A modified model for the LD pumped 2μm Tm YAG laser ... · A modified model for the LD pumped 2...

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A modied 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 b 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 article info Article history: Received 12 March 2014 Received in revised form 3 July 2014 Accepted 10 July 2014 Available online 22 July 2014 Keywords: 3-D modeling Rate equations Slab lasers Thermal analysis abstract A modied 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 inuence of coolant uid 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. 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 elds [13]. 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. Benet from the cross-relaxation process that can lead to a quantum yield of nearly two, high slope efciencies 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 scalabil- ities 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 [79]. 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 ow arising with non-uniform temperature distribution will change the tem- perature distribution acquired with 2-D heat ow assumption, and nally lead to a totally different prediction of the laser perfor- mance. 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 compre- hensive 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 modied version of that presented by Rustad [10], which has included the inuences of ground-state depletion, cross-relaxa- tion, upconversion and gain saturation. By replacing the Boltz- mann occupation factor, which was usually treated as a constant in previous works, with a temperature dependent form, the inuence 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 eld equations. With the help of an ANSYS package, a 3-D Finite- Element-Analysis (FEA) is then utilized to compute the tempera- ture distribution. This paper is organized as follows. In Section 2, a detailed introduction of the model is presented. The temperature distribu- tion in the laser crystal is then calculated by the ANSYS package. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/optcom Optics Communications http://dx.doi.org/10.1016/j.optcom.2014.07.037 0030-4018/& 2014 Elsevier B.V. All rights reserved. n Corresponding author. E-mail address: [email protected] (D. Shen). Optics Communications 332 (2014) 332338

Transcript of A modified model for the LD pumped 2μm Tm YAG laser ... · A modified model for the LD pumped 2...

Page 1: A modified model for the LD pumped 2μm Tm YAG laser ... · A modified model for the LD pumped 2 μm Tm:YAG laser: Thermal behavior and laser performance Xuan Liua, Haitao Huangb,

A modified model for the LD pumped 2 μm Tm:YAGlaser: 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, Chinab School of Physical Science and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, Chinac School of Electrical and Electronic Engineering, Nanyang Technological University, 639798 Singapore

a r t i c l e i n f o

Article history:Received 12 March 2014Received in revised form3 July 2014Accepted 10 July 2014Available online 22 July 2014

Keywords:3-D modelingRate equationsSlab lasersThermal analysis

a b s t r a c t

A modified three-dimensional (3-D) plane-wave model is developed to analyze the laser performanceand temperature distribution in an end-pumped Tm:YAG slab laser. A numerical iterative method is usedto calculate the stable distributions of laser intensity and temperature. This three-dimensional modelprovides a more practical method than traditional 2-D model to analyze laser systems with non-uniformtemperature distribution. Experiments using Tm:YAG ceramic slabs are carried out to examine theinfluence of coolant fluid temperature on laser output. The experimentally acquired data are found to bein reasonable agreement with the theoretical predictions.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Tm doped materials have attracted much attention as a lasermedium that produces a 2 μm laser. The eye-safe nature of 2 μmregion laser has guaranteed wide applications in medicinal, lidarand atmospheric sensing fields [1–3]. Tm doped solid state lasersusually adopt commercial high power GaAlAs laser diodes aspump sources due to its strong absorption bands around0.79 μm. Benefit from the cross-relaxation process that can leadto a quantum yield of nearly two, high slope efficiencies wellbeyond Stokes Limit(�39%) can be achieved in this pumpingscheme [4,5]. However, thermal effects such as thermal fractureand thermal distortions have greatly limited the power scalabil-ities of Tm doped solid state lasers [6]. Detailed analysis of thethermal behavior is necessary for further power scaling.

Although much work have been done on analyzing thermaleffects in Tm:YAG lasers in the past years, most of them havefocused on 2-D analyses, which assumed that the temperaturedistribution is symmetric in one dimension and built the thermalmodel on a cross-sectional area of the crystal [7–9]. The 2-Dassumption of temperature distribution is not always valid forpractical conditions, especially when the pump laser intensity isnot ideally uniform in one dimension. The 3-D heat flow arising

with non-uniform temperature distribution will change the tem-perature distribution acquired with 2-D heat flow assumption, andfinally lead to a totally different prediction of the laser perfor-mance. Besides, to acquire a description of the stable temperaturedistribution and laser performance, an iteration method is needed,which is not included in previous works [8,9]. A more compre-hensive understanding of the thermal behavior of Tm:YAG laserscalls for a 3-D coupled rate equation model combined with aniteration method.

