Abstract

The transient heat conduction and thermal effects in pulse end-pumped fiber laser are modeled and analytically solved. For the arbitrary temporal shape of pump pulse, a three-dimensional (3D) temperature expression is derived via an integral transform method, and the thermal stress field is deduced through solving the Navier displacement equations. The results show that pulse shape has an important influence on the peak thermal stress and transient phase shift induced by heating of the fiber. Reasonable design for pulse duration and period can reduce thermal effects and optimize the performance of high-power fiber laser.

© 2009 OSA

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References

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  1. C. Lecaplain, C. Chedot, A. Hideur, B. Ortac, and J. Limpert, “High-average power femtosecond pulse generation from a Yb-doped large-mode-area microstructure fiber laser,” Proc. of SPIE 6873, 68730S1–68730S5 (2008)
  2. M. Eichhorn and S. D. Jackson, “High-pulse-energy actively Q-switched Tm3+-doped silica 2 microm fiber laser pumped at 792 nm,” Opt. Lett. 32(19), 2780–2782 (2007).
    [CrossRef] [PubMed]
  3. Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).
  4. B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
    [CrossRef]
  5. Z. Y. Dai, Z. S. Peng, Y. Z. Liu, and Z. H. Ou, “Research on SBS and pulse pumped hybrid Q-switched Er3+/Yb3+ co-doped fiber laser,” Proc. of SPIE 6823, 68231C1–68231C4.
  6. S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).
  7. C. G. Ye, P. Yan, M. Gong, and M. Lei, “Pulsed pumped Yb-doped fiber amplifier at low repetition rate,” Chin. Opt. Lett. 3, 249–250 (2005).
  8. V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
    [CrossRef]
  9. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
    [CrossRef]
  10. M. K. Davis, M. J. F. Digonnet, and R. H. Pantell, “Thermal effects in doped fobers,” J. Lightwave Technol. 16(6), 1013–1023 (1998).
    [CrossRef]
  11. E. H. Bernhardi, A. Forbes, C. Bollig, and M. J. D. Esser, “Estimation of thermal fracture limits in quasi-continuous-wave end-pumped lasers through a time-dependent analytical model,” Opt. Express 16(15), 11115–11123 (2008).
    [CrossRef] [PubMed]
  12. W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44(7), 3162–3170 (1973).
    [CrossRef]
  13. F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).
  14. M. N. Özisik, Heat Conduction (Wiley, New York, 1980).
  15. Z. G. Li, X. L. Huai, L. Wang, and Y. J. Tao, “Influence of longitudinal rise of coolant temperature on the thermal strain in a cylindrical laser rod,” Opt. Lett. 34(2), 187–189 (2009).
    [CrossRef] [PubMed]
  16. Y. Takeuchi, Thermal Stress (Science, 1977).
  17. T. Liu, Z. M. Yang, and S. H. Xu, “3-Dimensional heat analysis in short-length Er3+/Yb3+ co-doped phosphate fiber laser with upconversion,” Opt. Express 17(1), 235–247 (2009).
    [CrossRef] [PubMed]
  18. Ansys Finite Element Software Package, http://www.ansys.com/
  19. C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
    [CrossRef]
  20. P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer 131(1), 011304–0113047 (2009).
    [CrossRef]

2009

B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
[CrossRef]

Z. G. Li, X. L. Huai, L. Wang, and Y. J. Tao, “Influence of longitudinal rise of coolant temperature on the thermal strain in a cylindrical laser rod,” Opt. Lett. 34(2), 187–189 (2009).
[CrossRef] [PubMed]

T. Liu, Z. M. Yang, and S. H. Xu, “3-Dimensional heat analysis in short-length Er3+/Yb3+ co-doped phosphate fiber laser with upconversion,” Opt. Express 17(1), 235–247 (2009).
[CrossRef] [PubMed]

P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer 131(1), 011304–0113047 (2009).
[CrossRef]

2008

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

E. H. Bernhardi, A. Forbes, C. Bollig, and M. J. D. Esser, “Estimation of thermal fracture limits in quasi-continuous-wave end-pumped lasers through a time-dependent analytical model,” Opt. Express 16(15), 11115–11123 (2008).
[CrossRef] [PubMed]

2007

M. Eichhorn and S. D. Jackson, “High-pulse-energy actively Q-switched Tm3+-doped silica 2 microm fiber laser pumped at 792 nm,” Opt. Lett. 32(19), 2780–2782 (2007).
[CrossRef] [PubMed]

F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).

