## Abstract

We discuss the deformation of a partially pumped active mirror amplifier as a free standing disk, as implemented in several laser systems. We rely on the Lucia laser project to experimentally evaluate the analytical and numerical deformation models.

©2011 Optical Society of America

## 1. Introduction

Diode–Pumped Solid–State Lasers (DPSSL) gained special interest in recent years with decreasing costs of laser diodes and the development of large size laser gain media, especially laser grade ceramics and horizontally grown (HDC) crystals [1]. DPSSLs with their high wall–plug efficiency opened the way to efficient power amplification in high energy laser chains. These last amplifying sections define to a great extent the overall efficiency and the average power of a laser beam line. They differ mainly in the way the gain medium is used to extract the stored energy: in pure transmission (slabs or rods) or coupled with internal reflections (active mirrors [2], zig-zag slabs [3], total reflection active mirror [4]). However, Amplified Spontaneous Emission (ASE), Laser Induced Damage (LID) and thermal effects remain the main bottlenecks to overcome in the quest for an efficient (> 10% wall–plug efficiency), high average power (> 100 *W*), energetic laser system (> 10 *J* pulse energy).

ASE reduces the energetic gain and LID limits the safely usable intensity of the laser pulse. Thermal effects can reduce the achievable gain and distorts the wave front, leading to unwanted (de)focusing while propagating through the amplifier system. Since the choice of the laser gain medium geometry depends to a great extend on the requested laser parameters, we will concentrate only on one specific amplifier class, the Active Mirror Amplifier (AMA).

The aspect ratio of an AMA is limited by both ASE considerations and thermal limitations. Scaling of such amplifiers lead to an optimum design point, where ASE and thermal effects are minimized to an acceptable level. A further increase in amplification performance can only be achieved by a change in design or a multiplication of amplifier stages.

A laser beam traveling through a pumped laser gain medium will experience unavoidable wave front deformations. The main origins are the deformation of the surfaces (mechanical lens effect), a nonuniform temperature distribution (thermal lens effect) and a nonuniform pump profile (electronic lens effect). The impact of these effects can be quantified in first order with equivalent focal lengths *f _{mech}*,

*f*and

_{th}*f*.

_{e}The first effect is related to the nonuniform temperature distribution, leading to a different expansion at different positions within the crystal/ceramic: the gain medium is deformed. This can, as we will see, lead to a “bi–metal” type of deformation in the case of a thin disk approximation. The thermal lens effect can be split up into several contributions. At first, a nonuniform spatial temperature distribution generates a change of the index of refraction. This impacts the phase of the transmitted laser beam and acts like a lens: this effect is therefore called thermal lens. Further, with the thermal expansion the material experiences stress and strain, affecting the optical indicatrix [5]. Polarization dependent effects, like bi–focusing, birefringence, etc. appear. A thorough discussion can be found in [6].

The electronic lens effect *f _{e}* is more subtle, as an excitation within the electronic level system changes the polarizability within the material and consequently the index of refraction [7, 8]. This effect is typically, at room temperature, one order of magnitude smaller than the other effects, but can become important at cryogenic temperatures. A comparison between thermal and electronic lens effect can be found in [9]. Nevertheless, a uniform pump distribution within the extraction surface can strongly reduce this effect.

In the case of a rod–like geometry, the thermal lens dominates usually over the surface deformation [10], therefore the surface deformation is often neglected in a first order approximation.

Considering an AMA, where the gain medium is close to the shape of the thin disk geometry, the main thermal variation takes place along the pump direction. A strong thermal variation perpendicular to the extraction direction is not expected as long as most of the total gain medium surface is pumped. Let us recall that we do not discuss the typical thin disk case, where the active medium is glued or soldered onto a heat sink, we are indeed studying an almost free standing geometry. For such a geometry a circulating liquid is used to cool the back side. In this case the mechanical lens effect can become stronger than the thermal lens effect.

We will derive in the following section a simple model estimating the thermally induced focal length and discuss its validity in the case of a partially pumped AMA. The design of the laser gain medium under study is defined by the Lucia laser program [11, 12]. Starting from a one–dimensional model estimating the thermally induced deformation, we will discuss its validity in the case of a partially pumped AMA. In a second phase we will consider a three dimensional numerical model and compare those results with an experimental validation.