In this paper, a 3-D model of high-power end-pumped Tm:YAGslab laser is developed to analyze the temperature distribution andlaser performance. The corresponding rate equations model is amodified version of that presented by Rustad [10], which hasincluded the influences of ground-state depletion, cross-relaxa-tion, upconversion and gain saturation. By replacing the Boltz-mann occupation factor, which was usually treated as a constant inprevious works, with a temperature dependent form, the influenceof temperature is examined. Instead of using a heat conversionfactor, which accounts for the fraction of absorbed pump powerdeposited as heat, to compute the heat distribution, this model hascalculated the heat load distribution by solving the set of couplefield equations. With the help of an ANSYS package, a 3-D Finite-Element-Analysis (FEA) is then utilized to compute the tempera-ture distribution.

This paper is organized as follows. In Section 2, a detailedintroduction of the model is presented. The temperature distribu-tion in the laser crystal is then calculated by the ANSYS package.

Contents lists available at ScienceDirect

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

Optics Communications

http://dx.doi.org/10.1016/j.optcom.2014.07.0370030-4018/& 2014 Elsevier B.V. All rights reserved.

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

Optics Communications 332 (2014) 332–338

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After several iterations, the stable temperature distribution andlaser output are obtained. In Section 3, the impact of the heatexchange coefficient in laser performance is analyzed based on themodel presented in Section 2. Then an infrared camera is used tomonitor the temperature distribution under non-lasing conditionsand the real value of the heat exchange coefficient in our experi-ment is calculated. At last, experiments are carried out to examinethe impact of coolant fluid temperature. The 3-D model describedin Section 2 is applied to simulate the laser performance andreasonable agreement is achieved.

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 indissipating heat load and reducing thermal induced strain andaberration [11,12]. Both face- and edge- pumped high power slablasers have been achieved [13–16]. In comparison, end-pumpedslab lasers have better mode matching efficiency and relativelysimpler mechanical design [17]. With higher thermal conductivityand better mechanical properties than many other mediums, YAGcrystals appear to have more advantages when used in high powerdiode-pumped solid state lasers. In recent years, transparent Tmdoped YAG laser ceramic materials with similar optical andthermal properties compared to single crystals have been fabri-cated [18,19]. Furthermore, they can be fabricated with large sizeand high concentration while keeping low cost and short fabrica-tion period, which can be advantageous when applied to highpower slab lasers. Hence our model is developed on the basis of aTm:YAG slab laser system, and the following experiments haveused Tm:YAG ceramic slabs to examine their thermal behaviorsand laser performances.

In order to include in the model the influence of the spatialdistributions of laser intensity and local temperature, the lasercrystal is divided into many small elements. Fig. 1 has shown asequence of elements along the length direction of the slab, whereIf i and Ibi are the laser intensities in each element, and thesubscript f and b represent that the laser propagates forward andbackward, respectively. L is the total length of crystal. The pumplaser is assumed to incident from one side in an end-pumpingscheme. Ip is the pump laser intensity in each element. The rate

equations of each element are then calculated after the followingtransitions are considered.

A typical energy level diagram of Tm ions is sketched in Fig. 2.The arrows with different colors indicate different transitionsconsidered in the laser model. According to Gunnar's work [10],the rate equations for each element can then be written as follows:

dN4ðx; y; zÞdt

¼ σabs�meanN1ðx; y; zÞIpðx; y; zÞ

hvp

�kCRN4ðx; y; zÞN1ðx; y; zÞþkETU1N22ðx; y; zÞ�

N4ðx; y; zÞτ4

ð1Þ

dN3ðx; y; zÞdt

¼ kETU2N22ðx; y; zÞþβ43

N4ðx; y; zÞτ4

�N3ðx; y; zÞτ3

ð2Þ

dN2ðx; y; zÞdt

¼ 2kCRN4ðx; y; zÞN1ðx; y; zÞ�2ðkETU1þkETU2ÞN22ðx; y; zÞ

�N2ðx; y; zÞτ2

þβ32N3ðx; y; zÞ

τ3þβ42

N4ðx; y; zÞτ4

� σ½f uN2ðx; y; zÞ� f lN1ðx; y; zÞ�ILðx; y; zÞ

hvLð3Þ

N1ðx; y; zÞ ¼NTmðx; y; zÞ� ∑4

i ¼ 2Niðx; y; zÞ ð4Þ

where Ni denotes the population desity for the ith energy mani-fold, whereas NTm is the thulium dopant concentration, σabs�mean

