## Abstract

We demonstrate an athermal, 100 mJ-energy class Yb^{3+}: YAG slab-based amplifier designed to operate in the range of temperatures from 20 to 200°C. The gain medium consists of an edge-pumped, thick slab with a folded laser beam-path, which is defined in a cat's eye cavity thanks to 6–12 elementary paths. Highly variable operating conditions are investigated, in a close connection with the induction of severe thermal penalties in the slab. These penalties concern the material's spectroscopic properties and the thermo-mechanical distortions, at the location of the laser faces. Looking at the shape of the optical-path-difference (OPD) along the slab, we evidence a strong dependency with the pump-dependent temperature cartography inside. This involves comprehensive fits between the measurement data and Finite Element Modeling (FEM) results. As a follow-up, by closely coupling a movable cylindrical lens along the slab, we validate the efficiency of an easy-to-implement correction process to cancel the OPD in the real time. Regardless of the operating regime, this enables fully updatable lasing conditions with reduced output beam distortions in the far field of the cavity output.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

This study is devoted to the demonstration of an athermal design for Yb^{3+}: YAG-based amplifiers at λ_{L }= 1030 nm, combining a high gain and a high output energy in a single stage, in the respective ranges of 15–20 dB and of 50–100 mJ. Potential applications of interest include recurrent multi 100 W-class multi-10 mJ laser systems. They are similar to those of more usual thin-disks, but with no need of a Pockels cell for regenerative amplification [1]. Generic needs to deal with temperature variations must be considered in recurrent laser systems as soon as they imply variable operating conditions, either in terms of the average pumping or of the lasing power. A number of industrial applications, for example, consist of micromachining and material processing [2,3]. The lasers involved usually need to be coupled and synchronized with scanners. Possible requirements may include the need to handle very flexible regimes of amplification in the QCW regime, regardless of the pulse repetition frequency (PRF) or pump duration, together with burst regimes of variable lengths. Due to the onset of variable and possibly deleterious thermal penalties in the active medium, this also implies a fine control of the beam quality for efficient focusing. Our optical design is based on the use of an edge-pumped, long and thick slab in a cat's eye cavity [4,5]. The thermal penalties take the form of a more or less severe temperature-dependent decrease of the emission cross section (σ_{eL}), reducing the net laser gain, and of strong thermo-optical-mechanical distortions at the location of the laser faces of the slab. During its propagation in the cavity, the distorted optical wave-front of the laser needs to be characterized by means of the Optical Path Difference (OPD). Wave-front quality degradation relative to thermally-induced OPD is a common limitation in any high-power laser system. This applies regardless of the geometry of the medium, in the form of a rod, a disk or a slab [6–9]. In our case of a slab, the applicable OPD is defined in terms of its local shape and magnitude, in the transverse plane to the beam propagation axis. The sizing parameters of primary concern are the pumping intensity (I_{P}) at the location of the pumping edge, the spatial distribution of pump power (P_{P}) in the slab, its dimensions and the cooling geometry. Rather than graduating the OPD versus the pumping power, as often proposed, we prefer to define the temperature (T) as the basic parameter of interest. This aims to avoid the need of more or less arbitrary assumptions, about the actual cooling efficiency through the top and bottom faces of the slab. This also helps to discriminate the contributing parameters following a more generic and rigorous approach.

We develop our argumentation in 5 steps: we first describe the optical design and associated setup. Second, we present various series of measurement results to quantify the gain and OPD penalties relative to the temperature gradients arising in the slab. Comprehensive FEM modelling and dedicated laser computations, in a third step, evidence very generic trends of interest with respect to the pumping conditions and measurement data. A good agreement will be evidenced between the experimental and numerical data. In the last fourth and fifth steps we present the principle of an efficient correction process to compensate the OPD in the cavity, by means of movable internal lensing, and its experimental validation within the frame of single-pass and double-pass cats’ eye configurations. The connection between the OPD and output beam distortions is investigated thanks to far field monitoring under various operating conditions. We will demonstrate the athermal behaviour of the design, in the real time, to ensure a convenient stabilization of the output energy and beam quality regardless of the operating regime and over the whole range of temperatures T = 20–200°C.

## 2. Optical design

The optical design comprises two building blocks, referred to as the pumping and the laser heads. The laser head comprises a long and thick slab of Yb^{3+}: YAG with four, broadband AR coated laser faces.

#### 2.1 Edge pumping configuration and packaging of the slab

The pumping head [10] involves a high power, single-pass and optically homogenized design, which makes use of a large stack of diode bars at λ_{P} = 932–937 nm. The bars are micro-lensed along their slow and fast axes (SA, FA) and placed in front of an external micro-lens array and of a couple of large, plano-convex focusing lenses (Fig. 1). This enables peak pump powers up to P_{stack} = 5 kW and a peak I_{P} up to 50 kW/cm^{2} in the QCW regime. The peak total value of the absorbed pump power (P_{absTot}) in the volume of the slab, a fractional value of P_{stack}, varies from 3 to 4 kW.

Given variations of the PRF and pumping durations in the ranges of 10–30 Hz and T_{P} = 1.5–10 ms, respectively, this implies variations of the average value of the total pump power in the slab (P_{absAvg}) up to more than 1 kW. An estimate of the thermal loading, in the first order, is defined by Q = η_{Q}.P_{absAvg} = η_{Q}.PRF.P_{absTot}.T_{P}. η_{Q }= 1-λ_{P} / λ_{L} ≅ 10% figures the quantum defect. In the following, the notations involved in a cartesian coordinate system of reference {O, x, y, z}, L_{S}, H_{S} and D_{S} figure the slab's length, thickness and width, respectively. The pump focal spot is located in the close vicinity of the pumping edge, in the front area of the slab. Its width and length can be ranged from Δ_{x} = 3–4 mm along the FA and Δ_{y} = 1–2 mm along the SA, respectively.

