## Abstract

We present a slab laser amplifier beam cleanup experimental system based on a 39-actuator rectangular piezoelectric deformable mirror. Rather than use a wave-front sensor to measure distortions in the wave-front and then apply a conjugation wave-front for compensating them, the system uses a Stochastic Parallel Gradient Descent algorithm to maximize the power contained within a far-field designated bucket. Experimental results demonstrate that at the output power of 335W, more than 30% energy concentrates in the 1x diffraction-limited area while the beam quality is enhanced greatly.

©2010 Optical Society of America

## 1. Introduction

Scaling solid-state lasers to quite high output powers while maintaining good beam quality (BQ) is a desired goal in many cases. Many kinds of architectures of solid-state lasers have been investigated to exploit the capability for achieving this goal, such as the rod lasers, fiber lasers, thin disk lasers, zigzag slab laser amplifiers and so on [1–3]. Among these architectures, zigzag slab laser amplifiers may have the greatest potential to realize the goal thanks to their high gain, convenient thermal management and other advantages [4–6]. Although zigzag slab laser amplifiers can overcome most of the disadvantages of conventional architecture lasers, thermal effects cannot be eliminated completely. Besides, the distortions in the optical elements and in the active area, the bifocus phenomenon and maladjustment of the cavity mirrors will all affect the BQ of the zigzag slab laser amplifiers. To make zigzag slab laser amplifiers output diffraction-limited beam quality, one promising approach is in use of an adaptive optics (AO) control system similar to those used in astronomy [7]. Usually, extra-cavity correction is employed for controlling the wave-front of lasers [8,9]. Outputs from zigzag slab laser amplifiers have been demonstrated up to 10 kW with good BQ achieved by removal of residual wave-front errors using AO system [8,9]. Most of the AO systems used in zigzag slab laser amplifiers employ gradient wave-front sensors to detect the distortions in wave-front and then apply conjugation wave-front for compensating them. However, in the strong scintillation regime or when the intensity of laser beam is rather non-uniform, conventional AO systems based on gradient wave-front sensors may not achieve outstanding performance [10,11].To offer a promising alternative, optimization-based adaptive optics control systems have been employed to improve the beam quality of many laser systems [12–16]. In this paper, we present an AO system based on Stochastic Parallel Gradient Descent (SPGD) algorithm which omits the measurement of the wave-front and can be implemented easily in a zigzag slab laser amplifier with non-uniform intensity output. This paper will be arranged in the following way: firstly, the experimental setup and theory will be presented and described in detail; and then the experimental results are presented, at last some conclusions and discussions are made at the end of this paper.

## 2. Experimental setup and theory

The optical layout of the AO system for beam cleanup of a zigzag slab laser amplifier is shown in Fig. 1
. It mainly consists of a slab laser amplifier, a deformable mirror (DM), and an industrial computer (IC). The IC is used to implement the SPGD algorithm for achieving aberrations correction. The seed laser that used as the signal laser is a continuous wave (CW) 1064nm Nd:YAG laser with an TEM_{00} mode output of 1.2W. The seed beam firstly passes through an expander which includes two cylinder lens and a slit, and then transforms into a 1.7 mm x10 mm rectangular beam. The beam then passes through a series of optical elements to achieve the goal that transmitting the slab laser amplifier back and forth two times which is represented by the blue arrow lines. M1 to M7 are seven reflectors all with high reflectivity (>99.9%) at 1064nm. Two lenses L1 and L2 (both with a focal length of 175 mm) are introduced to form a 4-f system. P1 and P2 are two quarter-wave plates which can convert the polarized orientation of seed beams to ensure that it passes through the slab laser amplifier 4 times. The seed beam will bounce 15 times for once passing through the slab amplifier. The diode end-pumped Nd:YAG slab amplifier has a dimension of 67 mm x 11 mm x 1.8 mm and can magnify the input beam power with a gain of 300. The slab amplifier is pumped by two 808 nm laser-diode (LD) arrays. The pump beams on both sides of the slab amplifier undergo total internal reflection at the slanted endface of the slab. This configuration ensures that the pump beams are absorbed along the entire length of the slab.

After traversing the slab laser amplifier, the amplified beam will take out the distortions generated in the slab. As is shown in Fig. 2
, the corrector employed for compensating for the distortions is a rectangular piezoelectric DM with 39 actuators. This DM is produced in our lab and has a continuous face plate with stacked piezoelectric actuators. The parameters of this DM are as follows: effective area 40mm x 40mm, maximum stroke ≈12μm, maximum voltage ± 500V, multilayer-dielectric coating with a high-damage-threshold (>2kW/cm^{2}). The circles of Fig. 2 represent the actuators. The actuators are designed as trigonal arrangement and the distance between each actuator is 8mm. Figure 3
is the original DM surface shape measured by a WYKO interferometer which can measure the wave-front with a precision of ± 1/20 PV and ± 1/120 RMS. The peak to valley (PV) and root-mean-square (RMS) of the original DM surface are about 0.16μm and 0.03μm respectively which denote that the optical quality of original DM surface is rather good.

