## Abstract

Optical aberrations induced in thin-disk laser elements with an undoped layer, performing as an anti-ASE cap, are analyzed. A numerical model, used for calculations of the optical path difference (OPD) induced by temperature distribution inside the laser element and by deformation of surfaces, was confirmed experimentally. Results of numerical modeling manifest that adding an undoped layer on the thin-disk has detrimental effect on the reflected laser beam brightness and scaling. It is also shown that brightness of a thin-disk laser may be enhanced by the use of the Gaussian pump beam profile.

©2013 Optical Society of America

## 1. Introduction

Thin-disk lasers are believed to allow power scaling maintaining high efficiency and good beam quality [1–3]. One of the limiting effects for power scaling is amplified spontaneous emission (ASE) which is more expressed in the Nd doped gain medium compared to the Yb doped media usually used in thin-disk geometries. Even if a thin-disk geometry is considered to have low optical aberrations, it is still a major limiting factor for obtaining a high power diffraction limited laser output [2,4]. In [4,5] a thin-disk configuration with an undoped layer, which acts as an anti-ASE cap, was proposed to reduce the effective ASE path. According to literature, it is expected that adding undoped layer on top of active layer should reduce temperature inside active element and reduce mechanical deformations of the thin disk due to increased stiffness [4–6]. This proposes that thermally induced optical aberrations of thin-disk laser element with anti-ASE cap should reduce also.

Typically, high power thin-disk lasers are generating in a highly multimodal regime where pump and generated beam intensity distributions are close to hat-top, thus edge effects in respect of optical aberrations do not affect laser performance significantly. A single mode regime requires optimization of the mode overlap for efficiency considering phase distortions of the laser beam to maintain a fundamental mode [1].

Analysis of thermal effects of thin disks with an undoped layer was performed in [6] where it was estimated that the undoped layer on thin-disk might drastically reduce temperatures inside the medium, thus reducing induced optical aberrations by the order of magnitude. Numerical modeling of a theoretical model (presented in [6]) using the finite element analysis gives similar shapes of temperature distributions, but values are very different and such a drastic drop of temperature in the thin disk with undoped layer was not obtained. Another important fact is that heat resistance at the boundary between a disk and a cooling medium was not taken into account by the author of [6] which gives underestimated temperatures. As the optical properties of the capped disk were analyzed in one pass geometry, deformation of reflecting surface was not taken into account which is very important as stated in [1,4].

Nevertheless, adding an undoped layer on the thin-disk changes temperature distribution inside the structure and character of deformations (Fig. 1). Mounting technology is also important as some deformations (such as flexing of the thin-disk) may be suppressed (Figs. 1(c) and 1(d)).

In this work a thermo-mechanical numerical model of a thin-disk, with or without an undoped layer on the top, was developed and confirmed experimentally. Then comparison of different geometries was performed by calculating quality degradation of the TEM_{00} beam reflected from such a structure. Scaling possibility was estimated as well.

## 2. Numerical model

Numerical modeling was performed using the finite-element analysis using COMSOL Multiphysics^{®} software. Model geometry was set to have radial symmetry, disk radius was set to be 20 mm which is large enough to consider an infinitely large disk as increasing radius further has no significant effect on the modeling results within the scope of pump and probe beam parameters being analyzed. Doped and undoped regions were not separated as different materials. A heat source was bounded by the pump radius (*R*_{p}) and the doped layer thickness (*d*) (Fig. 2). Two cases of boundary between the crystal and cooling plate were modeled. In the first case we set a heat transfer coefficient to be *h* = 10W/(cm^{2}·K) to a heat-sink with temperature *T*_{0} = 20°C and let this surface to deform freely (Fig. 2(a)). Such situation corresponds to direct water cooling or thermal grease as a bonding layer. In another case an additional indium layer (*d*_{In} *=* 100µm) was introduced. The opposite surface of indium layer was set to have constant temperature (*T*_{0} = 20°C) and was mechanically fixed. In this way deformation of the disk was partially suppressed (Fig. 2(b)), corresponding to soldering a thin disk to a stiff heat sink. The boundary between the crystal and indium was set to have heat resistance with a heat transfer coefficient chosen in the way that the efficient heat transfer coefficient was the same as in the previous case. All other surfaces were thermally insulated in both cases.