## 2. Partial pumping of an active mirror amplifier

Let us consider a geometry as shown in Figure 1. An active medium with lateral size *L* and thickness *D* is uniformly pumped over its surface along the z–axis. The top (front surface) is thermally insulated and the bottom (backside) is subject to cooling. The peripheral surfaces are considered to be thermally insulated as well and we can therefore consider this model to be in good approximation with a 1D approximation, at least along the central pumping axis of the gain medium. The resulting parabolic temperature distribution along this z–axis leads to an expansion difference between the front– and backside.

As a first order estimation we consider the deformation of our AMA to be small and the bulging between the surfaces to be negligible. With the definitions as shown in Figure 1 we get

*L*the lateral gain medium size,

*R*the gain medium radius of curvature and

*D*its thickness. The expansion with the temperature is Δ

*L*=

*α*Δ

_{T}*TL*with Δ

*T*the temperature difference between the large surfaces and

*α*

*the thermal expansion coefficient. We get by this:*

_{T}The gain medium holds a heat generation density *q̇*. Our interest is focused on the temperature difference between the surfaces in the single–side cooled AMA case. We get directly for this case Δ*T* = *q̇D*
^{2}/(2*k*). The volumetric heat generation *q̇* is expressed as *q̇* = *η _{h}η_{abs}P_{inj}*/

*V*, where the factor

*η*describes the fraction of the absorbed total injected average power

_{abs}*P*within the volume

_{inj}*V*and

*η*is the fraction of this absorbed power converted into heat [13].

_{h}As *q̇* was assumed to be uniform, the volume *V* is by this defined as *V* = *A _{beam}D* where

*A*is the beam section. Finally, the radius of curvature of the deformed gain medium is given by:

_{beam}*R*+

*D*), then experiencing a reflection on the back surface with a curvature

*R*and finally again a transmission through the first surface. Under the estimation of

*D*≪

*R*, a reflection on the front surface gives

*f*=

_{front}*R*/2. This is the same in the case of an internal reflection (

*f*=

_{mech}*f*=

_{int}*f*=

_{front}*R*/2) [14]. As

*P*/

_{inj}*A*corresponds to the average intensity

_{beam}*I*, the focal length is with these approximations:

_{avg}*k*and smaller thermal expansion coefficient

*α*will lead to longer focal lengths, while a higher dissipated power per area

_{T}*η*will shorten the focal length.

_{h}η_{abs}I_{avg}Another information which can be drawn directly out of Figure 1 is that in the case of the Lucia main amplifier we get preferably a negative focal length. This differs from what can be observed in rod–like geometries, where the positive thermal lens and the bulged surfaces generate a positive focal length in the case of a positive *dn*/*dT*.

Figure 2 (left) shows three separate cases. An Yb^{3+} doped Yttrium Aluminum Garnet (Yb^{3+}:YAG) crystal with a doping concentration of 2 *at*.% at ambient temperature (300 *K*) is characterized by *α _{T}* ≈ 6 × 10

^{−6}

*K*

^{−1}[15] and

*k*≈ 9

*Wm*

^{−1}

*K*

^{−1}[16], while at temperatures close to 150

*K*the thermally induced focal length increases by almost a factor of 4, since the thermal expansion decreases to

*α*≈ 3.5 × 10

_{T}^{−6}

*K*

^{−1}[15] and the thermal conductivity increases to

*k*≈ 20

*W m*

^{−1}

*K*

^{−1}[16]. A third case shown is a thick, highly doped crystal (20

*at*.%).

In such highly doped gain media, the heat generation can be significantly increased, leading to a fraction of absorbed power converted into heat as high as *η _{h}* ≈ 30 % [17]. In the case of a gain medium several times thicker than the reabsorption length (i.e. Yb

^{3+}:YAG at room temperature), additional heating occurs. This might further increase

*η*to values close to ≈ 50%.

