Abstract

The temperature, thermal stress, thermal strain, and optical path difference (OPD) in an orthotropic laser medium under Gaussian, top-hat, and uniform pumping schemes are solved both analytically and numerically. The results indicate that, provided the same total heat loading, the thermal effects under the top-hat pumping scheme are lower than under the Gaussian pumping scheme, whereas the thermal effects under uniform pumping are the least significant of all; in the absence of external forces, the orthotropic thermal properties have more significant effects on the thermal strain than on the thermal stress. The theoretical OPD agrees well with published experimental data and shows evident orthotropy.

© 2009 Optical Society of America

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References

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  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]
  2. X. Peng, L. Xu, and A. Asundi, “Thermal lensing effects for diode-end-pumped Nd:YVO4 and Nd:YAG lasers,” Opt. Eng. 43, 2454-2461 (2004).
    [CrossRef]
  3. H. Lindberg, M. Strassner, E. Gerster, J. Bengtsson, and A. Larsson, “Thermal management of optically pumped long-wavelength InP-based semiconductor disk lasers,” IEEE J. Quantum Electron. 11, 1126-1134 (2005).
    [CrossRef]
  4. Y. Chen, B. Chen, and M. Bass, “Calculation of three-dimensional thermal-gradient-induced stress birefringence in slab lasers,” Appl. Phys. B 81, 75-82 (2005).
    [CrossRef]
  5. J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385-396 (1992).
    [CrossRef]
  6. M. L. Baesso, J. Shen, and R. D. Snook, “Mode-mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732-3737 (1994).
    [CrossRef]
  7. Vargas and L. C. M. Miranda, “Photoacoustic and related photothermal techniques,” Phys. Rep. 161, 43-101 (1988).
    [CrossRef]
  8. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C. Malacarne, A. C. Bento, and M. L. Baesso, “Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials,” Opt. Lett. 33, 1464-1466 (2008).
    [CrossRef] [PubMed]
  9. F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: a theoretical study,” J. Appl. Phys. 104, 053520 (2008).
    [CrossRef]
  10. B. Li, S. Martin, and E. Welsch, “In situ measurement on ultraviolet dielectric components by a pulsed top-hat beam thermal lens,” Appl. Opt. 39, 4690-4697 (2000).
    [CrossRef]
  11. F. B. G. Astrath, N. G. C. Astrath, J. Shen, J. Zhou, L. C. Malacarne, P. R. B. Pedreira, and M. L. Baesso, “Time-resolved thermal mirror technique with top-hat cw laser excitation,” Opt. Express 16, 12214 (2008).
    [CrossRef] [PubMed]
  12. M. N. Özışik, Heat Conduction (Wiley, 1980).
  13. J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped CW Nd lasers,” IEEE J. Quantum Electron. 28, 1046-1056 (1992).
    [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. J. Wang, “Thermal analysis and thermal control of the high-power pumped laser medium,” Ph.D. dissertation (Tsinghua University, 2005).
  16. B. A. Boley and J. Weiner, Theory of Thermal Stress (Wiley, 1960).
  17. 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]
  18. W. Koechner, Solid-State Laser Engineering, 3rd ed. (Springer-Verlag, 1992).

2008

2005

H. Lindberg, M. Strassner, E. Gerster, J. Bengtsson, and A. Larsson, “Thermal management of optically pumped long-wavelength InP-based semiconductor disk lasers,” IEEE J. Quantum Electron. 11, 1126-1134 (2005).
[CrossRef]

Y. Chen, B. Chen, and M. Bass, “Calculation of three-dimensional thermal-gradient-induced stress birefringence in slab lasers,” Appl. Phys. B 81, 75-82 (2005).
[CrossRef]

2004

X. Peng, L. Xu, and A. Asundi, “Thermal lensing effects for diode-end-pumped Nd:YVO4 and Nd:YAG lasers,” Opt. Eng. 43, 2454-2461 (2004).
[CrossRef]

2003

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]

2000

1994

M. L. Baesso, J. Shen, and R. D. Snook, “Mode-mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732-3737 (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]

1992

J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped CW Nd lasers,” IEEE J. Quantum Electron. 28, 1046-1056 (1992).
[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. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385-396 (1992).
[CrossRef]

1988

Vargas and L. C. M. Miranda, “Photoacoustic and related photothermal techniques,” Phys. Rep. 161, 43-101 (1988).
[CrossRef]

Albers, P.