#### 2.2 Cat's eye cavity design

Our amplifying cavity belongs to the family of cat's eye optical schemes. This kind of scheme enables the implementation of a large number of optical passes in fairly compact and upgradable designs. It takes advantage of a high robustness to misalignment effects and thermo-mechanical penalties, to be expected at the location of the slab. For comparison, this is not the case with other optical architectures such as those used with thin-disks or angularly multiplexed designs. These designs usually require more complex optical schemes and, possibly, active mirror control. The whole cats’ eye cavity is embedded on a water-cooled laser head, around the slab. We use a couple of reflectors at the incidence of 45 degrees, which can be finely translated with respect to the other ones, in order to define a series of 6 closely spaced elementary paths along the slab. Starting from beam seeding throughout the input-output aperture, timely-referred ray tracing helps to figure the folded beam-path inside. The complete architecture of the cavity is shown in Fig. 2, to figure the alignment of the reflectors with respect to the slab, and movable optics of further interest for OPD compensation. Surface water-cooling, close to the top and bottom faces of the slab, operates via thermal contacting, thanks to a pair of multi-layered copper sheets. Copper has been preferred to more usual Indium, due to the un-consistency of a fusion temperature below 150°C. Given finely controlled screw-driving under calibrated tightening, we selected the option of multi-layered sandwiches for making sure of stress-free, very soft packaging conditions. This avoids any hazard of un-controlled mechanical constraints, while taking advantage of fairly stable heat transfer conditions through the metallic supports on both sides. Further brazing, or other alternatives to maximize the efficiency of heat transfer from the slab, should be possible. But this simple and flexible configuration appeared quite consistent with our prototyping requirements.

The lasing conditions of the setup may be finely adjusted thanks to monitoring the fluorescence in the slab. This helps to control the spatial distribution of absorbed pump during the pump sequence (Fig. 2-(b)) or make sure of the alignment of the different beam paths, during small-signal amplification (Fig. 2-(c)).

## 3. Thermally induced gain penalties and OPD variations along the slab

We basically need to discriminate the most critical temperature-dependent effects to be expected, due to the spectroscopy of Yb^{3+} ions in the host matrix and to the onset of thermo-mechanical distortions in the slab. They imply very different contributions to the global optical performance of the amplifier, which involve dedicated measurement procedures.

#### 3.1 Gain penalties

Starting with gain issues, we determine the variations of the spectral emission cross section [11] and of the usable optical bandwidth besides **λ _{L}**. This aims to make an idea of the material's behavior and of possible requirements for complementary spectral gain shaping when amplifying picosecond, or even femtosecond-long pulses. Then, normalizing the measurement results provided by spectral fluorescence data in the near field, it is worth to vary the plotting resolution (Fig. 3). This eases the determination of further tradeoffs, with respect to the benefit of elevated peak gains such as those enabled in our cat's eye design. When T ranges from 20 to 100°C, the peak emission cross section (σ

_{eL}) decreases by a factor of about 2. The bandwidth of the peak emission line around λ

_{L}broadens up noticeably. The FWHM is enlarged by a factor of 3–4 near T = 100°C, nearly up to 15 nm. More severe gain limitations arise in the range of 150 - 200°C. These results should be discussed in more details, in terms of the affordable range of operating temperatures, versus applicable criteria to be specified in the design of a MOPA system.

#### 3.2 Thermal monitoring

Prior going on with OPD issues, we also need to characterize the external temperature gradients in the slab. The thermal measurements are implemented in the transient regime, in order to make sure of the actual time of stabilization whatever the cooling conditions, in a variety of pumping conditions. An infrared camera Optris is triggered at the PRF of the pump (Fig. 4). Because of the high emissivity of the YAG in the MWIR band, however, temperature cartographies just give access to the slab's external surfaces. The internal temperature gradients may only be determined by a fitting process, between these cartographies and computational data. Provided PabsAvg = 300 W, for benchmark, we verify that the higher the doping content, the larger temperature gradients along the slab and the higher the peak temperature. This applies for constant values of the λ_{P} and I_{P}.

#### 3.3 Characterization of the OPD

The connection of pump-induced thermal effects with the subsequent variations of the OPD, along the laser faces of the slab, is established thanks to complementary Phasics-based measurements [12]. Starting from a planar seeding wave-front of reference upstream, to be precisely adjusted, the reconstruction of the distorted wave-front transmitted after crossing the slab makes use of a set of image-based interferograms with dedicated image processing (Fig. 5). For circular pupils, this involves Zernike-like decomposition and projections.

