SPIE Proceedings [SPIE LASE - San Francisco, California (Saturday 23 January 2010)] Solid State Lasers XIX: Technology and Devices - Dynamic 3D modeling of solid state laser resonators

Dynamic 3D modeling of solid state laser resonators using acoupled thermo-optical finite element analysis

Matthias Wohlmuth∗, Konrad Altmann†, J. Werner‡, Christoph Pflaum§

ABSTRACT

The complex behavior of the optical wave in laser resonators requires a comprehensive model of thermal lensingand the dynamic, 3-dimensional behavior of the laser beam. To this end, we perform a combined finite elementanalysis (FEA) of the optical wave and of thermal lensing. Here, the simulation of the optical wave is the mostchallenging task. Therefore, we also discuss another method, Dynamic Multimode Analysis, which is suitablefor a wide range of lasers. Finally, we present a complex heat model in order to analyze the interaction of heatgeneration, thermal lensing, laser dynamics, and the beam profile.

1. INTRODUCTION

The complex behavior of the optical wave in laser resonators requires a combined model of thermal lensing and thedynamic 3-dimensional behavior of the laser beam. First, we present a complex heat model1 based on informationobtained from a rate equation model. Second, we analyze thermal lensing depending on the given heat source,crystal and pump light configuration. Finally, we show how the thermal lens information can be included intothe model of the optical wave. To this end, we refer to two different approaches to simulate the optical wave:Firstly, a Dynamic Multimode Analysis (DMA)2 which is able to analyze mode competition and, secondly, animmediate finite element approach to the optical wave.3 DMA provides good results for output power andbeam quality of Q-switched lasers. However, DMA using gaussian modes only approximates aberrations due tothermal lensing (see Section 4). This delivers good results for a lot of resonator configurations, but especially incombination with a complex heat model (see Section 3), a more general approach can be necessary. In this case,we propose an immediate finite element analysis (FEA) of the transformed wave equation. From a numericalpoint of view, the simulation of the optical wave using finite elements is a challenging task. Section 5 will explainthe basic ideas of both the DMA and the FEA approach. By coupling the simulation of the optical wave toa dynamic model of thermal lensing and to space-dependent rate equations, we provide a simulation model toanalyze the dynamic interaction of heat generation and thermal lensing, mode competition, pulse generation,and other physical effects in the resonator. Fig. 1 shows an overview of the different modules of our simulationmodel.

2. DESCRIPTION OF INPUT DATA FOR NUMERICAL SIMULATION

In the software model depicted in Fig. 1, the description of the laser configuration is divided into three parts:resonator description, crystal description, and pump light description. In our opinion, the description of the laserconfiguration is already an important part of the overall simulation, because only a powerful, comprehensive,and flexible concept for the abstract laser description allows to obtain good results by any simulation technique.Sometimes the input data even can be considered as output data as well. On the one hand, the user has tospecify the laser configuration he is interested in. On the other hand, one wants to obtain simulation results inorder to improve the resonator setup. For example, our simulation could be used to perform parameter studieson the position of intra-cavity lenses. In this case, a resonator configuration with a certain optimality propertywould be a goal of the simulation. In case of similar studies on doping concentration, pumping set-up, and

∗[email protected],University Erlangen-Nuremberg, School in Advanced Optical Technologies (SAOT), Erlangen, Germany

†[email protected],LAS-CAD GmbH, Munich, Germany

‡[email protected],University Erlangen-Nuremberg, School in Advanced Optical Technologies (SAOT), Erlangen, Germany

§[email protected],University Erlangen-Nuremberg, School in Advanced Optical Technologies (SAOT), Erlangen, Germany

Solid State Lasers XIX: Technology and Devices, edited by W. Andrew Clarkson, Norman Hodgson, Ramesh K. Shori, Proc. of SPIE Vol. 7578, 75782H · © 2010