_{h}However, the Lucia experimental test bench does not comply with such an approximation of a fully pumped surface. Practically is the pump spot size limited by the achievable gain, geometrical considerations used to circumvent the deleterious impact of ASE and the gain medium mount. Consequently, a variation of the radial temperature distribution is expected. This will result in a thermal lens effect as well as additional bulging of the surfaces. The impact of the thermal lens effect may be approximated in first order by the model discussed by Cousins [18]. We use this model to estimate the average radial temperature distribution for an end–face cooled circular slab with a circular pumped area and additionally we approximate the global deformation with the derived simple model.

The calculated radial average temperature distribution *T̄*(*r*) is approximated to be a parabolic profile within a fixed observation aperture according to [18]. The pump aperture however may vary. Knowing the observation aperture, a temperature difference Δ*T̄* between the center and the radial extend of the aperture can be derived. Expressing the expected radial variation *n*(*r*) of the index of refraction *n*
_{0} in the form of *n*(*r*) = *n*
_{0} (1 − (2*r*
^{2})/*q*
^{2}), with *r* the radial position and *q* a fitting parameter, and using the relationship for the corresponding focal lens of the thermal length *f _{th}* ≃

*q*

^{2}/(4

*n*

_{0}

*D*) [10], we get:

*p*is the pupil radius,

*D*is the thickness of the laser gain medium and

*dn*/

*dT*the variation of the refractive index with temperature.

In the case of Yb^{3+}:YAG at room temperature (300 *K*) values for *dn*/*dT* are in the order of ≈ 8 × 10^{−6}
*K*
^{−1} and decreases almost linearly to ≈ 3 × 10^{−6}
*K*
^{−1} at 150 *K* [16]. The thermal lens will be encountered twice for an AMA architecture due to the internal reflection, effectively shortening *f _{th}* by a factor 2. We consider the mechanical and the thermal lens to be perfect thin lenses and the resulting total focal length

*f*will be:

_{tot}*f*is negative and

_{mech}*f*is positive. We expect a minimum

_{therm}*f*for a specific surface occupation of the pump. Consequently a reduction of the pumped surface for a given disk diameter will lead to a longer (negative) or even a positive focal length, compared to a fully pumped disk. This would be completely different in the case of a transmitted extraction beam, where both effects can positively add up to generate a converging lens (in the case of Yb

_{tot}^{3+}:YAG).

The impact of a partially pumped surface for an active mirror is shown in Figure 2 (right figure). The bold line gives the limiting case when a 60 *mm* diameter, 7 *mm* thick gain medium is fully pumped. It shows a focal length of approximately *f _{mech}* ≈ −16

*m*at an average intensity of 100

*W*/

*cm*

^{2}for a 2

*at*.% doped Yb

^{3+}:YAG crystal with

*η*= 0.1,

_{h}*η*= 0.9 and

_{abs}*α*= 6 × 10

_{T}^{−6}

*K*

^{−1}. Decreasing the pump spot diameter for a given pump intensity and keeping the remaining parameters constant, an increased focal length is found. The curves on the left side of Figure 2 shows the resulting focal length for pump spot apertures between 38

*mm*and 22

*mm*, whereas the observation aperture was taken to be 2

*mm*smaller in diameter.

In the case of an almost totally pumped surface (more than 2/3 of the diameter), no distinct impact on the total introduced wavefront deformation can be expected, while for smaller pumped surfaces the lateral extend becomes more and more important, finally resulting in a compensation of the (in our case) negative mechanical lens effect. A further reduction of the pumped surface will introduce a positive thermal lens, the total system will exhibit a converging lens like behavior.

## 3. Experimental results

In order to study the nature of the deformation on an AMA we used the setup shown in Figure 3. A cw laser source is imaged onto the gain medium under an angle of 25°, is refracted through the AR coated front surface, travels through the gain medium until it hits the HR coated backside. At this point, the beam gets reflected, leaves the laser gain medium along a symmetrical path and is finally imaged onto a wave front detector (SID4 sensor manufactured by Phasics). Adequately coated gain media also allow the measurement of the front surface deformation.