A particular optical scheme needs to be used, comprising a point source of reference at λ_{L} and two afocals (Fig. 5-(a)) with convenient magnification and imaging conditions. Provided pupil matching through the slab, the magnification factor of the second afocal is accounted for in the extraction of the phase map. This enables reliable OPD monitoring (Fig. 5-(b), (c)). Excepting in the first beam-path for N = 1 (Fig. 5(b)), where edge effects appear near to the pumping face, nearly plano-cylindrical profiles are evidenced along the whole slab’s length. This applies whatever the path number from N = 2 to 6 (Fig. 5-(c)). Due to the form factor and to axial symmetry of the geometry, this is consistent with a precise centering of the pump and symmetrical cooling from both sides. Further modeling will help to parameterize the alignment uncertainties and to complete the discussion. By scanning the circular pupil, we clearly evidence the peak-to-peak (PTV) variations of the OPD along the Z-axis and the influence of the doping content. When considering a set of three intermediate positions at 5-10-15 mm (Fig. 6) behind the pumping edge and a constant measurement pupil of diameter 2.8 mm, the PTV varies from 0.1 to 0.8 λ_{L} in the whole range of temperatures involved. This corresponds to variations of the equivalent focal length from 20 to 80 cm. For a millimetric beam-waist, the results are consistent with peak-to-peak distortions of 0.3–0.5 µm at the location of the laser faces.

To verify the influence of the doping content (N_{tot}), *e.g.* the value of the actual absorption depth (L_{abs}) in the presence of more or less saturation at λ_{P}, two kinds of slabs have been tested, with N_{tot} = 2 and 3%. The figure comprises rather large error bars, as estimates of the actual measurement uncertainties due to air turbulence. This arises due to deleterious convection near the hot faces of the slabs. Despite beam tubing everywhere, the experimental setup could not be enclosed in a large vacuum chamber. This should enable a reduction of the measurement noise down to 0.02–0.05 λ_{L}. Under these conditions, consistent trends and orders of magnitude essentially are determined in the whole range of T = 100–200°C. The peak value of the OPD at T = 200°C then varies from OPD_{P} = 0.1 to 0.6 λ_{L} with N_{tot} = 3%. It does not exceed 0.2 λ_{L} with N_{tot} = 2% doping. This figures the fact that the higher N_{tot}, the hotter front slab's area near the pumping face, and the shorter L_{abs}. In order to establish the connection between the variation of the temperature gradients and those of the OPD along the Y-axis, it is now time to look for applicable modeling.

## 4. Modeling the temperature-dependent variations of the cavity gain and OPD along the slab

The modeling topics of interest, to operate our design in Figs. 1,2 under highly variable thermal loading, imply two directions. They concern the gain penalties, in connection the results reported in Fig. 3, and the issue of OPD variations by comparison with those in Fig. 6. These experimental data clearly evidence that the temperature gradients and subsequent OPD are governed by the spatial distribution of the absorbed pump power P_{p}(x, y, z) in the slab.

#### 4.1 Energy extraction from the cavity

We determine the spatially distributed thermal source term Q(x, y, z) by considering the quantum defect (1 - λ_{L}/λ_{P}) ≅ 10%. The influence of complementary ASE-dependent source terms, possibly towards the external supports, can be neglected in the first order. Modeling the gain and energy extraction capabilities, at first, we have to deal with significant variations of the equivalent saturation fluence (F_{sat}) up to quite elevated values for T. This concerns the range of 100–200°C. Provided available peak pump powers up to P_{P} = 3.5 kW, our design takes advantage of large initial values of the small-signal gain, *e.g.* up to G = 15–20 dB with output energies in excess of E_{out }= 100 mJ [4,5]. But the ‘gain x energy’ product G x E_{out} obviously remains maximum as long as T can be kept in the range of 20–50°C, with PRFs below 10 Hz.

The implementation of peak emission cross sections values from Fig. 3-b in modified Frantz-Nodvik gain calculations, for P_{P} = 2 kW, then helps to verify consistent numbers with the actual rate of decrease of the cavity gain with T. The computational results are given in Fig. 7 (solid lines) and superimposed with experimental measurements (dotted points) in the whole range of T = 30–180°C. As shown for N = 6 paths, good fits are evidenced. Given that single-pass to double-pass switching just requires the addition of an input polarizer, a quarter-wave plate and a back reflector, this was verified for both single-pass and double-pass cavities.

In the presence of rather moderate saturation effects, the global gain decreases by a factor of 10–12 dB towards T = 150°C. Given a seed energy E_{seed} = 5 mJ, E_{out} decreases from 70 to 13 mJ when T increases up 180°C. This illustrates the effect of a strong decrease of σ_{eL} (1030 nm), from 2.15 to 0.6 10^{−20} cm^{2}, *e.g.* equivalent values of F_{sat} from 8 to 30 J/cm^{2}. The preservation of a constant E_{out} with the same P_{p}, even increasing T up to about 200°C, should only be made possible with a higher E_{seed} and/or of more internal paths in the cavity. The use of a longer slab with reduced N_{tot}, N = 8 to 10 paths, might be a valuable option.

#### 4.2 Generic modeling of the pumping cartography in the slab

The second modeling topic of a prime interest invokes the influence of the variations of thermo-mechanical properties in the YAG matrix at elevated temperatures. This already took place in a number of works in the field, in other optical configurations. Depending upon the measurement process, the doping content and the case of crystalline or ceramics-based options, more or less discrepancy may be noticed between the references. But the data summarized in Table 1 may be considered as widely accepted numbers [13–16]. They give the input data of interest to build a consistent thermo-opto-mechanical model, for application at elevated temperatures. Lets us then refer to Fig. 4 with N_{tot} = 3% and nearly total pump absorption, with L_{abs} << L_{S}. It is worth to notice that no cold area can be evidenced in the rear side of the pumped slab, even near the top and bottom faces. For consistent heat transfer modeling, the thermal interface between the large faces of the slab and its external metallic supports needs to be defined by means of its global contacting conductance *h _{C}* (W/m

^{2}/K). This parameter implies a uniform thermal transfer through the whole contacting surface, which essentially determines the local drop of temperature (ΔT

_{contact}) along the X-axis.