According to its principle, the DM deforms its surface when applying voltages on the actuators:

*φ(x,y)*relates to the mirror surface shape, ${v}_{\text{j}}$ is the voltage applied onto the jth actuator, k represents the relationship between voltage and the DM surface shape and can usually be ascertained in advance by measuring the surface shape deflection in the case of applying a unit voltage onto actuator. ${f}_{\text{j}}\text{(x,y)}$is the influence function of the jth actuator on the wave-front, j is the serial number of actuators. The influence function of the DM can be expressed as [17],

_{j}, y

_{j}) is the space position of the jth actuator, d is the distance between every two neighboring actuators and set at 8 mm, x and y represent respectively the value in x-coordinate and y-coordinate in the orthogonal coordinate plane.

At the input position of cylinder expander 1 (CE1), the dimension of the beam is 2 mm x 11 mm, CE1 can enlarge the beam length in Y-direction with a ratio of 2, therefore after CE1, the beam dimension is 2mm x 22mm. Cylinder expander 2 (CE2) is for enlarging the beam in X-direction with an adjustable expanding ratio of 11 to 15. The cylinder expander systems are used to accomplish two tasks: compensating for the bifocus of two orthogonal directions and enlarging the beam to cover the DM as many actuators as possible. During the experimental course, we found that 28 mm x 22 mm was a compromise optimum size, therefore, the expanding ratio of CE2 is set at 14. After being reflected by the DM, the laser beam is incident on a high reflector (HR) to attenuate the power. The reflective part is collected by a power meter (PM) for measuring the power while the transmitted part passes through a focus lens (L3) and a cuniform reflector (CR) in turn. Camera which is set on the focal plane of the L3 is a Swiss produced high speed CMOS camera (MV-D1024E-160-CL-12). This camera has a resolution of 1024x1024 pixels each with a size of 10.6μm, therefore the active optical area is 10.9 mm x 10.9 mm. One of the most advantages of this camera is that the active optical area of the camera can be artificially chosen for actual applications. Generally, the smaller the active optical area, the shorter the time cost for a frame acquisition.

Before correction, a measurement system is built to evaluate the light intensity distribution. Figure 4 shows the beam measurement configuration by use of a Hartmann-Shack (H-S) sensor. Just as the beam cleanup system (Fig. 1), the seed laser beam also passes through the amplifier four times (for compactness, many components are not drawn in Fig. 4), and then the size of amplified beam is expanded to 28 mm X 22mm by two cylinders (CE1 and CE2). A beam splitter (BS) is used to attenuate the beam power of which the transmitted part is for H-S measurement and the reflected part is collected by a power bucket. The H-S has a 32x32 lenslet, with a focal length of 4mm and the diameter of each sublens is 0.14 mm. The beam size is reduced to about 3.5mm x 2.8 mm by an inversion placed 8-x telescope to match the size of lenslet (4.5mm x 4.5mm). Figure 5 shows the far-field (far-field is defined as the beam distribution on the focal plane while near-field means the beam distribution before any focal lenses) intensity distribution on the camera at 70W output while Fig. 6 shows the wave-front reconstructed from Fig. 5 using a zonal reconstruction method. We can know that due to thermal and other deteriorated effects in the slab amplifier, even though the output power is not very high, the intensity of output laser beam is far from uniform distribution. In this case, it is difficult to use a gradient wave-front sensor such as H-S sensor to measure wave-front gradient precisely owning to the discontinuous distribution of light intensity. Therefore, the traditional AO systems which depend on parameters measured by gradient wave-front sensors may not be suitable for beam cleanup of this slab laser amplifier.

Accordingly, an AO system based on SPGD algorithm is employed in this system for achieving good outcomes. As a promising iterative optimization algorithm, SPGD algorithm has been used in many laser fields [18,19]. The principle of SPGD is described in detail in reference 18. In this paper, the power-in-a-bucket (PIB) of far-field spot on the focal plane of L3 (sampled from the CMOS camera) is taken as the metric J ^{(k)} = J (${u}^{k}{}_{1,},{u}^{k}{}_{2,},\mathrm{...},{u}^{k}{}_{39,}$) to optimize, where ${u}^{k}{}_{1,},{u}^{k}{}_{2,},\mathrm{...},{u}^{k}{}_{39,}$ are voltages generated by the IC at the kth generation iterative calculation. These voltages are firstly amplified by a 39-channel High-Voltage-amplifier (HVA) and then applied onto the 39 actuators of the DM. The iterative calculation cycle is as follows: Firstly, generate a group of statistically independent random small amplitude perturbation voltages$\delta {u}^{k}{}_{1,},\delta {u}^{k}{}_{2,},\mathrm{...},\delta {u}^{k}{}_{39}$, and then apply a “positive” perturbation array $({u}^{k}{}_{1}+\delta {u}^{k}{}_{1,},{u}^{k}{}_{2}+\delta {u}^{k}{}_{2},\mathrm{...},{u}^{k}{}_{39}+\delta {u}^{k}{}_{39})$ onto the 39 actuators, afterwards calculate the PIB${J}^{k}{}_{+}$. Similarly, apply a “negative” random perturbation array $({u}^{k}{}_{1}-\delta {u}^{k}{}_{1,},{u}^{k}{}_{2}-\delta {u}^{k}{}_{2},\mathrm{...},{u}^{k}{}_{39}-\delta {u}^{k}{}_{39})$ onto the same 39 actuators, and then calculate another PIB${J}^{k}{}_{-}$. Update voltages as: ${u}_{i}{}^{k+1}={u}_{i}{}^{k}+\gamma \delta {u}^{k}{}_{i}({J}^{k}{}_{+}-{J}^{k}{}_{-})$, i = 1 to 39, where *γ* is the gain coefficient, *δ* is the metric perturbation. The choices of *γ* and *δ* depend on the actual experimental system, in our experiments, the *γ*and *δ* are set at 0.18 and 0.15 respectively. The iterative steps run continuously until they are ended manually or some promising results are achieved.

