A modified model for the LD pumped 2 μm Tm:YAG laser: Thermal behavior and laser performance

Optics Communications 332 (2014) 332–338

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

A modified model for the LD pumped 2 μm Tm:YAG laser: Thermal behavior and laser performance Xuan Liu a, Haitao Huang b, Heyuan Zhu a, Yong Wang a,b, Li Wang b, Deyuan Shen a,n, Jian Zhang b, Dingyuan Tang c a

Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China School of Physical Science and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, China c School of Electrical and Electronic Engineering, Nanyang Technological University, 639798 Singapore b

art ic l e i nf o

a b s t r a c t

Article history: Received 12 March 2014 Received in revised form 3 July 2014 Accepted 10 July 2014 Available online 22 July 2014

A modified three-dimensional (3-D) plane-wave model is developed to analyze the laser performance and temperature distribution in an end-pumped Tm:YAG slab laser. A numerical iterative method is used to calculate the stable distributions of laser intensity and temperature. This three-dimensional model provides a more practical method than traditional 2-D model to analyze laser systems with non-uniform temperature distribution. Experiments using Tm:YAG ceramic slabs are carried out to examine the influence of coolant fluid temperature on laser output. The experimentally acquired data are found to be in reasonable agreement with the theoretical predictions. & 2014 Elsevier B.V. All rights reserved.

Keywords: 3-D modeling Rate equations Slab lasers Thermal analysis

1. Introduction Tm doped materials have attracted much attention as a laser medium that produces a 2 μm laser. The eye-safe nature of 2 μm region laser has guaranteed wide applications in medicinal, lidar and atmospheric sensing fields [1–3]. Tm doped solid state lasers usually adopt commercial high power GaAlAs laser diodes as pump sources due to its strong absorption bands around 0.79 μm. Benefit from the cross-relaxation process that can lead to a quantum yield of nearly two, high slope efficiencies well beyond Stokes Limit(
39%) can be achieved in this pumping scheme [4,5]. However, thermal effects such as thermal fracture and thermal distortions have greatly limited the power scalabilities of Tm doped solid state lasers [6]. Detailed analysis of the thermal behavior is necessary for further power scaling. Although much work have been done on analyzing thermal effects in Tm:YAG lasers in the past years, most of them have focused on 2-D analyses, which assumed that the temperature distribution is symmetric in one dimension and built the thermal model on a cross-sectional area of the crystal [7–9]. The 2-D assumption of temperature distribution is not always valid for practical conditions, especially when the pump laser intensity is not ideally uniform in one dimension. The 3-D heat flow arising

n

Corresponding author. E-mail address: [email protected] (D. Shen).

http://dx.doi.org/10.1016/j.optcom.2014.07.037 0030-4018/& 2014 Elsevier B.V. All rights reserved.

with non-uniform temperature distribution will change the temperature distribution acquired with 2-D heat flow assumption, and finally lead to a totally different prediction of the laser performance. Besides, to acquire a description of the stable temperature distribution and laser performance, an iteration method is needed, which is not included in previous works [8,9]. A more comprehensive understanding of the thermal behavior of Tm:YAG lasers calls for a 3-D coupled rate equation model combined with an iteration method. In this paper, a 3-D model of high-power end-pumped Tm:YAG slab laser is developed to analyze the temperature distribution and laser performance. The corresponding rate equations model is a modified version of that presented by Rustad [10], which has included the influences of ground-state depletion, cross-relaxation, upconversion and gain saturation. By replacing the Boltzmann occupation factor, which was usually treated as a constant in previous works, with a temperature dependent form, the influence of temperature is examined. Instead of using a heat conversion factor, which accounts for the fraction of absorbed pump power deposited as heat, to compute the heat distribution, this model has calculated the heat load distribution by solving the set of couple field equations. With the help of an ANSYS package, a 3-D FiniteElement-Analysis (FEA) is then utilized to compute the temperature distribution. This paper is organized as follows. In Section 2, a detailed introduction of the model is presented. The temperature distribution in the laser crystal is then calculated by the ANSYS package.

X. Liu et al. / Optics Communications 332 (2014) 332–338

After several iterations, the stable temperature distribution and laser output are obtained. In Section 3, the impact of the heat exchange coefficient in laser performance is analyzed based on the model presented in Section 2. Then an infrared camera is used to monitor the temperature distribution under non-lasing conditions and the real value of the heat exchange coefficient in our experiment is calculated. At last, experiments are carried out to examine the impact of coolant fluid temperature. The 3-D model described in Section 2 is applied to simulate the laser performance and reasonable agreement is achieved.