2006

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

2005

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

C. G. Ye, P. Yan, M. Gong, and M. Lei, “Pulsed pumped Yb-doped fiber amplifier at low repetition rate,” Chin. Opt. Lett. 3, 249–250 (2005).

2001

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

1998

1994

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

1973

W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44(7), 3162–3170 (1973).
[CrossRef]

Ballato, J.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Bass, M.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Bernhardi, E. H.

Bollig, C.

Brown, D. C.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

Chen, Y.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Davis, M. K.

Digonnet, M. J. F.

Dong, F. J.

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Dong, W.

F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).

Dong, X. Y.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Eichhorn, M.

Esser, M. J. D.

Forbes, A.

Gong, M.

Gong, M. L.

B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
[CrossRef]

Gruber, R.

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

Hoffman, H. J.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

Hu, S. L.

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Huai, X. L.

Huang, F.

F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).

Jackson, S. D.

Jain, P. K.

P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer 131(1), 011304–0113047 (2009).
[CrossRef]

Jia, W. W.

F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).

Kai, K. G.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Koechner, W.

W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44(7), 3162–3170 (1973).
[CrossRef]

Lei, M.

Li, Z. G.

Liu, Q.

B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
[CrossRef]

Liu, T.

Liu, Y. G.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Lu, F. Y.

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Lu, Y. F.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Mccomb, T.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Merazzi, S.

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

Pantell, R. H.

Peng, B.

B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
[CrossRef]

Pfistner, C.

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

Richardson, M.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Rizwan-uddin,

P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer 131(1), 011304–0113047 (2009).
[CrossRef]

Siegman, A. E.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Singh, S.

P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer 131(1), 011304–0113047 (2009).
[CrossRef]

Sudesh, V.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Sun, T. T.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Tao, Y. J.

Wang, H. J.

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Wang, L.

Wang, Y. F.

F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).

Wang, Z.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Weber, R.

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

Wever, H. P.

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

Xie, C. X.

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Xu, S. H.

Yan, P.

Yang, P.

B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
[CrossRef]

Yang, Z. M.

Ye, C. G.

Yuan, S. Z.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Zhang, C. S.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Zhang, S. M.

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Acta Photon. Sin.

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Acta Phys. Sin.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Appl. Phys. B

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Chin. Opt. Lett.

IEEE J. Quantum Electron.

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

J. Appl. Phys.

W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44(7), 3162–3170 (1973).
[CrossRef]

J. Heat Transfer

P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer 131(1), 011304–0113047 (2009).
[CrossRef]

J. Lightwave Technol.

Opt. Commun.

B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).

Other

M. N. Özisik, Heat Conduction (Wiley, New York, 1980).

C. Lecaplain, C. Chedot, A. Hideur, B. Ortac, and J. Limpert, “High-average power femtosecond pulse generation from a Yb-doped large-mode-area microstructure fiber laser,” Proc. of SPIE 6873, 68730S1–68730S5 (2008)

Y. Takeuchi, Thermal Stress (Science, 1977).

Ansys Finite Element Software Package, http://www.ansys.com/

Z. Y. Dai, Z. S. Peng, Y. Z. Liu, and Z. H. Ou, “Research on SBS and pulse pumped hybrid Q-switched Er3+/Yb3+ co-doped fiber laser,” Proc. of SPIE 6823, 68231C1–68231C4.

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Figures (11)

Fig. 1
Fig. 1

Schematic illustration of a pulse end-pumped short-length fiber.