The laser gain medium is pumped using the pump delivery system as shown in Figure 4. The laser diode emission of a laser diode array is concentrated onto the gain medium using deviation prisms (fast axis) and large concentration mirrors (slow axis). The laser diode array can host up to 88 laser diode stacks with 3 *kW* peak power each. The laser diode emission wavelength is centered close to 940 *nm* and has a duration of 1 *ms*. In the experimental setup only 41 laser diode stacks are used, resulting in an intensity of 16 *kW*/*cm*
^{2} during the 1 *ms* long pump pulse. The almost circular pump surface has ≈ 6.5 *cm*
^{2}. Each of the laser diode stacks can be individually spectrally shifted to ensure the central emission wavelength between single–shot and repetition rates up to 10 *Hz* [19].

In order to ensure a sufficient cooling over the large surface, a jet–plate cooling system was developed and implemented. The heat exchange coefficient on the backside of the laser gain medium was measured to be *h* ≈ 15,000*W*/*m*
^{2}/*K*. The cooling water temperature is held at 13°*C*. We can therefore consider the outer surfaces to be thermally insulated and we can estimate the heat flow for a fully pumped surface to be one–dimensional in a very good approximation. We will use Equation 4 to estimate the focal length due to the deformation of the gain medium.

As test objects, three different HDC grown Yb^{3+}:YAG crystals were used in our AMA. We used a pair of rectangular crystals with a doping of 20 *at*.% with a size of 41 *mm* × 37 *mm* × 4.7 *mm* (R1) and 2 *at*.% with a size of 40 *mm* × 35.8 *mm* × 8.4 *mm* (R2) as well as a circular crystal with 60 *mm* diameter and 7 *mm* thickness (C1).

For the rectangular crystals, a large fraction of the whole surface is covered by the pump. The pump spot is almost circular with the half–axes of the elliptically shaped pump of 16 *mm* and 13 *mm*. On the other hand, in the circular case, only ≈ 50% of the crystal diameter is covered by the pump. As discussed in the foregoing sections, we expect consequently for the circular case an increased focal lens, whereas for the rectangular crystals the experimental results should be close to Equation 4.

In order to estimate the thermal lens effect coupled together with the deformation of the surfaces, a three dimensional Finite–Element–Analysis (FEA) was performed using CAST3M for all the cases under observation.

The physical deformation of the laser gain medium being not directly accessible experimentally, an indirect access was obtained through the measurement of the reflected wavefront. Out of the analysis of the recorded wavefront deformation, the gain medium deformation on its own can be assessed as long as the impact of the thermal lensing can be neglected. The experimental setup is shown in Figure 3.

Crystals R1 and R2 were probed using a HeNe laser. The applied coatings on the large surfaces were anti–reflective for 633 *nm* on one and highly reflective on the other side. Both exhibited a reflectivity of less than 1 % for the pump wavelength at 940 *nm*. This offers the possibility to observe independently the front surface deformation and the whole wave front deformation in the AMA case just by switching the gain medium upside down. The crystal C1 was measured at 1064 *nm* as the design point of its coatings is 1030 *nm*. This ensures a good performance at the test wavelength in the active mirror design for both the AR and HR coating and yields the wave front deformation in the energy extraction case.

The analysis of the recorded wave front yields a decomposition on Zernike polynomials, where one of those polynomial describes the parabolic deformation (defocus) of the wave front. From its associated amplitude *Z _{def}*, the corresponding focal length

*f*can be calculated with the pupil radius

*r*and the measurement wavelength

_{p}*λ*:

*at*.% doping and 4.7

*mm*thickness), as it is expected to exhibit the strongest deformation for a given

*I*. For several incident average intensities

_{avg}*I*the wavefront deformation was measured. Out of this the corresponding focal length was computed using Equation 7 and the defocus polynomial amplitude

_{avg}*Z*.

_{def}Front surface deformation analysis results compared to the simple derived model as well as an FEA simulation are shown in Figure 5. The experimental results fit very well to the expectation out of the simple model and the 3D simulation under the estimation of a significantly increased thermal load (50 %). Catastrophic damage occurred at average intensities of about 30*W*/*cm*
^{2} and consequently no measurement was performed for a reflection in the active mirror configuration. The right image in Figure 5 shows the calculated temperature distribution at 30*W*/*cm*
^{2} for the case of R1. The temperature rise leads to a significantly increased stress of more than 100 *MPa* at the outer boundary. This explains the observed catastrophic damage.