SPIE · CCC code: 0277-786X/10/$18 · doi: 10.1117/12.841930

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cooling techniques, both crystal and pump light description would be a simulation result.In most cases, it is possible to provide accurate data for the resonator geometry as well as for the crystal geome-try. However, it is more difficult to specify all necessary material data which are needed to perform a thermal anddynamic analysis of the laser. For example, it is possible to simulate rate equation models for complex energylevel structures, but this would also introduce a lot of new parameters, for example branching ratios. Theseparameters can be very difficult to specify for some materials. Therefore, it is important for any simulation toolto achieve a good trade-off between complexity and usability of the chosen model by minimizing the numberof input parameters. Furthermore, the user has to understand which parameters are most important to specifyaccurately. For example consider a long end pumped rod.2 In this case a change of the absorption efficiency ofthe incident pump light would not only influence the output power, but also the beam quality. Because of thedivergence of the pump beam, a decreased absorption efficiency would lead to a broader absorbed pump lightdistribution. This would excite more high-order modes.The same example also showed that the simulation data has to be understood with respect to the accuracy pro-vided by the input data description.2 The simulation predicted a constant slope efficiency, but the experimentaldata did not. That behavior was caused by a power-dependent wavelength shift of the pump diodes. This leadto a change in the absorption efficiency, which was not considered in the simulation.If experimental data are available, some parameters can be calibrated to fit these values. However, this has tobe done very carefully, being aware of which results are reliable to calibrate to. For example in case of pulsedlasers, it has proven to be better to first calibrate to an accurate pulse duration and then to fit the output power.Finally, the description of the laser configuration can already require previous computations. In our simulation,we consider the absorbed pump light description as given input. However, for complex pump configurations,variable doping concentrations, and saturation effects, the computation of the absorbed pump light distributionitself would require a detailed analysis, for example by a ray tracing technique. In case of saturation effects, thecomputation of the absorbed pump light distribution even has to be coupled to the rate equation model.Let us now discuss how to compute the heat power density by the pump light distribution and a rate equationmodel.

3. HEAT MODEL

The first simulation step in our approach is the computation of the thermal lens by a 3-dimensional finite elementanalysis (FEA). Besides information about cooling and clamping of the crystal, the most important input datafor the FEA is the amount of generated heat in the crystal. In a first, simple model the heat power densityH(�x, t) can be assumed to be proportional to the absorbed pump light distribution. Then, their ratio definesthe heat efficiency ηh, which is bounded from below by the Stokes shift, i.e. the difference between pump andlaser wavelength. For Nd:YAG at 1064nm, heat efficiency values of about 0.3 have proven to be reasonable.Furthermore, it is well-known that the heat efficiency is different with and without laser extraction.4–7 It seemssufficient to chose a measured value for the heat efficiency in case of cw laser operation. Even pulsed operationat high repetition frequencies, such that a constant thermal lens can be assumed,8 can be modeled that way.However, as soon as we need a detailed time-dependent analysis of the system, for example in single pulseoperation, we have to simulate the dynamic behavior of the heat power density as well. A detailed model forthe produced heat can be obtained by analyzing the energy flow in the rate equations.1 The effect of a finitelower laser level lifetime can be also included.9 The goal is to obtain a comprehensive analysis of the interactionof pumping, generated heat in the crystal, thermal lensing, and laser extraction. To this end, we couple thecomputation of the heat power density both to rate equations and the simulation of the optical wave. The modelfor the optical wave and the used rate equations will be presented in Section 5.For a detailed description of the energy flow in a laser it is crucial to use an appropriate set of rate equations andto consider the fractions of different kinds of energy: heat (non-radiative), fluorescence (radiative), and lasing.For a first approach, it is enough to consider 5 energy levels, which can consist of several degenerated manifoldswith degeneracies and Stark-splittings. Let us introduce the following index notation (for more details refer toBrown1):

0 : ground level



F : “center of gravity” for radiative decay (lower level of average fluorescent decay), which can be computedby the branching ratios βu,J and energies kJ of the various radiative decays from the upper laser level

1 : lower laser level (terminal level)

2 : upper laser level (metastable level)

p : pump level

Let us now discuss the different heat contributions.

3.1 PumpingThe whole laser is driven by the absorbed pump power density Pp(t, �x) which leads to an excitation density Re

given by

Re(t, �x) =Pp(t, �x)hckp

. (1)

A non-unity quantum efficiency ηe of the pump level leads to a reduced increase of the population in the pumplevel. Since we assume an instantaneous decay from the pump to the upper laser level (with a unit quantumefficiency of the upper laser level), we obtain the following increase of the upper laser level population N2:

∂N2(t, �x)∂t

= +ηeRe(t, �x). (2)

The energy difference between absorbed pump light and the energy of the upper laser level population leads tothe following heat contributions:

• Non-unity pump level quantum efficiency:

He(t, �x) = (1 − ηe)Re(t, �x)hckp. (3)

• Quantum defect (part of Stokes shift) between pump and upper laser level:

