## Analytical investigation on transient thermal effects in pulse end-pumped short-length fiber laser

Optics Express, Vol. 17, Issue 15, pp. 12875-12890 (2009)

http://dx.doi.org/10.1364/OE.17.012875

Acrobat PDF (365 KB)

### 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

## 1. Introduction

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]

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]

## 2. Resolving of three-dimensional transient temperature field

*T*and

_{1}*T*are the temperatures in fiber core and cladding region, respectively.

_{2}*T*is the heat sink temperature.

_{h}*k*denotes the fiber thermal conductivity.

*h*is the convective coefficient.

*ω*is Gaussian radius of the pump light.

_{p}*η*is the fractional thermal loading,

*α*is the optical absorption coefficient,

*ρ*is the density of the fiber material and

*c*is its the specific heat,

*P*is the energy in each pulse and

_{in}*g*(

*t*) is laser pulse shape.

*J*and

_{0}*J*are the zero and first rank Bessel function of the first kind, respectively. More detailed deductions are shown in Appendix A.

_{1}## 3. Thermal stress and strain fields

*θ*can be obtained by summing up the thermal components

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]

*θ*of the temperature rise expression

_{1}*θ*is only a function of

*r*, the thermal stress caused by

*θ*in the absence of external forces can be obtained by the displacement method as [16]:

_{1}*E*,

*γ*and

*ν*are Young’s modulus, thermal expansion coefficient and Poisson’s ratio

*θ*of the temperature rise

_{2}*θ*is axisymmetric and varies in both

*r*and

*z*directions, the resulting thermal stress under the traction free condition can be determined by the thermoelastic displacement potential method [16].

## 4. Analytical results and discussions

*t*,

_{0}*T*and

_{0}*n*are pulse duration, pulse period and the number of pulse, respectively. Integrating the time term of

*θ*, and the expression is shown in Appendix B.

^{3+}/Yb

^{3+}co-doped phosphate glass fiber. The thermal properties [17

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

*k*= 0.55 W·m

^{−1}·K

^{−1},

*η*= 0.36,

*T*= 300 K,

_{h}*h*= 10 W·m

^{−2}·K

^{−1}and the other parameters are taken as

*ρ*= 3.2 g·cm

^{−3},

*c*= 960 J·kg

^{−1}·K

^{−1},

*γ*= 9.6 × 10

^{−6}K

^{−1},

*ν*= 0.27,

*E*= 56.4 GPa,

*r*= 2.7 μm,

_{1}*r*= 62.5 μm,

_{2}*l*= 1 cm,

*P*= 1W.

_{in}### 4.1. Transient temperature distribution

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

*T*= 0.1 s,

_{0}*t*= 0.01 s. From the graph 4(a), a transient thermal diffusive process with time is clearly displayed by the evolution of the ten curves. During the first pulse pump space time, as shown in Fig. 4(b) and Fig. 4(c), the temperature distribution in fiber tends to be flat and the temperature difference between the center and edge gradually decreases because of the air convection and the heat conduction of fiber medium. After the first pulse pump ends, the temperature of each position in fiber medium nearly reaches the same, but there is still about 2 °C temperature rise compared with the initial value due to the insufficiency of heat dissipation, which is deposited in laser medium as residual heat. The fiber core temperature will come back to 300 K if the first pulse pump space time is long enough, about 50 s, as shown in Fig. 4(d).

_{0}*T*= 0.1 s,

_{0}*t*= 0.01 s. Graph 5(b) and 5(c) are the enlarged drawings of the part of graph 5(a). As shown in figures, the temperature fluctuates with the time and the peak value gradually increases. After a long enough time (about 50 s), the peak temperature becomes stable and the temperature field makes a periodical change, as shown in Fig. 5(c). Figure 6 shows the same the transient temperature distribution under different pulse duration with the same pulse period 0.1 s. Compared with the short pulse duration, the long pulse duration induce a higher temperature rise because of the longer time of pump injection. Furthermore, if fiber medium is under steady-state pump, the maximum temperature will exceed 1000 K in the same conditions according to Liu et al [17

_{0}17. T. Liu, Z. M. Yang, and S. H. Xu, “3-Dimensional heat analysis in short-length Er^{3+}/Yb^{3+} co-doped phosphate fiber laser with upconversion,” Opt. Express **17**(1), 235–247 (2009). [CrossRef] [PubMed]

### 4.2. Transient thermal stress and strain fields

## 5. Thermal phase shift

*δn*

**/**

*δT*) and by axial strain field of the fiber.