Fig. 2
Fig. 2

The temperature distribution from the analytical solution at the time of 0.1 s. (a) The 3D temperature distribution; (b) The temperature distribution along the active fiber axial coordinate.

Fig. 3
Fig. 3

A time-dependent temperature finite element analysis at the time of 0.1 s. (a) The temperature distribution of fiber end surface at the pump side; (b) The temperature distribution along the active fiber axial coordinate.

Fig. 4
Fig. 4

The temperature distribution of fiber end surface at the pump side along the active fiber radial coordinate. (a) During the first pulse pump time; (b) The first 5 ms during the first pulse pump space time; (c) The last 10 ms during the first pulse pump space time; (d) The long enough pump space time and t0 = 0.01, interval: 1 s.

Fig. 5
Fig. 5

The transient temperature variation in the centre of fiber end surface as a function of time. Graph b and c are the enlarged drawings of the part of graph a.

Fig. 6
Fig. 6

The transient temperature under different pulse duration. Red: t0 = 0.06, Blue: t0 = 0.01.

Fig. 7
Fig. 7

The 3D radial, tangential and axial thermal stress distributions at the time of 0.1 s. (a) σ rr ; (b) σθθ; (c) σ zz

Fig. 8
Fig. 8

The 3D radial, tangential and axial thermal strains distributions at the time of 0.1 s. (a) εrr ; (b) εθθ ; (c) εzz

Fig. 9
Fig. 9

The 3D the radial, axial thermal stress and radial strain distribution at the time of 0.1 s under the plain-strain assumption. (a) σ rr ; (b) σ zz , (c) εrr

Fig. 10
Fig. 10

The radial thermal stress in the centre of fiber end surface as a function of time. (a)T0 = 0.1 s, t0 = 0.01 s; (b) T0 = 0.001 s, t0 = 0.0001 s; (c) T0 = 0.0001 s, t0 = 0.00001 s.

Fig. 11
Fig. 11

The phase shift rise induced by thermal effects as a function of time. Graph b is the enlarged drawing of graph a

Equations (98)