J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped CW Nd lasers,” IEEE J. Quantum Electron. 28, 1046-1056 (1992).
[CrossRef]

Astrath, F. B. G.

Astrath, N. G. C.

Asundi, A.

X. Peng, L. Xu, and A. Asundi, “Thermal lensing effects for diode-end-pumped Nd:YVO4 and Nd:YAG lasers,” Opt. Eng. 43, 2454-2461 (2004).
[CrossRef]

Baesso, M. L.

N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C. Malacarne, A. C. Bento, and M. L. Baesso, “Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials,” Opt. Lett. 33, 1464-1466 (2008).
[CrossRef] [PubMed]

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: a theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

F. B. G. Astrath, N. G. C. Astrath, J. Shen, J. Zhou, L. C. Malacarne, P. R. B. Pedreira, and M. L. Baesso, “Time-resolved thermal mirror technique with top-hat cw laser excitation,” Opt. Express 16, 12214 (2008).
[CrossRef] [PubMed]

M. L. Baesso, J. Shen, and R. D. Snook, “Mode-mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732-3737 (1994).
[CrossRef]

Bass, M.

Y. Chen, B. Chen, and M. Bass, “Calculation of three-dimensional thermal-gradient-induced stress birefringence in slab lasers,” Appl. Phys. B 81, 75-82 (2005).
[CrossRef]

Belancon, M. P.

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: a theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

Bengtsson, J.

H. Lindberg, M. Strassner, E. Gerster, J. Bengtsson, and A. Larsson, “Thermal management of optically pumped long-wavelength InP-based semiconductor disk lasers,” IEEE J. Quantum Electron. 11, 1126-1134 (2005).
[CrossRef]

Bento, A. C.

Boley, B. A.

B. A. Boley and J. Weiner, Theory of Thermal Stress (Wiley, 1960).

Chen, B.

Y. Chen, B. Chen, and M. Bass, “Calculation of three-dimensional thermal-gradient-induced stress birefringence in slab lasers,” Appl. Phys. B 81, 75-82 (2005).
[CrossRef]

Chen, Y.

Y. Chen, B. Chen, and M. Bass, “Calculation of three-dimensional thermal-gradient-induced stress birefringence in slab lasers,” Appl. Phys. B 81, 75-82 (2005).
[CrossRef]

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]

Frauchiger, J.

J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped CW Nd lasers,” IEEE J. Quantum Electron. 28, 1046-1056 (1992).
[CrossRef]

Gerster, E.

H. Lindberg, M. Strassner, E. Gerster, J. Bengtsson, and A. Larsson, “Thermal management of optically pumped long-wavelength InP-based semiconductor disk lasers,” IEEE J. Quantum Electron. 11, 1126-1134 (2005).
[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, 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]

Koechner, W.

W. Koechner, Solid-State Laser Engineering, 3rd ed. (Springer-Verlag, 1992).

Larsson, A.

H. Lindberg, M. Strassner, E. Gerster, J. Bengtsson, and A. Larsson, “Thermal management of optically pumped long-wavelength InP-based semiconductor disk lasers,” IEEE J. Quantum Electron. 11, 1126-1134 (2005).
[CrossRef]

Li, B.

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]

Lindberg, H.

H. Lindberg, M. Strassner, E. Gerster, J. Bengtsson, and A. Larsson, “Thermal management of optically pumped long-wavelength InP-based semiconductor disk lasers,” IEEE J. Quantum Electron. 11, 1126-1134 (2005).
[CrossRef]

Lowe, R. D.

J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385-396 (1992).
[CrossRef]

Malacarne, L. C.

Martin, S.

Mendes, R. S.

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: a theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[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, 1605-1615 (1994).
[CrossRef]

Miranda, L. C. M.

Vargas and L. C. M. Miranda, “Photoacoustic and related photothermal techniques,” Phys. Rep. 161, 43-101 (1988).
[CrossRef]

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]

Özisik, M. N.

M. N. Özışik, Heat Conduction (Wiley, 1980).

Pedreira, P. R. B.

Peng, X.

X. Peng, L. Xu, and A. Asundi, “Thermal lensing effects for diode-end-pumped Nd:YVO4 and Nd:YAG lasers,” Opt. Eng. 43, 2454-2461 (2004).
[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, 1605-1615 (1994).
[CrossRef]

Sato, F.

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: a theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

Shen, J.