is the mean effective absorption cross-section on the pumptransition considering the bandwidth of the pump source, and σis the atomic stimulated emission cross-sections at the laserwavelength. Ip and IL are the pump and laser intensities vp andvL represent the frequencies of pump photons and laser photons,respectively. kCR Describes the cross-relaxation process, kETU1describes the 3F4þ3F4-3H4þ3H6 energy transfer process andkETU2 describes the 3F4þ3F4-3H5þ3H6 process, τi is the lifetimeof energy level i, βij is the branching ratio of the spontaneoustransition i-j, f u and f l are the Boltzmann occupation factors ofthe upper and lower laser level, respectively. The populationdensity of the 3F4 manifold in each element can be calculatedthrough the following equation:

N2ðx; y; zÞ ¼NTm

κTm½ð1þ irðx; y; zÞþ IRðx; y; zÞÞ2þ2κTmðirðx; y; zÞ

þ f IRðx; y; zÞÞ�1=2�NTm

κTmð1þ irðx; y; zÞþ IRðx; y; zÞÞ ð5Þ

kTm ¼ 2NTmK∑Tm ð6Þ

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

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

X. Liu et al. / Optics Communications 332 (2014) 332–338 333

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where K∑Tm is the total upconversion parameter, η4 is the cross-relaxation efficiency, β4 is the total branching ratio for the3H4-

3F4 transition, IR and ir are the relative laser intensity andthe relative pump intensity, respectively [10]. Another importantfactor introduced in (5) is f , which is defined as

f ¼ f lf uþ f l

ð7Þ

The propagating laser fields can be calculated through the follow-ing formula:

∂If ðx; y; zÞ∂z

¼ σ½f uN2ðx; y; zÞ� f lN1ðx; y; zÞ�If ðx; y; zÞ ð8Þ

∂Ibðx; y; zÞ∂z

¼ �σ½f uN2ðx; y; zÞ� f lN1ðx; y; zÞ�Ibðx; y; zÞ ð9Þ

And the pump distribution is calculated through the followingequation:

∂Ipðx; y; zÞ∂z

¼ �σabs�meanN1ðx; y; zÞIpðx; y; zÞ ð10Þ

The total logarithmic single-pass gain G of the laser is obtained byan integration over the volume.

In G¼Z

σ½f uN2ðx; y; zÞ� f lN1ðx; y; zÞ�ψ sðx; y; zÞdV ð11Þ

therein ψ sðx; y; zÞ is the spatial distribution function of the single-pass gain. The pump beam is assumed to have a Gaussiantransverse distribution with a constant beam size through theentire length of the gain medium. A typical beam size of12 mm�0.8 mm in Nd-doped slab lasers [24] is chosen for thenumerical simulation. The divergence of the pump beam in thecrystal is neglected. This assumption is made based on the factthat for a pump laser with a M2 beam quality factor of 78, forexample, the beam radius of the pump laser only varies from400 μm to 460 μm in the crystal if a thermal lens of 50 mm istaken into consideration and the divergence could be smaller iffurther beam shaping techniques are used to improve the beamquality of the pump source. The laser beam is assumed to have anoverlapping distribution with the pump beam. The boundarycondition is then derived from self-consistency of circulating laserpower in the cavity:

ð1�TÞð1�LαÞG2 ¼ 1 ð12Þ

where T is the transmission of the output coupler and Lα is theround-trip cavity loss.

After the laser intensity distribution is calculated in the firststage, a 3-D thermal model is built for temperature distributioncalculation. The corresponding configuration is shown in Fig. 3.The Tm:YAG is sandwiched between two copper heatsinks. Indiumlayers of about 100 μm thickness are used for even thermalcontact. The Cu heatsinks are cooled from surface S1 and S2 bya coolant fluid, which is assumed to be de-ionized water in thispaper, with a heat exchange coefficient of h1. The other foursurfaces of Tm:YAG are assumed to be in thermal contact with airwith a heat exchange coefficient of h2. The stationary heatconduction equation that governed the thermal performance ofthe system is