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

2T(z,r,t)r2+1rT(z,r,t)r+2T(z,r,t)z2+Q(z,r,t)k=ρckT(z,r,t)t
             0<rr1
2T(z,r,t)r2+1rT(z,r,t)r+2T(z,r,t)z2=ρckT(z,r,t)t
r1<rr2
T=Th,                                                               t=0
kT/r=h(ThT),                                                 r=r2
T/z=0,                               z=0,                                                 z=l
T1=T2,                                       T1/r=T2/r,                                     r=r1
T/r=0                                                                                                     r=0
Q(z,r,t)=(2ηαPin/πωp2)exp(2r2/ωp2αz)g(t)
θ=TT0=p=1kρclNpJ0(βpr)exp(kρcβp2t)0tg0p(τ)exp(kρcβp2τ)dτ+                                 m=1p=12kρclNpJ0(βpr)cos(ηmz)exp[kρc(βp2+ηm2)t]0tgmp(τ)exp[kρc(βp2+ηm2)τ]dτ
g0p(τ)=0l0r1Q(z,r,t)kJ0(βpr)rdrdz
gmp(τ)=0l0r1Q(z,r,t)kcos(mπLz)J0(βpr)rdrdz
0tg0p(τ)exp(kρcβp2τ)dτ=2ηPin[1exp(βl)]kπωp20r1exp(2r2/ωp2)J0(βpr)rdr0tg(τ)exp(kρcβp2τ)dτ
0tgmp(τ)exp[kρc(βp2+ηm2)τ]dτ=2ηβ2l2Pin[1exp(βl)cos(mπ)]kπωp2(β2l2+m2π2)0r1exp(2r2/ωp2)J0(βpr)rdr0tg(τ)exp[kρc(βp2+ηm2)τ]dτ
   hJ0(βpr2)kβpJ1(βpr2)     =0
σθθ0=γE(1ν)[1r220r2T1rdr+1r20rT1rdrT(r)]=γE(1ν)p=1Apβp[J1(βpr2)r2+J1(βpr)rβpJ0(βpr)]
σrz0=0
Ap=kρclNpexp(kρcβp2t)0tg0p(τ)exp(kρcβp2τ)dτ
2ur1r2ur+112νr(1rr(rur)+zuz)=2(1+ν)12νγTr
2uz+112νr(1rr(rur)+zuz)=2(1+ν)12νγTz
2Φ=(1+ν)/(1ν)γθ2
22L=0
σrr1=E1+ν(2Φr22Φ)
σrr2=E1+ν[2r2(Lz)ν2(Lz)]
σθθ1=E1+ν(1rΦr2Φ)
σθθ2=E1+ν[1rr(Lz)ν2(Lz)]
σzz1=E1+ν(2Φz22Φ)
σzz2=E1+ν[2z2(Lz)(2ν)2(Lz)]
σrz1=E1+ν2Φrz
σrz2=E1+ν[2rz(Lz)(1ν)r(2L)]
σij=σij1+σij2
Φ=m=1p=1Cmpcos(mπlz)J0(βpr)
Cmp=1+ν1νγ2kL2ρclNp(m2π2+L2βp2)exp[kρc(βp2+ηm2)t]0tgmp(τ)exp[kρc(βp2+ηm2)τ]dτ
L=1+ν1νγp=1[D1pβpJ0(βpr)+D2prJ1(βpr)]exp(βpz)βp
σrr1=E1+νm=1p=1Cmp[βprJ1(βpr)+(mπl)2J0(βpr)]cos(mπlz)
σθθ1=E1+νm=1p=1Cmp[βp2J0(βpr)βprJ1(βpr)+(mπl)2J0(βpr)]cos(mπlz)
σzz1=E1+νm=1p=1Cmpβp2J0(βpr)cos(mπlz)
σθθ2=E1νγp=1[D1pJ1(βpr)r+D2p(12ν)βpJ0(βpr)]exp(βpz)
σrz1=E1+νm=1p=1Cmpβpmπlsin(mπlz)J0(βpr)
σrr2=E1νγp=1{D1p[βpJ0(βpr)+J1(βpr)r]+D2p[(12ν)βpJ0(βpr)βp2rJ1(βpr)]}exp(βpz)
σzz2=E1νγp=1{D1pβpJ0(βpr)+D2pβp[rβpJ1(βpr)2(2ν)J0(βpr)]}exp(βpz)
σrz2=E1νγp=1{D1pβpJ1(βpr)D2pβp[βprJ0(βpr)+2(1ν)J1(βpr)]}exp(βpz)
σrr=0                               σrz=0                                                           r=r2
D1pexp(βpz)=m=1Cmpkβpmπsin(mπz/l)/lh+[kβp2r2/h+2(1ν)]D2pexp(βpz)
D2pexp(βpz)=m=1Cmp[1/r2+km2π2/hl2]cos(mπz/l)[1/βpr2kβp/h]m=1Cmpkβpmπsin(mπz/l)/lh2(1ν)/βpr2βpr2k2βp3r2/h2
Cmp=2kL2ρclNp(m2π2+L2βp2)exp[kρc(βp2+ηm2)t]0tgmp(τ)exp[kρc(βp2+ηm2)τ]dτ