HQD(t, �x) = ηeRe(t, �x)hc(kp − k2). (4)

3.2 Stimulated emission and absorptionLet σe and σa be the effective emission and absorption cross-sections of the laser transition. Note that bothof them can be computed from the actual spectroscopic cross-section, taking into account crystal field leveldegeneracies and manifold populations. The latter are determined by the Boltzmann distribution and hencedepend on the absolute temperature.9,10 Using the abbreviations We(t, �x), Wa(t, �x) for stimulated emission andabsorption, the stimulated transitions lead to the following changes in the upper and lower laser levels populationsN1, N2:

∂N2(t, �x)∂t

= − (We(t, �x)N2(t, �x) − Wa(t, �x)N1(t, �x)) , (5)

∂N1(t, �x)∂t

= +(We(t, �x)N2(t, �x) − Wa(t, �x)N1(t, �x)) . (6)

In case of a quasi 3-level laser, this process does not produce any heat, because the lower laser level coincideswith the ground level. In case of 4-level systems, most models assume a zero lifetime for the lower laser level.However, it was shown9 that a short lower level lifetime τ1 has a certain effect in the pulsed regime. Of course,the lower level lifetime has to be short in comparison with the upper level lifetime. Then, we obtain

∂N1(t, �x)∂t

= −N1(t, �x)τ1

, (7)

and a heat power contribution of

H1(t, �x) = −N1(t, �x)τ1

hck1. (8)



3.3 Radiative Decay and Concentration Quenching

Let τf be the effective fluorescent lifetime for a given doping concentration. Then, τf can be chosen such that italready includes the effect of concentration quenching.1 The radiative decay occurs between the upper laser leveland the fluorescent “center of gravity ”. It leads to an exponential decay of the upper level population accordingto

∂N2(t, �x)∂t

= −N2(t, �x)τf

. (9)

The difference between the “center of gravity” and the ground level leads to the heat contribution

HF (t, �x) =N2(t, �x)

τfhckF . (10)

3.4 Resonator losses

Resonator losses include round-trip loss, Q-switch loss modulation, and the out-coupling process at the outputmirror. All these losses are assumed to produce no heat in the crystal. Therefore they only influence the photonrate equation by

∂Φ(t)∂t

= −Φ(t)τc

. (11)

3.5 Up-conversion

So far, we only consider simple up-conversion processes. That is, the amount of excited states which is transferredto higher energy levels by an Auger up-conversion process, is completely disposed as heat. This is assumed to beimportant if stimulated emission is absent, for example in giant pulse generation, when a high degree of inversionis reached.1 Note that, for example in Nd:YLF6 or Erbium-Ytterbium co-doped systems,11 up-converted ionsare not completely lost as heat, but interact with other effects like cross-relaxation. These systems require amore complicated rate equation model.In general, up-conversion is a quadratic-nonlinear process which diminishes the upper laser level population:

∂N2(t, �x)∂t

= −γUCN22 (t, �x). (12)

In case of a simple 4-level system, this results in an additional heat power density HUC

HUC(t, �x) = γUCN22 (t, �x)hck2 (13)

Altogether, we can summarize the different terms for the rate equations and the overall heat power density:

3.6 Rate equations and overall heat power density

The dynamic behavior of the photon number Φ and the laser level populations N1, N2 is given by

∂Φ(t)∂t

=∫

Ωa

(We(t, �x)N2(t, �x) − Wa(t, �x)N1(t, �x)) dV − Φ(t)τc

, (14)

∂N2(t, �x)∂t

= ηeRe(t, �x) − (We(t, �x)N2(t, �x) − Wa(t, �x)N1(t, �x)) − N2(t, �x)τf

− γUCN22 (t, �x), (15)

∂N1(t, �x)∂t

= +(We(t, �x)N2(t, �x) − Wa(t, �x)N1(t, �x)) − N1

τ1. (16)

Note that we will neglect the re-absorption term for 4-level systems in Section 5. Instead, we use the commondescription by only one equation for the population inversion (N2 − N1).12 A separate equation for the lower



laser level is only needed to discuss the effect of a finite lower laser level lifetime.9

Finally, the overall heat power density H(t, �x) is given by

H(t, �x) = He(t, �x) + HQD(t, �x) + H1(t, �x) + HF (t, �x) + HUC(t, �x) (17)

= (1 − ηe)Re(t, �x)�ωp + ηeRe(t, �x)hc(kp − k2) +N1(t, �x)

