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
Several applications require high average power laser systems and specifically high repetition rate ultra-intense and ultra-short pulse laser systems. In recent years, high power, short-pulse fiber lasers have been developed with ever increasing performances [1–3]. Owing to the excellent stability, low noise and all-fiber configurations, much attention has been paid to pulse pumped fiber lasers [4–7]. But during the operation of pulse laser, the fiber medium is simultaneously subject to heating by pulse pump radiation and cooling by surroundings, causing nonuniform temperature distribution followed by thermal stress, thermal birefringence and thermal lensing  detrimental to laser performance, and even inducing fiber facet damage . In addition, the time-dependent thermal effects caused by repetitive pump pulse more easily give rise to the fatigue fracture of laser medium and influence the light transmission characteristics.
Since thermal effects depend on the thermal profile inside the fiber, knowledge of the transient temperature distribution as a function of pump-pulse repetition rate, coolant flow, pump energy and laser material property permits one to predict thermal distortion and optimize a large variety of operating parameters. Existing analytical thermal models that describe the temperature and stress in fiber medium are restricted to special cases and approximations, such as continuous wave pump, steady–state conditions and radial two-dimensional (2D) heat flux approximation. 2D simulations for long fibers with small pump absorption coefficient constitute reasonable approximation to reality. However, such treatment may introduce a rather large calculation error in short-length fiber laser. The axial temperature changes significantly due to the exponentially varying thermal loading in the longitudinal direction, so the plain-strain approximation  calculating thermal stress is not fully satisfied for short-length fiber. A general analytical calculation describing the transient temperature distribution in a pulse-pumped fiber laser could be little found in literature except the Davis’s  research. But his heat conduction model is based on 2D heat flux approximation and excludes any heat source function. Until now a comprehensive analytical investigation on thermal stress in short-length fiber has not yet reported.
Up to now, analytical investigation on transient thermal effects is only available in solid-state lasers [11–13]. In view that the analytical methods possess many recognized advantages over numerical methods both in theory and in engineering application. In this paper, the three-dimensional (3D) transient heat conduction are modeled and solved analytically. The thermal stress and strain caused by the temperature filed in the absence of external forces are derived by solving the Navier displacement equations with the method of thermoelastic displacement potential. To the best of our knowledge, this is the first time that such a time-dependent analytical model has been reported in fiber lasers. We validate the model through a fully 3D finite element method (FEM), and apply the model to pulse end-pumped Er3+/Yb3+ co-doped phosphate glass fiber laser. The results show that the transient temperature and thermal stress fields in fiber make fluctuating distributions and their amplitudes are related to pulse duration and pulse period, which can be utilized to reduce the peak temperature and stress. Thus the thermal modeling can be further employed to optimize the performance of high-power fiber laser. Finally, the pump-induced transient thermal phase shift is analyzed. The calculated results demonstrate that the transient thermal effects will have an important influence on the light transmission characteristics and the beam quality of the laser medium.
2. Resolving of three-dimensional transient temperature field
Figure 1 is the schematic illustration of a pulse end-pumped short-length fiber butted against two planar mirrors.
The equation system (1a)-(1g) can be solved through the integral transform method introduced by Özisik . The 3D temperature rise expression in fiber is:
3. Thermal stress and strain fields
In the absence of external forces (e. g. extruded and stretched by piezoelectric ceramics as wavelength modulator), the thermal stress components caused by the temperature rise θ can be obtained by summing up the thermal components caused by each term of the temperature rise .
Since the first term θ1 of the temperature rise expression θ is only a function of r, the thermal stress caused by θ1 in the absence of external forces can be obtained by the displacement method as :
Since the second term θ2 of the temperature rise 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 .
The displacement field u(r, z) is governed by Navier displacement equations.