From the modeling results we obtained temperature distribution and deformations of surfaces. Using these data the optical path difference (OPD) of the reflected beam can be calculated taking into account a change of the refractive index due to temperature, deformation of the reflecting side and change of the thickness due to thermal expansion:

For evaluation of resulting wave front distortions, separation into spherical and aspherical parts is typically performed [1,2] which is not very intuitive especially when applying to a Gaussian beam. It is also tricky to compare several different aberrated wave fronts when the amplitude of distortions is similar but shapes are different.

We chose a different approach to calculate a beam propagation parameter (M^{2}) of a Gaussian beam after reflection from a thin disk, acting as an aberrated mirror. Calculation of the M^{2} parameter can be done numerically by simulating diffractive propagation using Fresnel integrals and calculating beam diameters at various distances (as it is performed in real M^{2} measurement systems according to ISO 11146 standard).

Increasing the pump power to the thin-disk results in an increased amount of generated heat in the active medium, thus the heat source density (heat power per volume) increases. A higher heat source density results in stronger wave front aberrations of the reflected beam, which reduces the quality (brightness) of an affected Gaussian beam (Fig. 3).

We set a level of the acceptable M^{2} parameter to be 1.05. Then we searched for the maximum heat source density *Q _{max}* fulfilling this condition (Fig. 3). This parameter indirectly represents capability of thin-disk geometry to manage a particular heat source, keeping aberrations under a certain level. In this way it is possible to compare different geometries of capped disks in the sense of thermal management. From the practical point of view this

*Q*parameter roughly represents the available output power from a laser oscillator or an amplifier with the predefined beam quality, as the output power is proportional to the absorbed pump power and part of the absorbed power is always converted to heat. It is important to note, that the

_{max}*Q*parameter depends not only on geometry of thin-disk, but on the heat source distribution and the laser beam size on the disk as well. Adding undoped layer does not change absorption, amplification or heat generation properties in active volume, thus different geometries can be compared under the same conditions of the heat source and the probing beam. It is also worth to note that level of the acceptable M

_{max}^{2}parameter is not strict and different values may be chosen, resulting in different calculated

*Q*parameters (Fig. 3).

_{max}## 3. Experimental validation of numerical model

To avoid drastic miscalculations (such as in [6]), experimental validation of the numerical model was performed.

A sample made of composite Nd:YAG/YAG ceramics was prepared. The sample size was 8x8x1.2 mm with the 0.2 mm thickness layer doped with Nd^{3+} of 2 at % concentration. The surface on the doped side was highly reflective for 808 nm and 1064 nm wavelengths, and the undoped side was antireflection coated. The sample was attached to the 8x8x0.5 mm CVD diamond plate and a water cooled copper plate. Bonding layers were “Antec Formula 7” thermal compound with the measured thicknesses of 25 µm. Besides good thermal conductivity of compound (8 W/m·K), heat resistance of the layer was estimated to be about 5 W/(cm^{2}·K).

The sample was pumped by the fiber coupled laser diode emitting up to 100W @ 808nm. The pump spot was formed by the magnified image relay of the fiber end. The pump spot diameter on sample was 2.2 mm. The collimated output of the single-mode fiber coupled laser diode emitting at 1064 nm was used for probing. Induced wave front changes were measured by relaying the sample image plane to the Shack-Hartmann wave front sensor using a 4F telescope (Fig. 4).

The dissipated heat power had to be estimated to compare experimental data with results of numerical modeling. This was done by solving steady state rate equations including cross-relaxation and upconversion effects [7]. High inversion levels were expected, which was manifested by good agreement of calculated and measured saturated pump absorption of the sample. High inversion levels in the Nd^{3+} doped medium give excessive conversion of the absorbed laser diode power to the heat power (it was estimated to reach 70% in our case). The estimated maximum heat source density was about 30 W/mm^{3}. With such a heat source power a nearly 200K maximum temperature rise in the sample was expected (Fig. 5), and nonlinear effects (temperature dependent heat conductivity and expansion coefficient) had to be taken into account. This is evident from comparison of experimental data with calculated values of OPD using linear approximation. The linear approximation resulted in reduced variation of OPD over radius and a different shape of OPD distribution (apex is less sharp) (Fig. 6(a)). Experimentally measured variation of OPD over radius (span between the minimum and maximum OPD value) also manifests non-linear dependence on the heat power (Fig. 6(b)).