Given the actual rate of heat transfer (∂Q/∂T, W), we get:

From the microscopic point of view, the overall value of *h _{C}* is given by the sum of 3 elementary contributors [17–20] in the form of ${h_C} = {h_{asp}} + {h_{{\mathop{\rm int}} }} + {h_{rad}}$. They figure, respectively, the contacting surface asperities (

*h*) on the surface involved, the interstitial air gaps (

_{asp}*h*) and the radiative heat transfer across the gaps (

_{int}*h*) when

_{rad}*ΔT*tends to exceed 100°C. Copper oxidation may lead to more or less complementary variations in the contributing surface emissivity to be considered, respectively some increase of

_{contact}*h*and some decrease of

_{asp}*h*. Paying a particular attention to minimizing the transfer of external constraints into the slab of our setup, we make use of rather low pressures for contacting. As verified in the following, this leads to a global

_{rad}*h*in range of 5 10

_{C}^{3}to 10

^{4}W/m

^{2}/K.

To figure the repartition of T(x, y, z) in the volume of the slab versus the pumping and cooling geometry, versus N_{tot} and actual edge-pumping conditions, we make use of representative FEM modeling using COMSOL [21]. This is done in the worst case of no pump depletion, prior beam amplification in the cavity. Our model is based on a consistent approximation for P_{p}(x, y, z), with respect to an average value of the effective absorption depth (L_{abs}) at λ_{P} and to the pump intensity (I_{P}(x, z)), at the location of the pumping edge. I_{P}(x, z) and L_{abs} are defined, respectively, at 1/e of the peak value of P_{P}(x, y, z), with respect to the size of the pump focal spot. Rather than referring to the usual Beer-Lambert coefficient, the parameterization of the longitudinal absorption by means of a specified L_{abs} looks more consistent with partially saturated absorption. Given a large local I_{P}(x, 0), the related absorption cross section of the material may not figure the actual L_{abs}. We just consider current single-pass pump absorption, as in the situation of our setup, due to the need of no possible damage hazards in the presence of uncontrolled optical feed-back into the diode stacks. Further double-pass operation, to optimize the overall optical efficiency, should involve similar principles as those to be presented in the following. Assuming pump focusing close to the front edge of the slab and integrating the contribution of pump divergence along L_{S}, in terms of I_{P} variations, P_{p}(x, y, z) can be expressed by means of a super-gaussian form of the 4^{th} order :

This approximation remains consistent because I_{P} does not vary by a factor of more than 2–3 along L_{S}, given the selected margins with respect to the optical aperture limitations of our pumping design. As defined in the (x-y) plane, the cross-sectional profile remains very close to pump monitoring results in the near field close to the pumping edge. Strictly speaking, the value of L_{abs} comprised in the first exponential term of (2) applies to pure linear pump absorption at λ_{P}. For I_{P} up to nearly 20 kW/cm^{2}, this implies that L_{abs} = σ_{abs}(λ_{P}) / N_{tot}. Conversely, in the situation of completely saturated absorption, with I_{P} in excess of 30 kW/cm^{2}, this first term should be replaced with a linear form such as (1 - z / L_{abs}) and a reduced value of L_{abs} [22,23]. In the intermediate range 20 < I_{P} < 30 kW/cm^{2}, of interest for actual intensity variations along the slab, the expression (2) still gives a relevant approximation. We fit L_{abs} accordingly to the definition of its average value from z = 0 to L_{s}. This was implemented by Ref. to [23] to ensure a representative approximation of the thermal source term, with no need of more complex and unusable expressions. Under these assumptions, it can be shown that P_{absAvg} can be written in terms of the Whittaker [24] function (W_{M}):

Where:

The expressions (3, 4) enable a relevant calibration of the thermal source term Q(x, y, z) in our FEM model, in terms of W/cm^{3}. This is done by reference to the value of P_{stack}, to the fraction of un-absorbed power behind the slab (P_{un-abs}) and to η_{Q}. P_{stack} and P_{un-abs} being easily measurable values using a calorimeter, we just have to identify the quantity P_{absTot} = P_{stack} - P_{un-abs} with its numerical counterpart from (3–4) to fit the FEM and experimental conditions (Fig. 8). Provided a set of input parameters {L_{S}, H_{S}, D_{S}, L_{abs}, Δ_{x}, Δ_{y}} = {20, 3, 8, 10, 1, 4} mm, to figure the application of (2–4) in the case of a slab with N_{tot }= 3% in the setup of Fig. 1, the repartition of P_{p}(x, y, z) takes the form of Fig. 8-(a). This gives the actual repartition of Q(x, y, z) of interest for thermal modeling. At this step, thanks to using the computational capabilities of COMSOL Multi-Physics, we have determined all the input data of interest for a comprehensive thermo-opto-mechanical model of the slab in the static regime. Thanks to combining the two modules referred to as ‘Heat Transfer’ and ‘Solid mechanics’, the internal gradients of temperature, stress and deformation can be determined at the same time versus the boundary conditions applied to the slab. We define D(x, y, z) the local deformation. The thermal contacts are supposed to be stress-free. We do not consider forced convection over the lateral faces. As already introduced to figure top and bottom cooling, we can verify that the value of *h _{C}* actually helps to parameterize the thermal gradients between the large faces of the slab and its mechanical supports [25–27].