N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C. Malacarne, A. C. Bento, and M. L. Baesso, “Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials,” Opt. Lett. 33, 1464-1466 (2008).
[CrossRef] [PubMed]

F. B. G. Astrath, N. G. C. Astrath, J. Shen, J. Zhou, L. C. Malacarne, P. R. B. Pedreira, and M. L. Baesso, “Time-resolved thermal mirror technique with top-hat cw laser excitation,” Opt. Express 16, 12214 (2008).
[CrossRef] [PubMed]

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: a theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

M. L. Baesso, J. Shen, and R. D. Snook, “Mode-mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732-3737 (1994).
[CrossRef]

J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385-396 (1992).
[CrossRef]

Snook, R. D.

M. L. Baesso, J. Shen, and R. D. Snook, “Mode-mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732-3737 (1994).
[CrossRef]

J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385-396 (1992).
[CrossRef]

Strassner, M.

H. Lindberg, M. Strassner, E. Gerster, J. Bengtsson, and A. Larsson, “Thermal management of optically pumped long-wavelength InP-based semiconductor disk lasers,” IEEE J. Quantum Electron. 11, 1126-1134 (2005).
[CrossRef]

Vargas,

Vargas and L. C. M. Miranda, “Photoacoustic and related photothermal techniques,” Phys. Rep. 161, 43-101 (1988).
[CrossRef]

Wang, J.

J. Wang, “Thermal analysis and thermal control of the high-power pumped laser medium,” Ph.D. dissertation (Tsinghua University, 2005).

Weber, H. P.

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]

J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped CW Nd lasers,” IEEE J. Quantum Electron. 28, 1046-1056 (1992).
[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]

Weiner, J.

B. A. Boley and J. Weiner, Theory of Thermal Stress (Wiley, 1960).

Welsch, E.

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]

Xu, L.

X. Peng, L. Xu, and A. Asundi, “Thermal lensing effects for diode-end-pumped Nd:YVO4 and Nd:YAG lasers,” Opt. Eng. 43, 2454-2461 (2004).
[CrossRef]

Zhou, J.

Appl. Opt.

Appl. Phys. B

Y. Chen, B. Chen, and M. Bass, “Calculation of three-dimensional thermal-gradient-induced stress birefringence in slab lasers,” Appl. Phys. B 81, 75-82 (2005).
[CrossRef]

Chem. Phys.

J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385-396 (1992).
[CrossRef]

IEEE J. Quantum Electron.

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]

H. Lindberg, M. Strassner, E. Gerster, J. Bengtsson, and A. Larsson, “Thermal management of optically pumped long-wavelength InP-based semiconductor disk lasers,” IEEE J. Quantum Electron. 11, 1126-1134 (2005).
[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]

J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped CW Nd lasers,” IEEE J. Quantum Electron. 28, 1046-1056 (1992).
[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. Appl. Phys.

M. L. Baesso, J. Shen, and R. D. Snook, “Mode-mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732-3737 (1994).
[CrossRef]

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: a theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[CrossRef]

Opt. Eng.

X. Peng, L. Xu, and A. Asundi, “Thermal lensing effects for diode-end-pumped Nd:YVO4 and Nd:YAG lasers,” Opt. Eng. 43, 2454-2461 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

Vargas and L. C. M. Miranda, “Photoacoustic and related photothermal techniques,” Phys. Rep. 161, 43-101 (1988).
[CrossRef]

Other

M. N. Özışik, Heat Conduction (Wiley, 1980).

W. Koechner, Solid-State Laser Engineering, 3rd ed. (Springer-Verlag, 1992).

J. Wang, “Thermal analysis and thermal control of the high-power pumped laser medium,” Ph.D. dissertation (Tsinghua University, 2005).

B. A. Boley and J. Weiner, Theory of Thermal Stress (Wiley, 1960).

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Figures (11)

Fig. 1
Fig. 1

Schematic of an end-pumped Nd: YVO 4 laser medium.

Fig. 2
Fig. 2

Temperature profiles in the pumping end surface ( z = 0 ) under different pumping schemes.

Fig. 3
Fig. 3

Temperature contours in the longitudinal central planes (K).

Fig. 4
Fig. 4

Comparison of temperature contours between orthotropic and isotropic laser medium (K).

Fig. 5
Fig. 5

Comparison of thermal stress components between analytical solution and numerical results under the Gaussian pump beam.

Fig. 6
Fig. 6

Contours of thermal stress components under the top-hat pump beam ( z = 0 ) (Pa).