τ1hck1 (18)

+N2(t, �x)

τfhck2 + γ2N

22 (t, �x)hck2. (19)

3.7 Results of the complex heat modelLet us discuss some results which show the importance of a detailed heat model as described above. All figuresrefer to a Nd:YAG laser either in cw-operation or in Q-switched mode with repetition rate 15kHz. Up-conversionand finite lower level lifetime are not considered in the following example. Fig. 2 shows various fractions of theheat power during the transient oscillations in cw-operation. The constant heat power fraction is caused by thedifference of upper pump level and upper laser level (see Section 3.1). The fluorescent heat power (see Section 3.3)increases during two spikes because of the increasing amount of population inversion. As soon as the next laserpulse extracts inversion, fluorescence decay and its associated heat power decreases. However, laser extractionduring the transient oscillation spikes leads to a large amount of heat due to excited states decaying from thelower laser level to the ground level (see Section 3.2).

As soon as the laser reaches its steady state behavior, all heat contributions converge to a constant value.

Figure 2. The generated heat of a laser in cw operation is shown. It can be seen that a significant amount of the heatpower is proportional to the output power.

Therefore, cw-operation can be modeled by a simple heat model using a heat efficiency factor.Let us now consider the same laser in Q-switched operation at repetition rate 15kHz. Compared to the timebetween two transient oscillations in cw operation, Fig. 3 shows a similar but more extreme behavior of the heatpower fractions. Furthermore, the time-dependent behavior of the effective thermal focal length in Fig. 4 showsthat the thermal lens increases especially during the laser pulses. Details about how the thermal focal length iscomputed in our simulation are explained in Section 4. Because of the high repetition rate, the thermal focallength converges to a constant value. This value corresponds to the value which can be obtained by assuming aconstant heat efficiency and conducting a time-independent thermal analysis. However, the detailed heat modelis able to predict the correct heat efficiency depending on the repetition rate. That way, the amount of heatwhich is lost due to fluorescent decay during the Q-switch load period can be estimated. Furthermore, these



Figure 3. The generated heat of a Q-switched laser at high repetition frequency is shown. Similar to the results forcw-operation in Fig. 2, a significant amount of heat occurs together with the laser pulse.

results suggest that a heat model coupled to rate equations is important for lasers with low repetition rate or insingle pulse operation.Note that this heat model also predicts a dynamic behavior of the shape of the heat power density. On the onehand, the heat generated immediately from pumping has the same shape as the pump light. This is also assumedfor a simple heat model using a heat efficiency factor. On the other hand, the heat produced after fluorescentdecay coincides with the spatial distribution of the population inversion. Finally, the heat due to decay from thelower laser level during laser extraction has the same spatial shape as the laser beam itself. Therefore, especiallyin multi-mode operation this could lead to time-dependent distortions in the thermal distribution. Hence, wewould notice unexpected aberrations during the laser pulse. To describe these aberrations, a non-paraxial modelof the optical wave is preferable. For this purpose, we propose a finite element approach described for the opticalwave as described in Section 5.2.

4. SIMULATION OF THERMAL LENSING

The first step in the simulation of thermal lensing is to choose between a stationary and a dynamic thermalanalysis. The stationary approach can be used for cw lasers and pulsed lasers with pulse period shorter then thethermal relaxation time. Furthermore, it requires a stationary heat source. In this case the heat equation forthe temperature T is given by

−div

⎛⎝ κx

κy

κz

⎞⎠∇T = H, in Ωa, (20)

T = Tref on ΓD, (21)∂T

∂�n= 0 on ΓN , (22)

κ∂T

∂�n+ β(T − Tref) = 0 on Γthird. (23)



Figure 4. The figure shows the time-dependent focal length of the Q-switched laser from Fig. 3. It can be seen that, dueto the huge amount of heat produced by the laser pulse, the focal length decreases after each pulse.