This is our quantity of basic concern in the model, for thermal fitting. Provided the inaccessibility of internal gradients, these fits only apply to the lateral faces of the slab along the Z-axis, thanks to comparing the cartographies of external temperatures. Computed cartographies are shown in Fig. 8-(b)-(c) in the case of representative conditions, with P_{absAvg} = 300 W and the rather moderate value *h _{C}* = 5 kW/m

^{2}/K. The peak temperature in the front part of the central plane in the slab is T

_{max}≅ 105°C. It is worth to underline the consistency between the experimental data in Fig. 4-(a) and modeling results in Fig. 8-(c), comparing the local gradients between the large surfaces of the slab and the metallic supports. Very similar temperature gradients ΔT

_{sr}= 35–40°C can be verified in the rear area of the slab, which actually help to justify the value of

*h*. Parameterized modeling versus ΔT

_{C}_{sr}then evidences that the precision in the determination of

*h*following our fitting approach is about +/- 15%. This helps to set up the basics and to make sure of the applicability of measurable temperature gradients, to benchmark the variations of the OPD.

_{C}#### 4.3 Variations of the temperature gradients and OPD along the slab with the contacting conductance

OPD modeling needs to refer to the measurements in Fig. 6-(a), with values of the PTV in the range of 0.2–0.5λ for N_{tot} = 3% and the appropriate pupil. We aim to verify that the experimental and modeling data may be efficiently linked using the straightforward connection between the profiles T(x, z) and OPD(x, z) along the laser faces. Let us denote OPD(x, z) = OPD_{z}(x) and T(y, z) = T_{y}(z), for convention.

The extraction of the transverse OPD profiles OPD(x) from conveniently parameterized COMSOL-based calculations gives the generic series of results in Fig. 9. We considered *h _{C}* from 2 to 100 kW/m

^{2}/°C, {L

_{S}, D

_{S}, H

_{S}} = {20, 8, 3} mm, N

_{tot }= 3%, P

_{absAvg }= 300 W and L

_{abs }= L

_{S}. This helps to figure the straightforward connection between the internal gradients of temperature (red lines) in the slab and the peak OPD per laser face (blue lines), along the y and x axes. When scanning the slab from the pumping edge to its rear area, the peak temperature (T

_{Py }= T

_{y}(z = 0)) in the central horizontal plane decreases from 180 to 80–90°C. Given that P

_{un-abs}/ P

_{stack}< 15%, despite gradients of the same signs along the Y-axis, the decreasing rate of T

_{Py}appears to be smaller than for the absorbed pump power. On the other step, the peak-to-peak value of the transverse gradient of temperature (ΔT

_{PPy }= T

_{y}(z = 0) - T

_{y}(z = L

_{s}/2)) along the Z-axis remains much smaller. We see an almost constant value of 30–35°C all along the slab, from the central vertical plane to the laser faces. The subsequent OPD

_{y}(x) undergoes rather large variations along the Y-axis, which follow the spatial distribution of P

_{p}(x, y, z). Due to the form factor of the geometry, as expected, the peak-to-peak gradients of the OPD (OPD

_{PPy}= OPD

_{y}(0) - OPD

_{y}(H

_{s}/2)) along the X-axis appear to be much larger than those along the Y-axis (OPD

_{PPx}= OPD

_{x}(0) - OPD

_{x}(L

_{s})). Comparing the OPD profiles near the pumping edge (OPD

_{0}(x)) and in the median area of the slab (OPD

_{-Ls/2}(x)), we can notice the same discrepancy as already stated in the experimental data from Phasics-based measurements of Fig. 5. Regardless of

*h*, the OPD

_{C}_{PP}remains in the order of 0.2 µm besides the pumping face. This must be underlined together with the fact that larger variations arise in the median area, from 0.2 up to 0.4 µm per laser face. The higher

*h*, the larger OPD

_{C}_{PPy}far from the pumping edge. Fitting these results with the experimental data in Fig. 4-(a) and Fig. 6-(a), we get an estimate of

*h*in the range of 4–6 kW/m

_{C}^{2}/°C. This applies to the operation of our setup in Figs. 1,2 with a PRF of 50 Hz and T

_{P}= 160°C. Very consistent numbers have been provided at this step, with a good agreement between the experimental and modeling data regardless of the location along the slab.

In order to complete the comparison of OPD data from Fig. 6-(a) and Fig. 9-(a), it is worth to underline that the current modeling is based on the assumption of stress-free and tension-free conditions in the plane of the large faces of the slab. The only surfaces assumed to be fixed and infinitely rigid are those of the cooling supports. This results in the fact that the computed OPD essentially describes the effect of natural distortions within the YAG material itself, purely due to thermo-mechanical loading. We did not consider the addition of parasitic stress transfers within the contacting layers, possibly induced through their own thickness. This combination of stress-free, tension-free conditions with fully flexible prototyping requirements aims to benchmark the minimum OPD with respect to other packaging conditions. This was not the topic of this work, but the impact of additional perturbations should also be considered to improve the cooling performance. With brazing, for example, the actual benefit to be expected from a possible increase of the above *h _{C}* by an order of magnitude and even more will have to be discussed in terms of additional OPD. Complementary issues then include the optimization of intermediate layers and of subsequent YAG's polishing requirements to avoid more ASE penalties.