Fig. 7
Fig. 7

Contours of thermal stress components under uniform pumping (MPa).

Fig. 8
Fig. 8

Contours of thermal stress components in an isotropic laser medium (under uniform pumping) for comparison (MPa).

Fig. 9
Fig. 9

Comparison of thermal strain components in an orthotropic laser medium (Unit: 1).

Fig. 10
Fig. 10

Comparison of thermal strain components between an orthotropic laser medium and an isotropic laser medium under uniform pumping (Unit: 10 5 ).

Fig. 11
Fig. 11

Δ OPD in different directions for the Gaussian and top-hat pumping schemes in comparison with published literature.

Equations (49)

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

2 θ x 2 + 1 ε 2 2 θ y 2 + 1 η 2 2 θ z 2 + q V ( x , y , z ) k x = 0 , 0 < x < a , 0 < y < b , 0 < z < L ,
θ / x = 0 , x = 0 ,
θ / x + H x θ = 0 , x = a ,
θ / y = 0 , y = 0 ,
θ / y + H y θ = 0 , y = b ,
θ / z = 0 , z = 0 ,
θ / z = 0 , z = L ,
q V ( x , y , z ) = 2 Q α π ω p 2 ( 1 e α L ) exp ( 2 x 2 + y 2 ω p 2 α z ) ,
q V ( x , y , z ) = { Q ( α L ) e α z π r 0 2 L ( 1 e α L ) , x 2 + y 2 r 0 2 0 , x 2 + y 2 > r 0 2 ,
q V ( x , y , z ) = Q / ( 2 a · 2 b · L ) .
θ ( β m , γ n , z ) = 0 b Y ( γ n , y ) d y 0 a X ( β m , x ) θ ( x , y , z ) d x ,
θ ( x , y , z ) = m = 1 n = 1 X ( β m , x ) Y ( γ n , y ) N ( β m ) N ( γ n ) θ ( β m , γ n , z ) ,
N ( β m ) = a ( β m 2 + H x 2 ) + H x 2 ( β m 2 + H x 2 ) , N ( γ n ) = b ( γ n 2 + H y 2 ) + H y 2 ( γ n 2 + H y 2 ) ,
d 2 θ ( β m , γ n , z ) d z 2 λ m n 2 θ ( β m , γ n , z ) = η 2 k x q V ( β m , γ n , z ) ,
d θ / d z = 0 , z = 0 ,
d θ / d z = 0 , z = L ,
q V ( β m , γ n , z ) = Q α 4 ( 1 e α L ) exp ( β m 2 + γ n 2 8 ω p 2 α z ) .
θ ( β m , γ n , z ) = A m n e λ m n z + B m n e λ m n z + C m n e α z ,
A m n = M m n α ( e λ m n L e α L ) 2 λ m n ( λ m n 2 α 2 ) sinh ( λ m n L ) ,
B m n = M m n α ( e λ m n L e α L ) 2 λ m n ( λ m n 2 α 2 ) sinh ( λ m n L ) ,
C m n = M m n / ( λ m n 2 α 2 ) ,
M m n = η 2 Q α 4 k x ( 1 e α L ) exp ( β m 2 + γ n 2 8 ω p 2 ) .
θ ( x , y , z ) = m = 1 n = 1 cos ( β m x ) cos ( γ n y ) N ( β m ) N ( γ n ) ( A m n e λ m n z + B m n e λ m n z + C m n e α z ) .
q V ( β m , γ n , z ) = Q α e α z π r 0 2 ( 1 e α L ) 1 γ n 0 r 0 cos ( β m x ) sin ( γ n r 0 2 x 2 ) d x .
P m n = η 2 Q α k x π r 0 2 ( 1 e α L ) 1 γ n 0 r 0 cos ( β m x ) sin ( γ n r 0 2 x 2 ) d x .
q V ( β m , γ n ) = Q 4 a b L sin ( β m a ) β m sin ( γ n b ) γ n .
θ ( β m , γ n ) = η 2 λ m n 2 Q 4 a b L k x sin ( β m a ) β m sin ( γ n b ) γ n .