## 3. Experimental results

At the pumping current of 60A and 80A (pumping power of bout 600W and 1000W), the slab laser amplifier can output 144W and 335W respectively in CW mode. At two cases, we have accomplished beam cleanup based on the SPGD AO system.

To evaluate the correction performance, some BQ metrics should be introduced. For example, the peak intensity of far-field spot or the residual error of the wave-front has been used commonly as the metric to evaluate the performance of a laser system. Generally, the definition of metrics should depend on the actual application situations. In this paper, it is defined as:

where*P*and

*P*(

_{DL}*DL*is the abbreviation of diffraction-limited) are, respectively, the fractions of laser power contained within a far-field radius of λ/D (λ is the wavelength and D is the beam aperture before the focus lens) for the actual beam and for a hypothetical diffraction-limited beam with a spatially uniform intensity and wave-front profile [4]. Since the dimension of beam on the DM is about 28 mm x 22 mm,thus, the ideal beam is defined approximately as a beam with spatially uniform intensity propagating from a 25 mm x 25 mm aperture and with a wavelength of 1064nm. The focus length of L3 is 350 mm. The size of the ideal 1x diffraction-limited spot in the CMOS camera can be calculated in combination with the focus length and beam dimension. According to the actual spot size, the optical area of camera used is 180 x 180 pixels which guarantee that the acquisition rate of the camera is larger than 600 frames per second (fps). For an optimization-based AO system, the aperture size of PIB which will affect the convergence rate and stability of system is a very important parameter. Too large or too small is neither wise for a successful beam cleanup. In fact, the optimum size of aperture should be found out in the actual experiments. In our experimental system, the aperture is a soft aperture (not an actual device but set by computer program) with a size of 11x11 pixels. During the experimental course, we also find that the change of the far-field spot is not rapid which indicates that the variety of phase distortions is slow. Since the SPGD AO system can achieve iterative calculation with a speed greater than 600 fps while 50 to 100 frames are sufficient for an effective close-loop control during the experiments, therefore the SPGD algorithm has a close-loop bandwidth >5 Hz and can keep up with the change rate of phase distortions.

Figure 7 to Fig. 10 shows the experimental results. Figure 7 shows that at 144W, when AO is on, 35% energy is within the 1x diffraction-limited area while BQ is enhanced from 3.7 to 1.5 calculated with Eq. (3). Figure 8 shows that at 335W, more than 30% energy concentrates in the 1x diffraction-limited area while BQ is enhanced from 4.1 to 1.7 when the beam cleanup is accomplished. Figure 9 and Fig. 10 show the horizontal and vertical lineouts of far-field spots at 144W and 335W of which the blue lines represent the closed-loop cases and the red lines stand for the open-loop cases.

## 4. Conclusion and discussions

We have presented the recent slab laser amplifier beam cleanup experimental results at the output powers of 144W and 335W; the achieved results have demonstrated that the SPGD AO system is available for application in the case of non-uniform intensity output beam cleanup. The acquired BQs are promising; however we also note that there are still distances between the close-loop results and the ideal 1x diffraction-limited BQ. There may be three reasons for this: on the one hand, the beam size (28 mm x 22 mm) doesn’t match the DM aperture (40 mm x 40 mm) appropriately, especially in the Y direction (22 mm) which cannot exploit the spatial resolving capability of DM completely; on the other hand, the bifocus phenomenon of the laser beam in X-direction and Y-direction cannot be corrected completely by CE1 and CE2, therefore it still will play some detrimental role in the correction; thirdly, thanks to the limited actuators of the DM, some high-order distortions may not be able to correct at present.

In the future, we will in one way design the beam matching system elaborately(to match the DM aperture more precisely), in another way, fabricate a high density and large stroke DM (with the number of actuators >100 and stroke range>20μm) for correcting high order and high deflection distortions. Besides, we still plan to improve the close-loop bandwidth of SPGD by embedding the program into a real-time Linux operating system to meet possible high bandwidth demands of higher output power zigzag slab laser amplifier systems.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China (NSFC) and the Chinese Academy of Sciences under grants 10974202, 60978049, A09K005 and CXJJ-10-S01.

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