Tref denotes the surface temperature, β the cooling power, κx, κy, κz the diffusion coefficients, and H the heatpower density. In the dynamic case, the first equation is replaced by

∂T

∂tcpρ = div

⎛⎝ κx

κy

κz

⎞⎠∇T + H, (24)

where ρ denotes the density and cp the specific heat capacity of the crystal material. A temperature changeleads to a change of the refractive index distribution n(t, �x), which is computed by

n(t, �x) = n0(t, �x) +∂n

∂T (t, �x)(T (t, �x) − T0), (25)

where ∂n∂T is the thermal index gradient and T0 is the room temperature. The above equation as well as the

following structural mechanics equations have to be evaluated after each temperature change, and at the end of thestationary analysis, respectively. The structural mechanics equations for the deformation �u(�x) = (ux, uy, uz)T (�x)



are given by

D�u =

⎛⎜⎜⎜⎜⎜⎜⎝

ααα000

⎞⎟⎟⎟⎟⎟⎟⎠

(T − T0) + C−1σ, (26)

div(σ) = 0, (27)

C =

⎛⎜⎜⎜⎜⎜⎜⎝

λ + 2μ λ λλ λ + 2μ λ 0λ λ λ + 2μ

2μ0 2μ

2μ

⎞⎟⎟⎟⎟⎟⎟⎠

, (28)

λ =νE

2(1 + ν)(1 − 2ν)(29)

μ =E

2(1 + ν). (30)

Here, ν is the Poisson ratio in, E the elasticity modulus, and α the coefficient of thermal expansion.

4.1 Input data and boundary conditions

In our model (see Fig. 1), the thermal analysis does not depend on the resonator description, but on the crystaland pump light description. The pump light description mainly determines the heat source H. The crystaldescription provides all material data and defines the boundary conditions which describe cooling and clampingof the crystal. Note that Eq. (23) is used to model fluid cooling, whereas Eq. (21) assumes cooling to a fixedtemperature.

4.2 Remarks on numerical algorithms

Especially, in case of a time-dependent analysis of the heat equation, fast numerical solvers for both problemsare needed. fast. For the time-iteration of the heat equation it is possible to use a simple implicit Euler schemetogether with a standard solver for the solution of the spatial problem (for example, a geometric multi-gridsolver13). More important, but also more difficult, is the solution of the structural mechanics equations whichrequire more advanced numerical techniques. In order to achieve the required performance, we use a specialmulti-grid algorithm in combination with a CG14 solver.

4.3 Parabolic fit

If the optical wave is modeled by the finite element approach described in Section 5.2, we can immediately usethe obtained results for the refractive index distribution and the deformation of the crystal. A gaussian modeanalysis, however, can only be applied for certain shapes of the refractive index distribution and of the crystaldeformation. In this case, both results have to be approximated by curved interfaces, gaussian ducts, or anequivalent lens. Nevertheless, we can approximate thermal aberrations by the following approach:Firstly, the crystal is divided into several thin slices along the resonator axis (see Fig. 5). Secondly, therefractive index distribution is fitted parabolically on each slice (see Fig. 6). Finally, the crystal deformationsat the end faces are approximated by a parabolically curved interface. Then, the crystal can be described as anagglomeration of curved interfaces at the end faces and gaussian ducts in between.





5.2 Finite element approach and two-wave ansatz

In this finite element model of the optical wave, we apply a two-wave ansatz and a slowly varying envelopeapproach to the wave equation. It was shown that curved interfaces, curved mirrors and lenses can be describedby this approach.3,16 The main equation of the model is a Schroedinger equation which governs the dynamicbehavior of the optical wave by

2jk2

f

ω

∂u

∂t= Δu − 2jkf

∂u

∂z− (k2

f − k2)u. (33)

To include gain into the above equation, we use a complex wave number k = kf + jki, which can be computedby

k = k0n + j

(σN

2− 1

2τcc0

), (34)

(35)

Note, that the solution of Equation (33) is still challenging and requires advanced numerical techniques.3

6. CONCLUSION

We presented a comprehensive model for the simulation of a solid-state laser resonator. The main parts of themodel are a coupled analysis of heat generation, thermal lensing, the optical wave, and the dynamic behavior ofthe laser. The model also contains a modular description of the laser configuration. The presented model is ableto analyze the dynamic, 3-dimensional behavior of the laser beam inside the resonator.

7. ACKNOWLEDGMENTS

We gratefully acknowledge funding of the Erlangen Graduate School in Advanced Optical Technologies (SAOT)by the German National Science Foundation (DFG) in the framework of the excellence initiative. The workpresented in this paper was supported by the InnoNet project SOL of the Federal Ministry of Economics andTechnology in Germany.

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SPIE Proceedings [SPIE LASE - San Francisco, California (Saturday 23 January 2010)] Solid State Lasers XIX: Technology and Devices - Dynamic 3D modeling of solid state laser resonators