θ ( x , y ) = m = 1 n = 1 cos ( β m x ) cos ( γ n y ) N ( β m ) N ( γ n ) η 2 λ m n 2 Q 4 a b L k x sin ( β m a ) β m sin ( γ n b ) γ n .
θ ( x , y , z ) = m = 1 n = 1 K m n ( z ) cos ( β m x ) cos ( γ n y ) ,
K m n ( z ) = A m n e λ m n z + B m n e λ m n z + C m n e α z N ( β m ) N ( γ n ) ,
K m n ( z ) = 1 N ( β m ) N ( γ n ) η 2 λ m n 2 Q 4 a b L k x sin ( β m a ) β m sin ( γ n b ) γ n .
θ ( x , y ) = Q 4 b L k x H x + Q 8 a b L k x ( a 2 x 2 ) Q 4 a b L k x m = 1 1 β m 3 1 N ( β m ) × cos ( β m x ) sin ( β m a ) cosh ( β m ε y ) β m ε H y sinh ( β m ε b ) + cosh ( β m ε b ) .
θ ( x , y ) = Q 4 b L H k + Q 8 a b L k ( a 2 x 2 ) + m = 1 cosh ( β m y ) cos ( β m x ) N ( β m ) × Q H 4 a b L k ( a cos ( β m a ) β m 2 sin ( β m a ) β m 3 ) Q 4 b L k sin ( β m a ) β m β m sinh ( β m b ) + H cosh ( β m b ) .
ε x x = 1 ν 2 E ( σ x x ν 1 ν σ y y ) + ( 1 + ν ) α x θ ,
ε y y = 1 ν 2 E ( σ y y ν 1 ν σ x x ) + ( 1 + ν ) α y θ ,
ε x y = 1 + ν E σ x y ,
σ x x = 2 ϕ y 2 , σ y y = 2 ϕ x 2 , σ x y = 2 ϕ x y ,
4 ϕ = E 1 ν ( α y 2 θ x 2 + α x 2 θ y 2 ) , 0 < x < a , 0 < y < b ,
ϕ = ϕ / x = 0 , x = a ,
ϕ = ϕ / y = 0 , y = b .
ϕ 1 = [ b 1 cosh ( β 1 y ) + b 4 y sinh ( β 1 y ) ] cos ( β 1 x ) ,
ϕ 2 = [ c 1 cosh ( γ 1 x ) + c 4 x sinh ( γ 1 x ) ] cos ( γ 1 y ) ,
ϕ 3 = m = 1 n = 1 R m n ( z ) cos ( β m x ) cos ( γ n y ) .
R m n ( z ) = E 1 ν K m n ( z ) ( α y β m 2 + α x γ n 2 ) ( β m 2 + γ n 2 ) 2 .
ϕ = [ b 1 cosh ( β 1 y ) + b 4 y sinh ( β 1 y ) ] cos ( β 1 x ) + [ c 1 cosh ( γ 1 x ) + c 4 x sinh ( γ 1 x ) ] cos ( γ 1 y ) + m = 1 n = 1 R m n ( z ) cos ( β m x ) cos ( γ n y ) ,
σ y y = [ b 1 cosh ( β 1 y ) + b 4 y sinh ( β 1 y ) ] β 1 2 cos ( β 1 x ) + { c 1 γ 1 2 cosh ( γ 1 x ) + c 4 [ 2 γ 1 cosh ( γ 1 x ) + γ 1 2 x sinh ( γ 1 x ) ] } cos ( γ 1 y ) m = 1 n = 1 β m 2 R m n ( z ) cos ( β m x ) cos ( γ n y ) ,
σ x x = { b 1 β 1 2 cosh ( β 1 y ) + b 4 [ 2 β 1 cosh ( β 1 y ) + β 1 2 y sinh ( β 1 y ) ] } cos ( β 1 x ) [ c 1 cosh ( γ 1 x ) + c 4 x sinh ( γ 1 x ) ] γ 1 2 cos ( γ 1 y ) m = 1 n = 1 γ n 2 R m n ( z ) cos ( β m x ) cos ( γ n y ) ,
σ x y = { b 1 β 1 sinh ( β 1 y ) + b 4 [ sinh ( β 1 y ) + β 1 y cosh ( β 1 y ) ] } β 1 sin ( β 1 x ) + { c 1 γ 1 sinh ( υ j x ) + c 4 [ sinh ( γ 1 x ) + γ 1 x cosh ( γ 1 x ) ] } γ 1 sin ( γ 1 y ) m = 1 n = 1 β m γ n R m n ( z ) sin ( β m x ) sin ( γ n y ) .
OPD ( x , y ) = 0 L n θ θ ( x , y ) d z + n 0 Δ u ( x , y ) + i , j 3 0 L n ε i j ε i j ( x , y ) d z ,

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