σij=σij0+σij1+σij2
εrr=[σrrν(σθθ+σzz)]/E+γθ
εθθ=[σθθν(σrr+σzz)]/E+γθ
εzz=[σzzν(σrr+σθθ)]/E+γθ
g(t)={1(n1)T0t(n1)T0+t00(n1)T0+t0tnT0
Δn(z,t)=(n/T)0θ(z,r,t)fs(r)2πrdr
fs(r)=1πr12exp(r2r12)
Δϕ1(t)=2πλsnT0l0r1θ(z,r,t)fs(r)2πrdrdz
Δϕ2(t)=2πλs0l0r1(n01)εzzdrdz
Δϕ(t)=Δϕ1(t)+Δϕ2(t)
Δϕ(t)=2πλsnTlπr12p=1kρclNpexp(kρcβp2t)0tg0p(τ)exp(kρcβp2τ)dτ0r1J0(βpr)exp(r2r12)2πrdr
2θi(z,r,t)r2+1rθi(z,r,t)r+2θi(z,r,t)z2+gi(z,r,t)=ρckθi(z,r,t)t            i=1,2
θi=0,                 t=0
kθ2/r+hθ2=0,                             r=r2
θi/z=0,             z=​ ​ ​ ​ ​0,                     z=L
θ1=θ2,                 θ1/r=θ2/r,                 r=r1
θ1/r=0                               r=0
θi(z,r,t)=m=0Z(ηm,z)θim(r,t)/N(ηm)
2θim(r,t)r2+1rθim(r,t)rηm2θim(r,t)+gim=ρckθim(r,t)t
θim=0,                                                               t=0
kθ2m/r+hθ2m=0,                                                 r=r2
θ1m=θ2m,                                       θ1m/r=θ2m/r,                                     r=r1
θ1m/r=0                                                                                                     r=0
gim=0LZ(ηm,z)gidz
ri1ri[ρckθim(r,t)t]rRim(r)dr=ri1ri[d2Rim(r)dr2+1rdRim(r)drηm2Rim(r)]rθim(r,t)dr+[rRim(r)θim(r,t)rrθim(r,t)dRim(r)dr]ri1ri+ri1rirRim(r)gimdr
d2Rim(r)dr2+1rdRim(r)dr+(ηm2+λ2)Rim(r)=0
kR2m/r+hR2m=0,                                                 r=r2
R1m=R2m                                                       r=r1
R1m/r=R2m/r                                                 r=r1
R1m/r=0                                                                                                     r=0
Rimp(βpr)=aimpJ0(βpr)+bimpY0(βpr)
a2mp=1
b2mp=0
   hJ0(βpr2)kβpJ1(βpr2)     =0
ri1ri[ρckθim(r,t)t+λ2θim(r,t)]rRimp(r)dr=[rRimp(r)θim(r,t)rrθim(r,t)dRimp(r)dr]ri1ri+ri1rirRimp(r)gimdr
i=12ri1ri[ρckθim(r,t)t+λ2θim(r,t)]rRimp(r)dr=i=12ri1rirRimp(r)gimdr
θmp(t)=i=12ri1rirRimp(r)θim(r,t)dr
gmp(t)=i=12ri1rirRimp(r)gimdr=0r1rR1mp(r)g1mdr
ρckdθmpdt+λ2θmp=gmp
θmp(t)=θmp(0)exp(kρcλ2t)+exp(kρcλ2t)0tkρcgmpexp(kρcλ2t)dτ=kρcexp(kρcλ2t)0tgmp(τ)exp(kρcλ2t)dτ
θim(r,t)=p=1cmp(t)Rimp(r)
cmp(t)=i=12ri1rirRimp(r)θim(r,t)drNmp=θmp(t)Nmp
θi(z,r,t)=m=0p=1Z(ηm,z)N(ηm)θmp(t)NmpRimp(r)
0tg(τ)exp[kρc(βp2+ηm2)τ]dτ=n=2N(n2)T0(n2)T0+t0exp[kρc(βp2+ηm2)τ]dτ+(n1)T0texp[kρc(βp2+ηm2)τ]dτ
(n1)T0t(n1)T0+t0
0tg(τ)exp[kρc(βp2+ηm2)τ]dτ=n=1N(n1)T0(n1)T0+t0exp[kρc(βp2+ηm2)τ]dτ
(n1)T0+t0tnT0
σθθ=γE(1ν)[1r220r2θrdr+1r20rθrdrθ]
εrr=(1ν2)E(σrrν1νσθθ)+(1+ν)γθ
εθθ=(1ν2)E(σθθν1νσrr)+(1+ν)γθ
εzz=0

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