm composite Nd:YAG laser," J. Optoelectron. Laser 17, 171-174 (2006) (in Chinese).

Yura, H. T.

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

Zhang, Q. D.

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, B. Ozygus, and H. Weber, "Study on thermal lens of Nd3+:YAG laser pumped by a laser diode," Chin. J. Lasers 27, 777-781 (2000) (in Chinese).

Zhang, X. Y.

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, B. Ozygus, and H. Weber, "Study on thermal lens of Nd3+:YAG laser pumped by a laser diode," Chin. J. Lasers 27, 777-781 (2000) (in Chinese).

Zhao, S. Z.

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, B. Ozygus, and H. Weber, "Study on thermal lens of Nd3+:YAG laser pumped by a laser diode," Chin. J. Lasers 27, 777-781 (2000) (in Chinese).

Zuo, T. C.

H. R. Yang and T. C. Zuo, "Investigation on thermal effects and output of all-solid-state miniature Yb:YAG laser," Acta Photonica Sin. 32, 907-910 (2003) (in Chinese).

Acta Photonica Sin.

H. R. Yang and T. C. Zuo, "Investigation on thermal effects and output of all-solid-state miniature Yb:YAG laser," Acta Photonica Sin. 32, 907-910 (2003) (in Chinese).

Appl. Phys. B

U. Brauch, "Temperature dependence of efficiency and thermal lensing of diode-laser pumped Nd:YAG laser," Appl. Phys. B 58, 397-402 (1994).
[CrossRef]

Appl. Phys. Lett.

M. E. Innocenzi, H. T. Yura, C. L. Frucher, 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

Z. G. Li, Z. J. Xiong, W. L. Huang, T. Chen, N. Moore, and G. C. Lim, "Study of high power laser diode end-pumped composite crystal lasers," Chin. J. Lasers 32, 297-300 (2005) (in Chinese).

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, B. Ozygus, and H. Weber, "Study on thermal lens of Nd3+:YAG laser pumped by a laser diode," Chin. J. Lasers 27, 777-781 (2000) (in Chinese).

IEEE J. Quantum Electron.

B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, "Characterization of the heat loading of Nd-doped YAG, YOS, and GGG excited at diode pumping wavelengths," IEEE J. Quantum Electron. 29, 1457-1459 (1995).

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Optimization in scaling fiber-coupled laser-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect," IEEE J. Quantum Electron. 33, 1424-1429 (1997).
[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, 1605-1615 (1994).
[CrossRef]

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

T. Y. Fan, "Heat generation in Nd:YAG and Nb:YAG," IEEE J. Quantum Electron. 29, 1457-1459 (1993).
[CrossRef]

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

J. Optoelectron. Laser

Y. Yang, D. G. Xu, J. O. Yao, H. F. Li, Y. Y. Wang, and P. Wang, "Experimental research of an all-solid-state high power 1064 nm composite Nd:YAG laser," J. Optoelectron. Laser 17, 171-174 (2006) (in Chinese).

J. Xi'an Jiaotong University

L. Li, P. Shi, and J. T. Bai, "Semi-analytical thermal analysis of single end-pumped laser crystal temperature distribution," J. Xi'an Jiaotong University 38, 369-372 (2004) (in Chinese).

Opt. Commun.

T. Graf, E. Wyss, M. Roth, and H. P. Weber, "Laser resonator with balanced thermal lenses," Opt. Commun. 190, 327-331 (2001).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Thermal model of a Nd:YAG rod by an end-pumped laser diode.

Fig. 2
Fig. 2

(Color online) Temperature field distribution diagram of a Nd:YAG rod by an end-pumped laser diode.

Fig. 3
Fig. 3

Isotherm diagram of a Nd:YAG rod by an end-pumped laser diode.

Fig. 4
Fig. 4

Thermal focus of a Nd:YAG rod as a function of laser diode pump power.

Equations (33)