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]

*f*(

_{s}*r*) is the signal mode intensity, and can be approximated by a Gaussian distribution.The instantaneous thermal phase shift caused by the varying index of refraction is obtained by integrating

*△n*along

*z*.The instantaneous thermal phase shift caused by axial strain field is gotten as [19

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]

*n*is the refractive index of fiber,

_{0}*λ*is the signal wavelength.

_{s}^{−6}K

^{−1}, and

*λ*= 1.53 μm. The phase shift rise induced by heating as a function of time is shown in Fig. 11 when

_{s}*T*= 0.1 s,

_{0}*t*= 0.01 s. Graph 11(b) is the enlarged drawing of graph 11(a). The large temperature rise results in a non-negligible phase shift. Subsequently, the varying thermal phase shift will have an influence on the light transmission characteristics and the beam quality of the laser medium. It is pointed out that the maximum value of transient phase shift is about one order of magnitude smaller than the stationary-state one owing to the higher temperature rise under steady state.

_{0}## 6. Conclusions

^{3+}/Yb

^{3+}co-doped phosphate glass fiber as an exemplary laser medium, we calculate the transient temperature, thermal stress and strain distributions. The results show that pulse shape has an important influence on the peak thermal stress, and reasonable design for the pulse duration and period can be utilized to reduce thermal stress and optimize the performance of high-power fiber laser. At last, the pump-induced transient thermal phase shift is analyzed. The calculated results demonstrate that the varying thermal phase shift will affect the light transmission characteristics and the beam quality of the laser medium.Appendix A

*θ=T-T*, Eq. (1a)-(1g) are simplified as: Inversion formula is:followed by integral transform:whereThe integral transform of the system (7) by the application of the transform (8b) yields: whereMultiplying Eq. (9a) by

_{h}*R*(

_{im}*r*) in the above equation is chosen to satisfy Solutions of the Eq. (11e) are eigenfunctions

*R*(

_{im}*r*) corresponding to the eigenvalues

*β*where

_{p}*R*(

_{imp}*r*) satisfy the following orthogonality condition [20

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]

*a*equal to unity, according to Eq. (11b)-Eq. (11d),

_{1mp}*a*,

_{2mp}*b*, and

_{2mp}*β*can be obtained: In view of Eq. (11a), Eq. (10) can be written as

_{p}## Acknowledgement

## References and links

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 |

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. |

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 Er |

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. |

5. | Z. Y. Dai, Z. S. Peng, Y. Z. Liu, and Z. H. Ou, “Research on SBS and pulse pumped hybrid Q-switched Er |

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. |

7. | C. G. Ye, P. Yan, M. Gong, and M. Lei, “Pulsed pumped Yb-doped fiber amplifier at low repetition rate,” Chin. Opt. Lett. |

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 |

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. |

10. | M. K. Davis, M. J. F. Digonnet, and R. H. Pantell, “Thermal effects in doped fobers,” J. Lightwave Technol. |

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 |

12. | W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. |

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 |

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. |

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

17. | T. Liu, Z. M. Yang, and S. H. Xu, “3-Dimensional heat analysis in short-length Er |

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. |

20. | P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer |

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: May 26, 2009

Revised Manuscript: June 30, 2009

Manuscript Accepted: July 2, 2009

Published: July 13, 2009

**Citation**

T. Liu, Z. M. Yang, and S. H. Xu, "Analytical investigation on transient thermal effects in pulse end-pumped short-length fiber laser," Opt. Express **17**, 12875-12890 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12875

Sort: Year | Journal | Reset

### References

- 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)
- 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]
- 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).
- 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. 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.
- 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).
- 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]
- 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]
- 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]
- 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]
- W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44(7), 3162–3170 (1973). [CrossRef]
- 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).
- M. N. Özisik, Heat Conduction (Wiley, New York, 1980).
- 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]
- Y. Takeuchi, Thermal Stress (Science, 1977).
- 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]
- Ansys Finite Element Software Package, http://www.ansys.com/
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.