It is now time to look at the capability of the cats’ eye cavity to be operated under varying thermal loading. We will take advantage of the latter analysis of the elementary phenomena for to discuss the results.

## 5. Process of internal OPD correction in the cat's eye cavity

Due to deleterious cascading effects, multi-passing always suffers from a particular sensitivity to the onset of OPD in the active medium. The larger N in the cavity, the more significant issues when summing the path-per-path wave-front distortion inside.

#### 5.1 Principle of OPD correction

With N = 6 to 12, in the situations of the single-pass or double-pass cat's eye designs in Fig. 2, the preservation of a consistent beam quality at the output of the cavity implies path-per-path, typical distortions of no more than λ/10-λ/20. But in the situation of a low T at very low PRFs, this is not the case. Taking advantage of the modeling of local OPD variations path per path, in a first step, it is worth to determine some orders of magnitude of interest at the output of the cavity. The estimates of the total OPD to be expected are given with respect to the thickness of the slab, while accounting for the constant fractional overlap between the propagating beam and the selected thickness. For fully generic modeling and further discussion, we also refer to more usual thin slabs, denoted as Innoslabs [28]. Ranging H_{S} from 1 to 5 mm and considering *h _{C}*

_{ }= 5 10

^{3}W/m

^{2}/°C, P

_{absAvg}= 300 W and N = 6 paths, we get the results in Fig. 10. They apply to the pumping conditions of Fig. 8, with N = 6 equally spaced paths along the slab. These results reveal that the peak-to-peak value of the total output OPD

_{PPy}should not exceed 4 µm (Fig. 10-(a)). The ortho-normal components of the peak total OPD from the cavity (Fig. 10-(b), red lines), as defined by OPD

_{PPx}and OPD

_{PPy}along the x and y axes, respectively, range up to 4 µm and 21 µm. Provided the form factor of the geometry and the enhancement of the distortion phenomena by multi-passing, this verifies the fact that the component OPD

_{PPy}does not contribute to output distortions in a significant manner.

Considering OPD_{PPx}, as long as H_{S} does not exceed 3–3.5 mm, a strongly nonlinear dependency is evidenced with H_{S}. The rate of increase of the global OPD in thicker slabs, Δ(OPD_{PPx}) / Δ(H_{S}), stabilizes around the constant value of + 2 µm/mm. The characteristic OPD_{PPx}(H_{S}) evidences a threshold-like behavior. The rate of increase of the peak internal temperature, Δ(T_{P}) / Δ(H_{S}), remains nearly constant, approximately +40°C/mm. A couple of experimental data have been superimposed in the figure, to fit the computational data with {T_{P}, OPD_{PPx}} measurement results for single-pass cavities using 3 and 4mm-thick slabs. These data involve conditions as those of Fig. 4, when monitoring T_{P}. But Phasics-based measurements as those of Fig. 5 were not possible. More deleterious environmental perturbations, due to air turbulence over the whole operating system, could not be avoided. We made use of rougher estimates, provided thanks to more easy-to-implement measurements of the equivalent thermal focal lens (F_{th}) at the output of the cavity. This just involves the definition of the thermal lens (Fth) versus the OPD, along the X-axis. The conversion of OPD_{PPx} in terms of F_{th} also helps to refer to the cavity geometry and to the length of the beam-paths inside, which enables further benchmarking with respect to other works in the field.

#### 5.2 Thermal lensing and basics of OPD compensation

We make use of the spherical approximation. In the direction of the X-axis of a given laser face, we consider a circular profile of radius *R _{N}*. Denoting

*R*the radius of curvature, the focal length of the elementary YAG-based plano-convex lens associated in the

_{N}*N*beam-path is given by ${F_N} = \frac{{{R_N}}}{{2({n_{YAG}} - 1)}}$. The relationship between the peak value (

^{th}*Δ*) of the OPD in the same beam-path writes as $R{}_N = \frac{{\omega _o^2}}{{2{\Delta _N}}}$, within a pupil of diameter 2xω

_{N}_{o}, implies a focal length in the form of:

The equivalent thermal lens of the complete cavity can be expressed by $\frac{1}{{{F_{th}}}} = \sum\limits_{i = 1}^6 {\frac{1}{{{F_{th\_i}}}}} $. Given that H_{s}/4 < ω_{o} < H_{s}/3, typically, this remains consistent as long as the total propagation length inside does not exceed the Rayleigh length of the propagating beam. To give orders of magnitude for pupils with an average diameter of 3 mm, the former ranges of OPD_{PP} of interest, typically 4–0.5 µm, must be related to thermally induced focal lengths from 30 cm to 2–3 m. The experimental OPD data in Fig. 10 have been determined thanks to (5), by monitoring the beam pinching phenomena at the output of the amplifier. Despite significant uncertainties in the presence of varying beam distortions, in the order of +/−10 to +/−15%, we verified that these results evidence a fairly good agreement between the modeling and experimental data. Here again, the definition of a consistent value for *h _{C}* remains the most critical parameter for fitting.

Let us now complete this study with the topic of OPD compensation. We aim to preserve the output beam quality within acceptable limits with respect to pure TEM_{00}, regardless of the operating regime and temperature. As already introduced, the idea consists of adding internal lensing into the cavity to finely shape the global phase map of the OPD at its output. The global OPD being governed by the summation of elementary OPD from the individual pupils, we must generate opposite phase profiles with the suitable PTV through a given number of those pupils. This is made possible thanks to a cylindrical lens with a negative focal length (F_{O}), which is closely coupled to the slab and translated along Z-axis. The compensation of local OPD penalties is based on the principle of the addition of some amount of complementary OPD in excess, along a variable fraction (N_{O}) of the N beam-paths in the cavity.