The results for the front surface and the internal reflections in the case of the 2 *at*.% crystal R2 are shown in Figure 5 as well. Again, a very good agreement compared with the simple model is found and furthermore reflections on the front and back side show the same focal length in first order approximation, which validates Equation 4. Considering measurement uncertainties, it can be noticed, that the experimentally found induced lens effect is the same for a reflection on the front or back surfaces. FEA results (not displayed) exhibit a roughly similar behavior. The error bars in Figure 5 for the low doped case were ommited for the sake of readability of the graph, but are in the same relative order than for the higher doped case.

Obviously for those AMA cases, where most of the gain medium surface is covered by the pump and the gain medium architecture is almost close to a thin disk, the simple relation 4 can be used in good approximation.

Let us now consider the case of the Lucia laser AMA: the circular crystal C1 of 60 *mm* diameter and 7 *mm* thickness is pumped on a ≈ 6.5 *cm*
^{2} surface. Average intensities reach values up to 70*W*/*cm*
^{2}. As shown on the left of Figure 6, experimental results differ significantly from the mechanical lens model. Thermal lensing becomes important and must be taken into account in order to predict the total lens effect. The right image shows a cut through the 3D simulation for an intensity of 75*W*/*cm*
^{2}. The calculated wave front deformation out of the 3D model (the expected focal length) is in good agreement with the experimental results.

The spatial temperature distribution is responsible for a significant increase of the focal length. It introduces a non negligible thermal lens effect, partially compensating the negative focal lens introduced by the deformation of the gain medium. However such a more complex deformation will introduce higher order aberrations, like astigmatism.

As the propagation in our setup is not perpendicular to the surfaces, a strong defocus will also lead to higher aberrations. Despite the parabolic wave front deformation is strongly reduced, the astigmatism is not necessarily minimized in the same way. Figure 7 shows a compilation of the experimental and simulation results for all the studied cases as well as the computed wave front deformation in a central, horizontal lineout for the three crystals.

The horizontal lineouts through the discussed cases are mainly of parabolic nature for R1 and R2, but deviates from such an approximation for C1. Consequently the compensation of such higher order wave front deformations becomes more important. Nonetheless, such a significant reduction of *Z _{def}* reduces the risks for laser induced damages, especially on the transport optics.

## 4. Conclusion

For large size, partially pumped Active Mirror Amplifiers (AMA) the deformation of the surfaces is of greater importance compared to the classical rod design. We introduced a simple model capable of estimating the introduced focal power depending only on the material properties as long as most of the surface is subject of the pumping process and the lateral extend of the pump spot is sufficiently larger than the thickness of the gain medium. First order considerations and an experimental benchmark showed a good agreement of this model for a pump spot covering more than 2/3 of the gain medium surface, with a ratio of the pump spot diameter to the gain medium thickness of about 4:1 or higher.

A further reduction of the pumped area on the gain medium (e.g. for ASE management purpose) increases the resulting thermal lens effect. We showed, that such an additional thermal lens can be used to minimize the focal power of a free standing AMA in high power lasers. This is of particular interest to reduce an unwanted (de)focus. The defocus can become as weak as the higher order distortions. High power laser systems usually include a wave front control loop based on Deformable Mirrors (DM) to compensate for such higher orders wave front distortions. Minimizing the defocus as much as possible by other means is of great advantage when running these servo–loops. It allows the DM to use its dynamic range for higher phase distortions.

In the Lucia AMA case under study here, a careful design of the laser gain medium can greatly increase the focal length introduced by the parabolic deformation. Furthermore, being now experimentally benchmarked and validated, the derived models will be useful for further energetic and size scaling of AMA based laser systems, like in the HiPER laser fusion facility definition [20, 21].

## Acknowledgments

This work was supported by the Institut franco-allemand Saint-Louis (ISL). The research leading to these results has received funding from the EC’s Seventh Framework Program ( LASERLAB-EUROPE, grant agreement n 228334).

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