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

I i ( r , 0 ) = I 0 e 2 ( r 2 / w 2 ) ,
I ( r , z ) = I i ( r , z ) + I r ( r , z ) = I 0 e 2 ( r 2 / w 2 ) × ( e β z + ζ R e β L e β ( L z ) ) .
q V ( r , z ) = η I ( r , z ) β = I 0 η β e 2 ( r 2 / w 2 ) × ( e β z + ζ R e β L e β ( L z ) ) ,
T ( r , φ , z ) | r = R = 0 ,
T ( r , φ , z ) z | z = 0 = T ( r , φ , z ) z | z = L = 0 .
2 T r 2 + 1 r T r + 1 r 2 2 T φ 2 + 2 T z 2 = q v λ ,
T ( r , z ) = n = 1 m = 0 A n m cos ( m π L z ) J 0 ( μ n ( 0 ) R r ) ,
n = 1 m = 0 A n m { cos ( m π L z ) [ J 0 ( μ n ( 0 ) R r ) + 1 r J 0 ( μ n ( 0 ) R r ) L 2 m 2 π 2 J 0 ( μ n ( 0 ) R r ) ] } = q V λ .
0 c cos m π L z cos k π L z d z = L 2 δ m k .
n = 1 A n m { J 0 ( μ n ( 0 ) R r ) + 1 r J 0 ( μ n ( 0 ) R r ) m 2 π 2 L 2 J 0 ( μ n ( 0 ) R r ) } = 2 L 1 λ 0 L q V cos ( m π L z ) d z .
d x = μ n ( 0 ) R d r ,
d d r J 0 ( μ n ( 0 ) R r ) = μ n ( 0 ) R d d x J 0 ( x ) ,
d 2 d r 2 J 0 ( μ n ( 0 ) R r ) = ( μ n ( 0 ) R ) 2 d 2 d x 2 J 0 ( x ) .
n = 1 A n m ( μ n ( 0 ) R ) 2 { d 2 J 0 ( x ) d x 2 + 1 x d J 0 ( x ) d x m 2 π 2 R 2 L 2 ( μ n ( 0 ) ) 2 J 0 ( x ) } .
d 2 J 0 ( x ) d x 2 = d J 1 ( x ) d x = J 1 ( x ) x + J 2 ( x ) .
n = 1 A n m ( μ n ( 0 ) R ) 2 { J 2 ( x ) 2 J 1 ( x ) x m 2 π 2 R 2 L 2 ( μ n ( 0 ) ) 2 J 0 ( x ) } .
n = 1 A n m m 2 π 2 R 2 + L 2 ( μ n ( 0 ) ) 2 L 2 R 2 J 0 ( x ) .
n = 1 A n m m 2 π 2 R 2 + L 2 ( μ n ( 0 ) ) 2 L 2 R 2 J 0 ( x ) = 2 λ L 0 L q V cos ( m π L z ) d z .
n = 1 A n m J 0 ( μ 0 ( 0 ) R r ) = 2 L R 2 λ [ m 2 π 2 R 2 + L 2 ( μ n ( 0 ) ) 2 ] × 0 L q V cos ( m π L z ) d z ,
2 I 0 η β L R 2 λ [ m 2 π 2 R 2 + L 2 ( μ n ( 0 ) ) 2 ] e 2 ( r 2 / w 2 ) 0 L ( e β z + ζ R e β L e β ( L z ) ) × cos ( m π L z ) d z .
0 L e β z cos m π L z d z = β L 2 ( 1 e β L cos m π ) β 2 L 2 + m 2 π 2 ,
0 L e β ( L z ) cos m π L z d z = β L 2 ( e β L + cos m π ) β 2 L 2 + m 2 π 2 ,
n = 1 A n m J 0 ( μ 0 ( 0 ) R r ) = 2 I 0 η β 2 L 3 R 2 [ ( 1 e β L cos m π ) + ζ R e β L ( e β L + cos m π ) ] λ ( β 2 L 2 + m 2 π 2 ) [ m 2 π 2 R 2 + L 2 ( μ n ( 0 ) ) 2 ] e 2 ( r 2 / w 2 ) .
0 R J 0 ( μ k ( 0 ) R r ) J 0 ( μ n ( 0 ) R r ) r d r = R 2 2 J 1 2 ( μ n ( 0 ) ) δ n k .
A n m = 4 I 0 η β 2 L 3 [ ( 1 e β L cos m π ) + ζ R e β L ( e β L + cos m π ) ] λ ( β 2 L 2 + m 2 π 2 ) [ m 2 π 2 R 2 + L 2 ( μ n ( 0 ) ) 2 ] J 1 2 ( μ n ( 0 ) ) × 0 R e 2 ( r 2 / w 2 ) J 0 ( μ n ( 0 ) R r ) r d r .
I 0 = P 2 π 0 e 2 ( r 2 / w 2 ) r d r
ε i j = [ ε r 0 0 0 ε θ 0 0 0 ε z ] .
n r ε r = n 0 3 12 [ 3 p 11 + 3 p 12 + 6 p 44 ] = 0.2154 ,
n r ε θ = n 0 3 12 [ p 11 + 5 p 12 2 p 44 ] = 0.0701 ,
n r ε z = n 0 3 12 [ 2 p 11 + 4 p 12 4 p 44 ] = 0.1127 .
dOPD = dOPD 1 + dOPD 2 + dOPD 3 .
OPD ( r ) = dOPD ( r ) .
f = r e 2 2 [ OPD 0 OPD ( r e ) ] ,

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