Starting from fully representative computational data with respect to the former data, the Fig. 11 gives an illustration of the shaping process. Just adding a plano-concave lens with F_{O} = −400 mm, we made use of the input data from Figs. 8, 9 and 12 under the assumptions of P_{absAvg} = 300 W and h_{C} = 5 kW/m^{2}/K. As stated in Fig. 2-(a), to make sure of no supplementary phase shift modulo 2π, this is made possible thanks to the addition of a neutral plate in the plane of the moving lens with the same thickness in its central area. By figuring the actual phase map of interest for each of the N beam paths (Fig. 11-(a)), as defined by the global phase map of the OPD along the laser faces (Fig. 11-(b)), these modeling conditions clearly evidence the process of OPD compensation. They show the longitudinal variations of the equivalent focusing lens, of which the positive focal length decreases while varying the position z from 0 to L_{S}. The addition of a compensating lens with F_{O }= −1 m along the last N_{O }= 4 beam pupils then helps to generate complementary phase maps (Fig. 11-(c)) of which the sum has been minimized. In more generic terms, the adjustment of the set of values {F_{O}, N_{O}} for a given T_{O} enables an efficient compensation of the global OPD in a specified cat's eye cavity.

## 6. Optical performance of the athermal design from 20 to 200°C

Accordingly to the simple process above, various series of experimental configurations are implemented in the manual mode of operation. As a follow up of the identification of the suitable combinations, in the generic terms of {P_{absAvg}, N_{O}(N), F_{O}}, they will be experienced in the real time.

#### 6.1 Static monitoring in the far field to quantify the output beam distortions

To get a global signature of the efficiency of OPD compensation, we need to make use of beam monitoring in the far field of the cavity output. The cavity is operated in a MOPA configuration with a variable PRF. This involves synchronous seeding and pumping in the QCW regime, while triggering a Wincam-based camera system. Given a propagation length of the amplified beam of 2 m downstream the output of the cavity, the operating temperature in the slab is ranged from T_{P} = 30 - 200°C. A series of pictures in Fig. 12 show the natural variations of the output beam quality in the case of the un-compensated cavity. They give an illustration of the beam monitoring results which were used to plot the experimental data in Fig. 10. Starting from to OPD-free conditions at T_{P} = 30°C (Fig. 12, left-hand side) with nearly single-mode conditions, the PRF is gradually increased up to 50 Hz.

The measured T_{P} then reaches 180°C. The amplified beam undergoes significant distortions, with spurious displacements along the X-axis. These displacements arise due to the combination of residual alignment uncertainties in the vertical plane, in the presence of pump centering uncertainties with respect to the central horizontal plane of the slab. Additional uncertainties come from residual shifts along the slab, beam-path to beam-path. Even paying a great attention to minimize tilt errors when aligning the cavity reflectors, the current prototyping conditions do not easily help to reduce the peak-to-peak uncertainty below 200–500 µm. This remains affordable, in the whole range of temperatures involved. In the single-pass configuration, earlier internal beam distortions then arise towards T_{P} = 125–150°C. They evidence a strong astigmatism in the {x, y} plane, as expected, with the shorter focal length in the direction of the X-axis. The most deleterious contribution (OPD_{PPx} t) to the total OPD is originated in the direction of the thickness of the slab. The formerly stated features about the curvature of laser faces are verified, even though this is not so evident in the double-pass configuration. Much more critical misalignment uncertainties are noticed. The onset of output beam distortions is experienced for lower temperatures, near T = 100°C. The fact that these distortions do not evidence the same shape as those in single-pass can be explained due to mixed contributions. Single-pass OPD effects arise, with an enhanced contribution of the distortions along the Y-axis. This might be discussed with respect to a more detailed analysis of the alignment uncertainties in the vertical plane and the size of the input-output aperture of the cavity. Complementary effects result from a slight, global axial shift between the forward and backward beam-paths of the double pass design. An estimate of the peak shift was about 100–200 µm. In order to quantify the natural sensitivity of the output beam quality to highly variable thermal loading, these results and numbers give representative orders of magnitude.

Introducing a plano-concave cylindrical lens, with F_{O} = −1 m in the cavity, and just facing the last 4 paths in the slab, we get the second set of pictures in Fig. 13. By reference to the computational results in Fig. 11, and comparing with the latter beam shapes in Fig. 12 of reference, this defines a particular configuration of interest to verify the efficiency of OPD compensation.

In this situation, there is no OPD compensation along the last 2 paths. It is worth to notice that minimal output beam distortions now arise far from the room temperature. This applies to the single-pass and to the double-pass configurations, as well, when crossing a couple of particular temperature areas. Comparing with the initial beam quality from the single-pass cavity to be operated near the room temperature, internal lensing then enables an efficient compensation of the global OPD in the range of T_{P} = 150–160°C. Given similar lensing conditions, the double-pass cavity takes advantage of a similar output performance in the range of T_{P} = 110–130°C. Out of these ranges of interest, we do verify the enlargement of the laser beam along the X-axis, due to un-consistent lens divergence. The same demonstration might be done for N_{O} = N = 6 with a longer F_{O}, if available. The reduction of the output beam size from the double-pass cavity, to be noticed with respect to its single-pass counterpart, basically results from the choice a retro reflector with a curvature radius of 2 m.