∇2Tðx; y; zÞ ¼ Pheatðx; y; zÞkYAG

in Tm : YAG ð13Þ

and the boundary conditions are

∂T∂ni

¼ h1

kCuðT coolant�T si Þ; in S1 and S2 surfaces ð14Þ

∂T∂nj

¼ h2

kYAGðTair�T sj Þ; in other four surrounding surfaces of Tm : YAG

ð15Þwhere kCu and kYAG are thermal conductivities of copper heatsinksand Tm:YAG, respectively. ni and nj are the normal direction to thecorresponding surfaces. Pheatðx; y; zÞ is the heat density generatedinside the crystal in each element and it is calculated through thefollowing equation:

Pheatðx; y; zÞ ¼∂Ipðx; y; zÞ

∂z�∂½If ðx; y; zÞ� Ibðx; y; zÞ�

∂z�N2hυL

τ2radð16Þ

In the equation above, τ2rad is the radiative life time of Tm ions on3F4 manifold. It should be noted that the effects of energy transferupconversion processes are not included in the equation above.The effect of the 3F4þ3F4-3H4þ3H6 process is largely compen-sated by the cross-relaxation process in Tm-doped systems. As forthe 3F4þ3F4-3H5þ3H6 process, the excited ions in 3H5 manifoldsare mainly de-excited by non-radiative decay to 3F4 manifoldswith a radiative quantum efficiency of 0.003 [26]. The fluorescencephotons emitted in the energy transfer upconversion processescan be neglected, thus it is reasonable to assume that no power isextracted from the Tm:YAG crystal in these processes. Hence theeffects of the energy transfer upconversion processes areneglected in Eq. (16). With the help of ANSYS package, a 3-Dfinite-element-analysis (FEA) is performed to compute the tem-perature distribution inside the crystal. The active medium isdivided into a mesh of finite elements with 50 elements in everydirection. This meshing is in accordance with that used in the rateequation model.

Then the influence of temperature distribution on laser perfor-mance is included in the rate equation model by changing thecorresponding occupation factors, which was treated as constantsin the first step, to temperature dependent forms:

f uðx; y; zÞ ¼exp � E21

KTðx;y;zÞ

� �

∑9i ¼ 1 exp � E2i

KT x;y;zð Þ

� � ð17Þ

f lðx; y; zÞ ¼exp � E17

KT x;y;zð Þ

� �

∑10j ¼ 1 exp � E1j

KTðx;y;zÞ

� � ð18Þ

where E2i is the energy of the ith Stark level in the excitedmanifold, E1j is the energy of the jth Stark level in the groundmanifold, Tðx; y; zÞ is the temperature in the corresponding

Fig. 3. Thermal model configuration.

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element, and K is Boltzmann's constant. The absorption andemission crosssections are also influenced by the temperature,but because of the lack of the corresponding data from currentavailable papers, the influence of temperature on these parametersare not included in our model. However, it should be noted that,from the result of a simulation of high power Yb:YAG slab lasers byLiu et al. [26], it can be seen that most part of the influence oftemperature on laser performance could be explained by thetemperature dependent nature of Boltzmann occupation factor.In the simulation, the output power with 1.2 kW pump sourcedrops from 760.1 W to 583.5 W by changing the Boltzmannoccupation factor into a temperature dependent form. Furtherconsideration of the temperature influences on the spectroscopicvalues 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, themodel presented in this paper could give an comprehensivedescription of the laser performance of the Tm:YAG slab laserBoltzmann occupation factor.

The temperature distribution in the crystal is calculatedthrough the 3-D FEA performed with ANSYS package, then theresult is used to modify the laser output in the first stage. Afterseveral iterations, both the laser output and the temperatureconverged to a stable distribution and the laser performance withtemperature 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 tosimulate the laser performance, a key factor that must be calcu-lated in the first step is the heat exchange coefficient, which isusually presumed as a constant in previous theoretical simulations[8,9,20,21]. But in experiment, this factor can differ from case tocase according to different heat-sink designs. For example, theheat exchange coefficient for laminar flow in tubes is in the orderof 4000 W/m2 �K, and for turbulent flow in micro-channel heat-sinks, this factor can increase to the order of 10,000 W/m2 �Kor higher [22]. By deliberately designing the parameters of micro-channel heatsinks, the heat exchange coefficient can be

augmented in a large scale. The impact of heat exchange coeffi-cient on laser performance in Tm:YAG slab laser system isexamined using our numerical model presented in Section 2. Theparameters used are listed in Table 1.