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equations of each element are then calculated after the following transitions are considered. A typical energy level diagram of Tm ions is sketched in Fig. 2. The arrows with different colors indicate different transitions considered in the laser model. According to Gunnar's work [10], the rate equations for each element can then be written as follows: I p ðx; y; zÞ dN4 ðx; y; zÞ ¼ σ abs  mean N1 ðx; y; zÞ hvp dt  kCR N 4 ðx; y; zÞN 1 ðx; y; zÞ þ kETU1 N 22 ðx; y; zÞ 

N 4 ðx; y; zÞ

τ4

ð1Þ 2. Numerical model Slab laser is a conventional choice for high power laser design. Compared to rod-like laser media, slabs have larger aspect ratio, which allowed this kind of laser to have excellent performance in dissipating heat load and reducing thermal induced strain and aberration [11,12]. Both face- and edge- pumped high power slab lasers have been achieved [13–16]. In comparison, end-pumped slab lasers have better mode matching efficiency and relatively simpler mechanical design [17]. With higher thermal conductivity and better mechanical properties than many other mediums, YAG crystals appear to have more advantages when used in high power diode-pumped solid state lasers. In recent years, transparent Tm doped YAG laser ceramic materials with similar optical and thermal properties compared to single crystals have been fabricated [18,19]. Furthermore, they can be fabricated with large size and high concentration while keeping low cost and short fabrication period, which can be advantageous when applied to high power slab lasers. Hence our model is developed on the basis of a Tm:YAG slab laser system, and the following experiments have used Tm:YAG ceramic slabs to examine their thermal behaviors and laser performances. In order to include in the model the influence of the spatial distributions of laser intensity and local temperature, the laser crystal is divided into many small elements. Fig. 1 has shown a sequence of elements along the length direction of the slab, where I f i and I bi are the laser intensities in each element, and the subscript f and b represent that the laser propagates forward and backward, respectively. L is the total length of crystal. The pump laser is assumed to incident from one side in an end-pumping scheme. I p is the pump laser intensity in each element. The rate

dN3 ðx; y; zÞ N 4 ðx; y; zÞ N 3 ðx; y; zÞ ¼ kETU2 N22 ðx; y; zÞ þ β 43  dt τ4 τ3 dN2 ðx; y; zÞ ¼ 2kCR N 4 ðx; y; zÞN1 ðx; y; zÞ  2ðkETU1 þ kETU2 ÞN22 ðx; y; zÞ dt N 2 ðx; y; zÞ N 3 ðx; y; zÞ N 4 ðx; y; zÞ þ β32 þ β42 

τ2

τ3

τ4

I L ðx; y; zÞ  σ ½f u N 2 ðx; y; zÞ  f l N 1 ðx; y; zÞ hvL 4

N1 ðx; y; zÞ ¼ NTm ðx; y; zÞ  ∑ Ni ðx; y; zÞ i¼2

ð3Þ

ð4Þ

where Ni denotes the population desity for the ith energy manifold, whereas NTm is the thulium dopant concentration, σ abs  mean is the mean effective absorption cross-section on the pump transition considering the bandwidth of the pump source, and σ is the atomic stimulated emission cross-sections at the laser wavelength. I p and I L are the pump and laser intensities vp and vL represent the frequencies of pump photons and laser photons, respectively. kCR Describes the cross-relaxation process, kETU1 describes the 3F4 þ 3F4-3H4 þ 3H6 energy transfer process and kETU2 describes the 3F4 þ 3F4-3H5 þ 3H6 process, τi is the lifetime of energy level i, β ij is the branching ratio of the spontaneous transition i-j, f u and f l are the Boltzmann occupation factors of the upper and lower laser level, respectively. The population density of the 3F4 manifold in each element can be calculated through the following equation: N2 ðx; y; zÞ ¼

NTm

κ Tm

½ð1 þ ir ðx; y; zÞ þ I R ðx; y; zÞÞ2 þ 2κ Tm ðir ðx; y; zÞ

þf I R ðx; y; zÞÞ1=2  kTm ¼ 2N Tm K ∑ Tm

Fig. 1. Schematic drawing of the slab divided along the length direction for modeling.