#### 6.2 Real-time OPD correction for an athermal design

Let us finally go on with the demonstration of an athermal behavior, by varying the PRF and scanning the complete range of temperatures of interest. Global experimental results are summarized in Fig. 14, which describes convenient updating conditions to optimize internal lensing. They comprise the suitable control margins, to make sure of an output beam with the optimized quality. Rough estimates of the M^{2} [29] are given, thanks to the measurement of beam divergence in the far field using step-by-step scanning. The related numbers are given on the right axis of the figure using color-coded polyhedras. The polyhedra aperture figures the measurement uncertainty by reference to the standard formula, ${M^2} = \frac{{\pi {\omega _o}{\theta _{farfield}}}}{{2\lambda }}$. Five zones of temperature must be distinguished (color-coded line segments) within the total range explored, *e.g.* 30 - 200°C, during a complete warm-up sequence associated to the onset of the pump at t = t_{start}. The total sequence duration of 70 sec figures the complete stabilization of the thermal regime. Each of these five zones involves specific internal lensing conditions, in terms of the number of beam-paths facing the movable lens. The M^{2} does not give any information about the actual modal content of the output beam and the fractional power content in terms of the gaussian fundamental mode. But its variations help to put some numbers on the output beam quality, in connection with those of former intensity cartographies. As shown, they remain negligible up to about T = 150°C. They become more significant in the range of 150–200°C, mainly due to un-corrected OPD along the direction of the length of the slab. This is consistent with beam distortions in Fig. 10. We superimposed the output energy variations along each zone of temperature, to relate the experimental conditions in Fig. 14 with those of E_{out} in Fig. 7 in the range of 20–90 mJ. They apply to E_{seed} = 5 mJ and P_{absTot} in the order of 3 kW.

An ultimate proof of concept would require the combination of the upgradable capabilities for OPD correction, at the same time and in the real time, together with gain and / or seed-energy adjustments. This will require a comprehensive remote control, with appropriate control sequences to vary P_{stack}. Due to the lack of motorized remote control, at the time of this work, manual operations are not sufficient to complete comprehensive feedback requirements. Consistent margins then remain to be determined versus the peak power available and / or of the seed energy. However, the latter demonstration of the efficiency of moderate rates of variation already gives the proof of the consistency of the whole process in the real time. By manually switching from a given zone of temperature to the others, we have verified the effectiveness of displacement times in the order of a fraction of second. As long as T_{P} does not exceed 180–200°C, the implementation of the Phasics-based analyzer or of a pre-calibrated thermal sensor in a computer-controlled feedback loop appears to be an efficient solution to stabilize the output beam quality. This gives basics for the development of fully athermal systems in the QCW or burst regimes, in the presence of adjustable pump powers, pump durations and / or PRFs.

## 7. Conclusions

We have investigated the most critical thermal issues of interest to operate a slab-based cat's eye cavity as a flexible amplifier, which produces highly energetic pulses at 1030 nm regardless of the lasing conditions. This was done in the whole range of temperatures T = 20–200°C. Paying attention to the spatial repartition of the absorbed pump power, we evidenced a straightforward connection between the repartition of the absorbed pump power in the slab and the shape of OPD cartographies along its laser faces. Good fits have been evidenced between the experimental data and FEM modeling results, regarding the variations of the local OPD. This involves thermal lensing conditions with local focal lengths as short as 20–80 cm. Complementary fits of interest also apply to gain variations and to the amplified energy, in the presence of very significant penalties regarding the emission cross section. Given such a large variety of coupled phenomena, from a more generic point of view, it is worth to emphasize the very global agreement between the experimental and modeling results.

On the other step, an easily implementable process has been demonstrated to make the cavity fully athermal, when operated in highly variable temporal regimes. Taking advantage of the cats’ eye principle with retro-reflectors at 45 degrees, this is made possible thanks to a movable, unique plano-concave lens. The lens essentially needs to be closely coupled and translated along the slab. Given convenient adjustments with respect to the operating conditions, nearly zero global OPD was evidenced at the output of the cavity in the whole range of temperatures explored. This applies in the real time, in consistency with peak thermal transients in the order of 20°C/s in the slab. Thanks to adjusting the pumping power and updating the OPD compensation in a consistent manner, the output beam quality and output energy can be stabilized at the same time. This was verified, for applicability to further remote control.

Our study should be of interest for the development of further laser systems with kW-class pump powers, using more or less slabs with optimized packaging conditions. The replacement of bulk contacting, as here involved for pumping powers up to 300 W, may enable an increase the contact conductivity through the large faces of the slab by a significant factor. This can be expected using brazing, straightforward water-based cooling through the large faces of the slab, or a combination of heat pipes. By reference to our current packaging conditions in this study, the validation of contact conductivities up to 10^{5}−10^{6} W/m^{2}/°C should be consistent with average pumping powers up to 2–4 kW. These numbers involve nearly ideal thermal conduction. They will open the route towards KW-class cats’ eye cavities, to be QCW pumped with PRFs in the range of 100 Hz, or in the CW regime at the expense of gain limitations here specified.

## Acknowledgements

We wish to thank our colleague Dr Fl. Leroi for valuable discussions and comments. We also wish to thank J-F. Gleyze, from the French Atomic Energy Commission, for the loan of electronic equipment.

## Disclosures

The authors declare no conflict of interest.

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