A particular solution of Eq. (2) is represented by the thermoelastic displacement potential Φ satisfying:Eq. (2), i.e., the solution of Eq. (2) when the right-hand sides are zero, will be represented by the Love function L satisfying:
The functions Φ and L shall be so constructed that they combine to satisfy all the boundary conditions and thus completely define the displacement field. Since the stresses are specified on the boundary, they need to be expressed in terms of Φ and L, and can be derived, respectively, from:Eq. (3), we will obtain the particular solution as:
4. Analytical results and discussions
To apply the above analytical solution, the pulse laser temporal shape is assumed as square incident laser irradiance:
An exemplary laser medium is Er3+/Yb3+ co-doped phosphate glass fiber. The thermal properties  are k = 0.55 W·m−1·K−1, η = 0.36, Th = 300 K, 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, r1 = 2.7 μm, r2 = 62.5 μm, l = 1 cm, Pin = 1W.
4.1. Transient temperature distribution
Figure 2 shows the temperature distribution from the analytical solution at the time of 0.1 s with the pulse duration more than 0.1 s. As a verification of the analytical thermal model, a time-dependent 3D temperature finite element analysis was implemented in a commercial software package Ansys , as shown in Fig. 3 . From the comparison of Fig. 2 (b) with Fig. 3 (b), it is noted that the results from our analytical thermal model are consistent with those from numerical solutions of a 3D finite element analysis within the calculation error.
Figure 4 shows the temperature distribution of fiber end surface at the pump side along the active fiber radial coordinate at different times with the interval of 1ms when T0 = 0.1 s, t0 = 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).
The transient temperature variation in the centre of fiber end surface as a function of time is plotted in Fig. 5 when T0 = 0.1 s, t0 = 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 . So, compared with the stationary state, the pulse pumping configuration displays more advantages to reduce heat deposition, and selecting appropriate pulse duration can efficiently avoid fiber fracture.
4.2. Transient thermal stress and strain fields
Figure 7 and 8 show the 3D thermal stress and strain fields from the analytical solution at the time of 0.1 s with the pulse duration more than 0.1 s. Comparing our analytical model with the plain-strain approximation (see Appendix C), as shown in Fig. 9 , we found that the radial thermal stress distribution does not induce any changes due to the small thermal gradient between the center and the edge of the cladding region. But the axial thermal stress distribution makes a great change due to considering of axial strain in our model from the comparison of two models, which correspondingly induces differences of strain distributions. The axial stress is about 10 times of the radial one in our analytical model, which owe to the significantly varying temperature field caused by the high gain coefficient of active fiver in the longitudinal direction. So our analytical model shows better accuracy of evaluating thermal stress and strain in fiber medium, and can be used to investigate end effects caused by heat disposition because of consideration of axial strain.
The radial thermal stress in the centre of fiber end surface as a function of time is plotted in Fig. 10 under different pulse laser temporal shapes. From three figures, we found that the pulse duration and period have an important influence on the peak thermal stress. Reasonable design for the pulse duration and period can effectively reduce thermal stress and optimize the performance of high-power fiber laser.
5. Thermal phase shift
Heating of the fiber induces a phase change in the signal via two effects, namely, by changing the index of refraction (coefficient δn / δT) and by axial strain field of the fiber.
The change in the effective index of the signal mode is given by :19]:
So the total instantaneous thermal phase shift rise experienced by the signal is given as:formula (5) with (6), we found that thermal phase shift caused by axial strain field amounts to less than 1% of the one caused by the varying index of refraction. We only include the thermal index change, which induces the thermal phase shift as follows:Fig. 11 when T0 = 0.1 s, t0 = 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.
In conclusion, a time-dependent analytical solution is derived to investigate the transient temperature and thermally induced stress and strain in pulse end-pumped fiber laser. Assuming the pulse laser temporal shape as square incident laser irradiance and using Er3+/Yb3+ 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 AEq. (9a) by and using integration of parts twice on the first term on the right hand side, one gets:Eq. (11e) are eigenfunctions Rim(r) corresponding to the eigenvalues βp
The eigenfunctions Rimp(r) satisfy the following orthogonality condition Eq. (11b)-Eq. (11d), a2mp, b2mp, and βp can be obtained:Eq. (11a), Eq. (10) can be written asEq. (9b)-Eq. (9d) and summing up the resulting expression from fiber core to cladding, one gets:Eq. (12) is
According to the plain-strain approximation, the thermal stress and strain can be expressed as:
This research was supported by the Guangdong Science and Technology Program (2005A10602001), the Guangzhou Science and Technology Program (2006Z2-D0161).
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