ð2Þ

N Tm

κ Tm

ð1 þ ir ðx; y; zÞ þ I R ðx; y; zÞÞ

ð5Þ ð6Þ

Fig. 2. Schematic view of the four lowest energy manifolds of Tm ions. CR—Crossrelaxation, ETU—Energy-Transfer Upconversion, and NR—Non-Radiative decay.

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where K ∑ Tm is the total upconversion parameter, η4 is the crossrelaxation efficiency, β4 is the total branching ratio for the 3 H4-3F4 transition, I R and ir are the relative laser intensity and the relative pump intensity, respectively [10]. Another important factor introduced in (5) is f , which is defined as f¼

fl f u þf l

ð7Þ

The propagating laser fields can be calculated through the following formula: ∂I f ðx; y; zÞ ¼ σ ½f u N 2 ðx; y; zÞ  f l N1 ðx; y; zÞI f ðx; y; zÞ ∂z

ð8Þ

∂I b ðx; y; zÞ ¼  σ ½f u N 2 ðx; y; zÞ  f l N1 ðx; y; zÞI b ðx; y; zÞ ∂z

ð9Þ

And the pump distribution is calculated through the following equation: ∂I p ðx; y; zÞ ¼  σ abs  mean N 1 ðx; y; zÞI p ðx; y; zÞ ∂z

ð10Þ

The total logarithmic single-pass gain G of the laser is obtained by an integration over the volume. Z ð11Þ In G ¼ σ ½f u N 2 ðx; y; zÞ  f l N 1 ðx; y; zÞψ s ðx; y; zÞdV therein ψ s ðx; y; zÞ is the spatial distribution function of the singlepass gain. The pump beam is assumed to have a Gaussian transverse distribution with a constant beam size through the entire length of the gain medium. A typical beam size of 12 mm  0.8 mm in Nd-doped slab lasers [24] is chosen for the numerical simulation. The divergence of the pump beam in the crystal is neglected. This assumption is made based on the fact that for a pump laser with a M2 beam quality factor of 78, for example, the beam radius of the pump laser only varies from 400 μm to 460 μm in the crystal if a thermal lens of 50 mm is taken into consideration and the divergence could be smaller if further beam shaping techniques are used to improve the beam quality of the pump source. The laser beam is assumed to have an overlapping distribution with the pump beam. The boundary condition is then derived from self-consistency of circulating laser power in the cavity: ð1  TÞð1 Lα ÞG2 ¼ 1

ð12Þ

where T is the transmission of the output coupler and Lα is the round-trip cavity loss. After the laser intensity distribution is calculated in the first stage, a 3-D thermal model is built for temperature distribution calculation. The corresponding configuration is shown in Fig. 3. The Tm:YAG is sandwiched between two copper heatsinks. Indium layers of about 100 μm thickness are used for even thermal contact. The Cu heatsinks are cooled from surface S1 and S2 by a coolant fluid, which is assumed to be de-ionized water in this paper, with a heat exchange coefficient of h1 . The other four surfaces of Tm:YAG are assumed to be in thermal contact with air with a heat exchange coefficient of h2 . The stationary heat conduction equation that governed the thermal performance of the system is ∇2 Tðx; y; zÞ ¼

P heat ðx; y; zÞ kYAG

in Tm : YAG

in S1 and S2 surfaces

∂T h2 ¼ ðT air  T sj Þ; ∂nj kYAG

in other four surrounding surfaces of Tm : YAG

ð15Þ where kCu and kYAG are thermal conductivities of copper heatsinks and Tm:YAG, respectively. ni and nj are the normal direction to the corresponding surfaces. P heat ðx; y; zÞ is the heat density generated inside the crystal in each element and it is calculated through the following equation: P heat ðx; y; zÞ ¼