Fig. 4(a) and (b) depict the laser performances for threedifferent values of heat exchange coefficients. From Fig. 4, it canbe seen that as the heat exchange coefficient increases from5000 W/m2 �K to 50,000 W/m2 �K, the maximum temperature risein the crystal is greatly reduced from around 600 1C to a muchlower temperature of around 180 1C. Due to the quasi-three-levelnature of Tm ions, high temperature will lead to heavier reabsorp-tion of 2 μm laser and reduction of population inversion, whichwill lead to the deterioration of laser performance. With thedecrease in temperature, the corresponding laser performance isgreatly improved then. The maximum output power that can beachieved with 800 W pump power increased from 57 W to 337 W.It should be noted that for the heat exchange coefficient of as lowas 5000 W/m2 �K, severe saturation occurs when the pump powerscales up. The temperature as high as 619 1C at 800 W pumppower not only limits the laser output, but also brings higher risksof thermal damage of the laser crystal. A high heat exchangecoefficient is essential for power scaling of slab lasers. It should bementioned that the thermal conductivity of Tm:YAG crystalswould change with temperature. According to the work of Satoet al. [28], the thermal conductivity of 5 at% Yb:YAG crystaldecreases from 6.82 W/(m�K) at 25 1C to 5.36 W/(m�K) at200 1C. By applying these thermal conductivities to the wholesample in our model, the influence of the temperature depen-dence 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 to219.5 W with the decrease in thermal conductivity. As the Tm andYb ions are nearly identical in size, similar effects should exist inTm:YAG crystals. Thus it is reasonable to conclude that in thismodel, which did not include the temperature dependence of thethermal conductivity of Tm:YAG crystals, the maximum errorbounds would be around 10.9% for heat transfer coefficient largerthan 10,000 W/m2 �K. As to laser systems with lower heat transfercoefficient such as 5000 W/m2 �K shown in Fig. 4, the maximumtemperature rise would be much larger and the saturation wouldbe much heavier than that shown in Fig. 4(a). Besides, the

Table 1Parameters used in the numerical simulation.

Symbol Quantity Value

/ Doping concentration 4 at%H�W� L Crystal dimension 1 mm�12 mm�14 mmτ2 Upper level lifetime 12 ms [10]σ Atomic stimulated emission cross-section 0.5�10�20 cm2 [10]σabs�mean Mean effective pump absorption cross-section 2.8�10�21 cm2 [19]ψp Pump laser profile Gaussianψ L 2 μm Laser profile Gaussianλ Pump wavelength 787 nmΔλ FWHM of pump laser 3 nmωHωV Beam radius of 2 μm laser 0.4 mm�6 mmβ4 Total branching ratio 3H4-

3F4 0.6 [10]η4 Cross-relaxation efficiency 0.97 [10]η3 Cross-relaxation efficiency 0 [10]K∑Tm Total upconversion parameter 3�10�18 cm3/s [10]Toc Transmission of output couple 10%Lα Resonator round-trip loss 1.5%h2 Heat-exchange coefficient 10 W/m2 K [20]kCu Heat conductivity of copper heat-sink 396 W/m/K [25]kYAG Heat conductivity of Tm:YAG 7 W/m/K [27]kindium Heat conductivity of indium 81.8 W/m/KT coolant Temperature of coolant fluid 290 KTair Temperature of air 300 K

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absorption of the pump laser would decrease to a large extent atsuch a high temperature and the laser performance would furtherdeteriorate.

Typical 3-D temperature distributions in the crystals with800 W of pump radiation are shown in Fig. 5 for Gaussian pumpdistribution and Top-Hat pump distribution assumptions. Fig. 6(a) and (b) shows the corresponding temperature profile on thehorizontal and vertical cross-sections in the center of the slab,respectively.

It can be seen that the highest temperature arises at the centerof the incident surface of the crystal when the Gaussian pumpdistribution assumption is used. In comparison, for the Top-Hatpump distribution assumption, one-dimensional temperature dis-tribution could be observed on the incident surface, as can be seenin Figs. 5(b) and 6(a). This is attributed to the large discrepancybetween the heat exchange coefficients of the two large surfaces(10,000 W/m2 �K) and that of the side-surfaces (10 W/m2 �K). Thisfact suggests that conventional one-dimensional heat flow

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 withdifferent 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 h1¼10,000 W/m2 K and (b) temperature distributions in the verticalcross-sections in the center of the slabs with h1¼10,000 W/m2 K.

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assumption in slab lasers is valid only when the pump distributionis ideally uniform in the horizontal direction. For more practicalconditions, in which the shape of the pump beam can usually bedescribed by a super-Gaussian function, a 3-D thermal model is ofessential 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 affectthe laser performance, which is mentioned before, it is necessary tomeasure the heat exchange coefficient experimentally.