For temperature dependent (non-linear) calculations heat conductivity dependence on temperature was set to be inversely proportional to absolute temperature which is a widely used approximation [8]. Thermal expansion coefficient dependence on temperature of YAG was set to be used from COMSOL Multiphysics^{®} database.

## 4. Results and discussion

As the numerical model gave results consistent with experimental data, this thermo-mechanical model of the thin-disk was used for evaluation of the influence of the undoped layer thickness on resulting optical aberrations.

All further calculations were performed in linear approximation, meaning that heat conductivity and thermal expansion coefficients were set to be temperature-independent. Motivation for the linear approximation is mainly related to computer resources. It has been found that the maximum deviation of calculation results of linear approximation from results of non-linear modeling is less than 12% for the presented data. Even lower errors were found in regions of interest. Despite the fact that linear approximation does not allow an accurate evaluation of wave fronts or the beam quality, it still gives sense on the influence of various sample and pump/probe parameters such as the thickness of the undoped layer, the size of the pumped region or mode overlap (fill factor). Meanwhile the calculation time may be reduced drastically. Amplification distribution was also not taken into account.

The thickness of the doped (active) layer was *d* = 0.2 mm (Fig. 2) for all cases. The thickness of the undoped layer was altered by setting the overall structure thickness (*D* in Fig. 2), thus 0.2 mm represents the uncapped thin-disk. The thickness was increased by the factor of 2 up to 3.2 mm (0.2 mm, 0.4 mm, 0.8 mm, 1.6 mm and 3.2 mm).

Two types of heat source density distributions over pump spot radius were modeled: hat-top (cylindrical) and Gaussian. A size of Gaussian distribution is defined as the diameter at 1/e^{2} level. A size of hat-top distribution is defined as the diameter of boundaries. Heat source distribution in *z* direction was constant, bounded by the doped layer thickness. It is worth noting that at such definitions of heat source sizes the total heat power of hat-top distribution is two times higher than that of Gaussian distribution if the maximum value of the heat source density is fixed the same.

In the first computer modeling run, the laser beam size was fixed to be 2 mm. The maximum heat source density dependence on the heat source size for which M^{2} of the reflected laser beam was less than 1.05 (*Q _{max}* parameter), is presented in Fig. 7. A general tendency is that by increasing the heat source (pump) size, while keeping the laser beam size fixed, samples of all modeled thicknesses can operate with a good beam quality at higher heat loads. The larger pump size, the larger the heat load is acceptable for a good beam quality. This could be expected as temperature and deformations are distributed more evenly over an effective area of the laser beam. But it comes at the cost of efficiency as the pumped area (thus required pump power) is a function of the transverse size squared. Another drawback of an increasing pump spot size is that ASE also increases and a thicker undoped layer is needed [5], which in most cases has detrimental effects on optical aberrations. A substantially larger pump spot size than that of the laser beam size demands special care in designing laser resonators to suppress higher order resonator modes (hard apertures have to be introduced) [4].

An interesting region for freely deformable disk is when the size of the heat source with Gaussian distribution is similar to the laser beam size (Fig. 7(c)). Theoretically it seems to be the region where the efficiency in conjunction with a good beam quality may be achieved as flexing of the disk is partially compensating aberrations due to the refractive index change and smaller scale surface deformations. But from the practical point of view this region is unstable as it highly depends on the heat source distribution (Figs. 7(a) and 7(c)). It also depends on the size of the laser beam which can be seen by comparing Fig. 7(c) with Fig. 8(c) where the same plot is presented for the laser beam size of 4 mm. Here (Fig. 8(c)) the uncapped freely deformable thin disk becomes the worst case scenario and a thin-disk with the overall thickness of 0.4 mm becomes favorable.

When flexing of the disk is partially suppressed (in our case by indium solder, assuming that other surface of indium is mechanically fixed), results become more predictable and controllable which is desired in realistic applications (Figs. 7(b) and 7(d)). Anyway, it is obvious that adding an undoped layer on a thin disk has detrimental effects in respect of optical aberrations as higher heat loads (higher pump powers) can be managed by a thin-disk without or with a thinner undoped layer. In other words, adding an anti-ASE cap worsens the beam quality at the same heat load.