∂I p ðx; y; zÞ ∂½I f ðx; y; zÞ  I b ðx; y; zÞ N 2 hυL   ∂z ∂z τ2rad

ð16Þ

In the equation above, τ2rad is the radiative life time of Tm ions on 3 F4 manifold. It should be noted that the effects of energy transfer upconversion processes are not included in the equation above. The effect of the 3F4 þ 3F4-3H4 þ 3H6 process is largely compensated by the cross-relaxation process in Tm-doped systems. As for the 3F4 þ 3F4-3H5 þ 3H6 process, the excited ions in 3H5 manifolds are mainly de-excited by non-radiative decay to 3F4 manifolds with a radiative quantum efficiency of 0.003 [26]. The fluorescence photons emitted in the energy transfer upconversion processes can be neglected, thus it is reasonable to assume that no power is extracted from the Tm:YAG crystal in these processes. Hence the effects of the energy transfer upconversion processes are neglected in Eq. (16). With the help of ANSYS package, a 3-D finite-element-analysis (FEA) is performed to compute the temperature distribution inside the crystal. The active medium is divided into a mesh of finite elements with 50 elements in every direction. This meshing is in accordance with that used in the rate equation model. Then the influence of temperature distribution on laser performance is included in the rate equation model by changing the corresponding occupation factors, which was treated as constants in the first step, to temperature dependent forms:   E21 exp  KTðx;y;zÞ   f u ðx; y; zÞ ¼ ð17Þ 2i ∑9i ¼ 1 exp  KT ðEx;y;z Þ

ð13Þ

  17 exp  KTEðx;y;z Þ   f l ðx; y; zÞ ¼ E1j ∑10 j ¼ 1 exp  KTðx;y;zÞ

ð14Þ

where E2i is the energy of the ith Stark level in the excited manifold, E1j is the energy of the jth Stark level in the ground manifold, Tðx; y; zÞ is the temperature in the corresponding

and the boundary conditions are ∂T h1 ¼ ðT  T si Þ; ∂ni kCu coolant

Fig. 3. Thermal model configuration.

ð18Þ

X. Liu et al. / Optics Communications 332 (2014) 332–338

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Table 1 Parameters used in the numerical simulation. Symbol

Quantity

Value

/ HWL τ2 σ σ abs  mean ψp ψL λ Δλ ωH ωV β4 η4 η3 K ∑ Tm Toc Lα h2 kCu kYAG kindium T coolant T air

Doping concentration Crystal dimension Upper level lifetime Atomic stimulated emission cross-section Mean effective pump absorption cross-section Pump laser profile 2 μm Laser profile Pump wavelength FWHM of pump laser Beam radius of 2 μm laser Total branching ratio 3H4-3F4 Cross-relaxation efficiency Cross-relaxation efficiency Total upconversion parameter Transmission of output couple Resonator round-trip loss Heat-exchange coefficient Heat conductivity of copper heat-sink Heat conductivity of Tm:YAG Heat conductivity of indium Temperature of coolant fluid Temperature of air

4 at% 1 mm  12 mm  14 mm 12 ms [10] 0.5  10  20 cm2 [10] 2.8  10  21 cm2 [19] Gaussian Gaussian 787 nm 3 nm 0.4 mm  6 mm 0.6 [10] 0.97 [10] 0 [10] 3  10  18 cm3/s [10] 10% 1.5% 10 W/m2 K [20] 396 W/m/K [25] 7 W/m/K [27] 81.8 W/m/K 290 K 300 K

element, and K is Boltzmann's constant. The absorption and emission crosssections are also influenced by the temperature, but because of the lack of the corresponding data from current available papers, the influence of temperature on these parameters are not included in our model. However, it should be noted that, from the result of a simulation of high power Yb:YAG slab lasers by Liu et al. [26], it can be seen that most part of the influence of temperature on laser performance could be explained by the temperature dependent nature of Boltzmann occupation factor. In the simulation, the output power with 1.2 kW pump source drops from 760.1 W to 583.5 W by changing the Boltzmann occupation factor into a temperature dependent form. Further consideration of the temperature influences on the spectroscopic values only lead to an ultimate value of output power of 519.1 W. Due to the same quasi-three level nature of Tm-doped lasers, the model presented in this paper could give an comprehensive description of the laser performance of the Tm:YAG slab laser Boltzmann occupation factor. The temperature distribution in the crystal is calculated through the 3-D FEA performed with ANSYS package, then the result is used to modify the laser output in the first stage. After several iterations, both the laser output and the temperature converged to a stable distribution and the laser performance with temperature influences included is obtained.