The experimental arrangement is shown in Fig. 7. Two cylind-rical lenses are used to focus the diode laser beam into a spot sizeof �1.6 mm�0.6 mm. The resonator used in the following laserexperiments is configured with a flat input mirror (FM) and aplano-concave output coupler (OC). The output coupler has aradius of curvature of 200 mm and a transmission of 5% at2015 nm. The Tm:YAG ceramic sample is sandwiched betweentwo copper heatsinks. Indium foils of about 0.1 mm thick are usedto offer even thermal contact for the samples and the heatsinks.The contact surfaces of the laser crystal and the copper heatsinksare all polished, and certain external forces are applied to theheatsink-indium-crystal volume to guarantee an effective thermalcontact. The cooling water temperature is kept at 15 1C. The Tm:YAG ceramic sample used has a doping concentration of 4 at% andthe dimensions are 1.29 mm�10 mm�15 mm.

An infrared thermal camera (FLIR SC655) is used to monitor thetemperature distribution on the incident surfaces of the ceramicsunder non-lasing conditions (without the input and ouput mir-rors). Fig. 8 shows the temperature distribution obtained with15 W of pumping laser. The temperature profiles on the two linesare shown in Fig. 8(b).

A 3-D thermal model is then built and the temperaturedistribution is calculated with ANSYS package. By carefully fittingthe heat exchange coefficient, a good agreement could be reachedbetween the simulation results and the temperature distributionobtained with the infrared thermal camera, as is shown in Fig. 8(b). The heat-exchange coefficient of our heatsinks is then

calculated to be around 14,000 W/m2 �K. This value is used inthe following numerical calculations.

3.3. Laser performance and thermal behavior

The influence of coolant temperature on laser performance isanalyzed. The laser performances of the Tm:YAG sample fordifferent cooling water temperatures are shown in Fig. 9. Thelaser spectrum is monitored by an optical spectrum analyzer(AQ6375, Yokogawa) and the spectrum is centered at 2013.6 nmfor both conditions. A round trip loss of 5.7% was used innumerical simulation to show good agreement with the experi-mental data. The real round trip cavity loss is also measured usingthe Caird analysis method, which determined the resonator loss bydetermining 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 numericalsimulation. The large round trip loss may be attributed to the largediffraction loss introduced by the existence of high-order lasermodes. Besides, the imperfect coatings of the cavity mirror alsointroduces additional losses. As can be seen in Fig. 9, when thetemperature increases from 15 1C to 25 1C, the slope efficiencyexperiences a drop of 2% in our experiment, whereas the numer-ical results illustrate about 34% of the difference. The discrepancymay be attributed to the uneven thermal contact in our experi-ment, which means that the average heat exchange coefficient inthe experiment may be lower than that we have calculated in the

Fig. 7. Experimental setup.

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.

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

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former step (14,000 W/m2 �K). Besides, this model did not includethe influence of temperature on absorption and stimulated cross-sections as well as the temperature dependent nature of thethermal conductivities of YAG and copper. Because of the lack ofthese parameters from current available papers, these factors arenot included in this paper. Consideration of these factors may leadto a better match between the numerical and the experimentalresults. This model should be used in a moderate temperaturerange, where the usages of the spectroscopic values and thethermal conductivities in this paper are suitable. The modelpresented here provides a method for comprehensively under-standing of the laser and thermal performance of high power Tm:YAG slab lasers.

4. Conclusion

A three-dimensional plane-wave thermal model has beendeveloped to analyze the couple field of laser and temperaturedistribution for high power end-pumped slab lasers. A 3-D FEAcombined with an iterative method is applied to calculate thestable temperature and laser distribution in the slab crystal.Experiments are carried out to analyze thermal behaviors of theTm:YAG ceramic slab lasers. An infrared thermal camera is used tomonitor the temperature of slabs under non-lasing condition togive an accurate value of the heat exchange coefficient. Anelevation of 10 1C in the cooling water temperature results in adecrease of 2% in slope efficiency. The corresponding numericalsimulation results can give reasonable illustrations of these phe-nomena. This model provides a more accurate method to dealwith problems that has 3-D inhomogeneous or asymmetric dis-tributions of lasers and heat loads.

Acknowledgments

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

the Jiangsu Key Laboratory of Advanced Laser Materials andDevices for support.

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