There are some effects emphasized by undoped layer which are not affecting the M^{2} parameter but might be important to note in practical applications. One of undesirable effects of a thick undoped layer is thermal lensing. When a thick undoped layer is applied, deformation of each surface and temperature distribution in the medium mutually acts as a positive lens (illustrated in Fig. 5). Meanwhile flexing of uncapped (or with thin undoped layer) disk tends to form a convex mirror which may compensate or even overcome positive lens due to temperature distribution and bulging of the disk. Another detrimental effect of a thick undoped layer observed during experiments is depolarization. This effect was not thoroughly analyzed, but could be explained by induced radial stresses due to an emerging radial heat flow, which is usually kept low in the uncapped thin-disk case.

According to [9] the scaling procedure is as follows: for doubling the output power, double the pump power and the mode area on the disk, while keeping the disk thickness and the output coupler transmission constant. This defines that the pump intensity (thus the heat source density) is kept constant and the pump spot (heat source) size is mutually increased with the laser beam size.

For estimation of the power scalability the heat source density was fixed at Q = 5 W/mm^{3} and the M^{2} parameter dependence on the heat source size was plotted (Fig. 9). The laser beam size was kept the same as the heat source size (fill factor 1). According to the data in Fig. 9, scaling is limited by degradation of the reflected laser beam quality. Generally, the M^{2} parameter is increasing as the sizes of the heat source and the laser beam are mutually increased. Moreover, freely deformable disks show some regions of low aberrations but demand a certain distribution of heat source (Figs. 8(a) and 8(c)). When deformations are suppressed by indium solder, the beam quality can be improved (or the pump power can be increased). A further improvement may be achieved by switching from the hat-top heat source (pump) distribution to the Gaussian one. The Gaussian distribution of the pump intensity is also desirable in fundamental mode laser resonators as it helps to maintain a fundamental mode [4]. In case of the heat source with the Gaussian distribution the uncapped or thin disk with a thin undoped layer seems to satisfy the power scalability concept as an increase of the M^{2} parameter tends to saturate as sizes of the pump and laser beam are increasing (Fig. 9(d)). Nevertheless, increasing of the thickness of the undoped layer diminishes this effect (Fig. 9(d)).

As mentioned earlier, by sacrificing efficiency, the increasing heat source (pump) size in respect of the laser beam size improves the reflected laser beam quality (or higher power levels may be achieved). Nevertheless, characteristics of the M^{2} parameter over scaling remain similar to those presented in Fig. 9.

## 5. Summary

A numerical model of heat flows in a thin disk with an undoped layer on the top was developed and confirmed experimentally. Resulting temperature distribution and deformations of surfaces were used to calculate the optical path difference distribution for the laser beam reflected from the disk. Aberrations were analyzed in respect of the Gaussian beam quality (focusability) changes after reflection from the thin-disk element.

We introduced a method to rate different optical phase distortions by calculating resulting M^{2} parameter of an affected diffraction limited laser beam. In contrast to widely used separation into spherical and aspherical parts, such evaluation is unambiguous when diameter of probing Gaussian beam is defined and has higher value in practice. We think that approach used in this work for comparison of thin-disk active elements of different configurations can be adapted for other configurations of active elements too. It could be a valuable tool for evaluation of influence of design changes of active elements on the final laser system parameters before manufacturing.

In this work modeling was simplified by linear approximation, thus evaluation of acceptable pump power is ambiguous and it would depend on the choice of active ions or host material. Nevertheless, as undoped layer does not affect absorption, amplification or heat generation in active volume it is still possible to predict the influence of undoped layer to optical properties of thin-disk laser element.

Results show that adding an undoped layer on a thin-disk laser element deteriorates the reflected Gaussian beam quality in most cases. In particular cases, for disks with unsuppressed deformations, there are parameter sets where aberrations are low (good beam quality may be achieved at higher power levels), but such configurations seem to be hardly controllable in practice. When disk deformations are partially suppressed (e.g. soldering it to stiff surface), aberrations become more predictable, thus easier to control in practice.

Scaling possibility is also diminished as a thicker undoped layer is added on the thin disk laser element. Improvements in scalability, the laser beam quality or the achievable power are expected by use of the Gaussian or Gaussian-like pump profile instead of the hat-top profile.

In conclusion, according to our numerical model, thin disks with undoped caps should be used only when there is a real need for ASE suppression or mechanical stability of the laser element.

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