3. Experiments and results 3.1. Impact of heat exchange coefficient In order to use the numerical model developed in Section 2 to simulate the laser performance, a key factor that must be calculated in the first step is the heat exchange coefficient, which is usually presumed as a constant in previous theoretical simulations [8,9,20,21]. But in experiment, this factor can differ from case to case according to different heat-sink designs. For example, the heat exchange coefficient for laminar flow in tubes is in the order of 4000 W/m2  K, and for turbulent flow in micro-channel heatsinks, this factor can increase to the order of 10,000 W/m2  K or higher [22]. By deliberately designing the parameters of microchannel heatsinks, the heat exchange coefficient can be

augmented in a large scale. The impact of heat exchange coefficient on laser performance in Tm:YAG slab laser system is examined using our numerical model presented in Section 2. The parameters used are listed in Table 1. Fig. 4(a) and (b) depict the laser performances for three different values of heat exchange coefficients. From Fig. 4, it can be seen that as the heat exchange coefficient increases from 5000 W/m2  K to 50,000 W/m2  K, the maximum temperature rise in the crystal is greatly reduced from around 600 1C to a much lower temperature of around 180 1C. Due to the quasi-three-level nature of Tm ions, high temperature will lead to heavier reabsorption of 2 μm laser and reduction of population inversion, which will lead to the deterioration of laser performance. With the decrease in temperature, the corresponding laser performance is greatly improved then. The maximum output power that can be achieved with 800 W pump power increased from 57 W to 337 W. It should be noted that for the heat exchange coefficient of as low as 5000 W/m2  K, severe saturation occurs when the pump power scales up. The temperature as high as 619 1C at 800 W pump power not only limits the laser output, but also brings higher risks of thermal damage of the laser crystal. A high heat exchange coefficient is essential for power scaling of slab lasers. It should be mentioned that the thermal conductivity of Tm:YAG crystals would change with temperature. According to the work of Sato et al. [28], the thermal conductivity of 5 at% Yb:YAG crystal decreases from 6.82 W/(mK) at 25 1C to 5.36 W/(mK) at 200 1C. By applying these thermal conductivities to the whole sample in our model, the influence of the temperature dependence of thermal conductivity on laser performance is estimated. When the heat transfer coefficient is set to be 10,000 W/m2 K, the laser output at 800 W pump power drops from 246.5 W to 219.5 W with the decrease in thermal conductivity. As the Tm and Yb ions are nearly identical in size, similar effects should exist in Tm:YAG crystals. Thus it is reasonable to conclude that in this model, which did not include the temperature dependence of the thermal conductivity of Tm:YAG crystals, the maximum error bounds would be around 10.9% for heat transfer coefficient larger than 10,000 W/m2 K. As to laser systems with lower heat transfer coefficient such as 5000 W/m2 K shown in Fig. 4, the maximum temperature rise would be much larger and the saturation would be much heavier than that shown in Fig. 4(a). Besides, the

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Fig. 4. (a) Laser output versus incident pump power with different heat exchange coefficients and (b) maximum temperature in the crystal versus incident pump power with different heat exchange coefficients.

Fig. 5. (a) 3-D temperature distribution for Gaussian pump distribution and (b) 3-D temperature distribution for Top-Hat pump distribution.

Fig. 6. (a) Temperature distributions in the horizontal cross-sections in the center of the slabs with h1 ¼ 10,000 W/m2 K and (b) temperature distributions in the vertical cross-sections in the center of the slabs with h1 ¼10,000 W/m2 K.

absorption of the pump laser would decrease to a large extent at such a high temperature and the laser performance would further deteriorate. Typical 3-D temperature distributions in the crystals with 800 W of pump radiation are shown in Fig. 5 for Gaussian pump distribution and Top-Hat pump distribution assumptions. Fig. 6 (a) and (b) shows the corresponding temperature profile on the horizontal and vertical cross-sections in the center of the slab, respectively.

It can be seen that the highest temperature arises at the center of the incident surface of the crystal when the Gaussian pump distribution assumption is used. In comparison, for the Top-Hat pump distribution assumption, one-dimensional temperature distribution could be observed on the incident surface, as can be seen in Figs. 5(b) and 6(a). This is attributed to the large discrepancy between the heat exchange coefficients of the two large surfaces (10,000 W/m2  K) and that of the side-surfaces (10 W/m2  K). This fact suggests that conventional one-dimensional heat flow

X. Liu et al. / Optics Communications 332 (2014) 332–338

assumption in slab lasers is valid only when the pump distribution is ideally uniform in the horizontal direction. For more practical conditions, in which the shape of the pump beam can usually be described by a super-Gaussian function, a 3-D thermal model is of essential need for a better understanding of the thermal behavior, and hence the laser performance. 3.2. Calculation of heat exchange coefficient As the value of the heat exchange coefficient could greatly affect the laser performance, which is mentioned before, it is necessary to measure the heat exchange coefficient experimentally. The experimental arrangement is shown in Fig. 7. Two cylindrical lenses are used to focus the diode laser beam into a spot size of
1.6 mm  0.6 mm. The resonator used in the following laser experiments is configured with a flat input mirror (FM) and a plano-concave output coupler (OC). The output coupler has a radius of curvature of 200 mm and a transmission of 5% at 2015 nm. The Tm:YAG ceramic sample is sandwiched between two copper heatsinks. Indium foils of about 0.1 mm thick are used to offer even thermal contact for the samples and the heatsinks. The contact surfaces of the laser crystal and the copper heatsinks are all polished, and certain external forces are applied to the heatsink-indium-crystal volume to guarantee an effective thermal contact. The cooling water temperature is kept at 15 1C. The Tm: YAG ceramic sample used has a doping concentration of 4 at% and the dimensions are 1.29 mm  10 mm  15 mm. An infrared thermal camera (FLIR SC655) is used to monitor the temperature distribution on the incident surfaces of the ceramics under non-lasing conditions (without the input and ouput mirrors). Fig. 8 shows the temperature distribution obtained with 15 W of pumping laser. The temperature profiles on the two lines are shown in Fig. 8(b). A 3-D thermal model is then built and the temperature distribution is calculated with ANSYS package. By carefully fitting the heat exchange coefficient, a good agreement could be reached between the simulation results and the temperature distribution obtained with the infrared thermal camera, as is shown in Fig. 8 (b). The heat-exchange coefficient of our heatsinks is then

Fig. 7. Experimental setup.

337

calculated to be around 14,000 W/m2  K. This value is used in the following numerical calculations. 3.3. Laser performance and thermal behavior The influence of coolant temperature on laser performance is analyzed. The laser performances of the Tm:YAG sample for different cooling water temperatures are shown in Fig. 9. The laser spectrum is monitored by an optical spectrum analyzer (AQ6375, Yokogawa) and the spectrum is centered at 2013.6 nm for both conditions. A round trip loss of 5.7% was used in numerical simulation to show good agreement with the experimental data. The real round trip cavity loss is also measured using the Caird analysis method, which determined the resonator loss by determining the slope efficiency using different output couplers [23]. The real round trip loss is then estimated to be around 5.61% which is in reasonable agreement with that used in the numerical simulation. The large round trip loss may be attributed to the large diffraction loss introduced by the existence of high-order laser modes. Besides, the imperfect coatings of the cavity mirror also introduces additional losses. As can be seen in Fig. 9, when the temperature increases from 15 1C to 25 1C, the slope efficiency experiences a drop of 2% in our experiment, whereas the numerical results illustrate about 34% of the difference. The discrepancy may be attributed to the uneven thermal contact in our experiment, which means that the average heat exchange coefficient in the experiment may be lower than that we have calculated in the

Fig. 9. Laser performance of sample 1 for different coolant temperatures.

Fig. 8. (a) Infrared thermal image of the incident surface under non-lasing condition and (b) temperature distributions on the vertical line and the horizontal line.

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former step (14,000 W/m2  K). Besides, this model did not include the influence of temperature on absorption and stimulated crosssections as well as the temperature dependent nature of the thermal conductivities of YAG and copper. Because of the lack of these parameters from current available papers, these factors are not included in this paper. Consideration of these factors may lead to a better match between the numerical and the experimental results. This model should be used in a moderate temperature range, where the usages of the spectroscopic values and the thermal conductivities in this paper are suitable. The model presented here provides a method for comprehensively understanding of the laser and thermal performance of high power Tm: YAG slab lasers.

4. Conclusion A three-dimensional plane-wave thermal model has been developed to analyze the couple field of laser and temperature distribution for high power end-pumped slab lasers. A 3-D FEA combined with an iterative method is applied to calculate the stable temperature and laser distribution in the slab crystal. Experiments are carried out to analyze thermal behaviors of the Tm:YAG ceramic slab lasers. An infrared thermal camera is used to monitor the temperature of slabs under non-lasing condition to give an accurate value of the heat exchange coefficient. An elevation of 10 1C in the cooling water temperature results in a decrease of 2% in slope efficiency. The corresponding numerical simulation results can give reasonable illustrations of these phenomena. This model provides a more accurate method to deal with problems that has 3-D inhomogeneous or asymmetric distributions of lasers and heat loads.

Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC 61078035, 61177045 and 61308047), the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). The authors wish to acknowledge

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