## Abstract

An optimization-based correction method is developed to control simultaneously two deformable mirrors in a wavefront-sensor-less adaptive beam cleanup system, where the wave-front aberrations could not be compensated by a single deformable mirror. Stochastic parallel gradient decent algorithm is chosen as the optimization algorithm. In this control method, different aberrations are assigned to each deformable mirror according to their different correction quality. The method is proved to be effective by numerical simulations as well as experiments. Experimental results showed that the area containing 84% energy of the laser beam in the far-field can reach 3.0 times diffraction limited.

©2012 Optical Society of America

## 1. Introduction

High beam quality is one of the most important issues in solid-state laser applications. However, it is hard to realize because of the wave-front distortions within laser cavities [1]. Adaptive Optics (AO) is an effective way to improve beam qualities of solid-state lasers by correcting the wave-front distortions [2–7]. Normally, it is difficult to achieve large stroke and high spatial resolution with a single deformable mirror (DM), thus more than one DM is needed in many AO systems for wave-front compensation. Usually, the first DM with a large stroke corrects large-scale low-order aberrations and the second one with high a spatial resolution compensates for small-scale high-order aberrations. These applications of double-DM or even multi-DM could be found in manipulating eye’s monochromatic aberrations or conducting high-resolution retinal imaging *in vivo* [8–13], laser beam cleanup etc [14–18]. The control methods for multi-DM have attracted a lot of attentions. In the above double-DM systems, aberrations detected by wave-front sensors (mostly Hartmann-Shack wave-front sensor) are assigned to different DMs according to correction performances. However, in strong-scintillation conditions or when the intensity of laser beam is rather non-uniform, AO systems based on wave-front sensors may not work well [19, 20]. Besides, the control architecture of wave-front sensor-based AO systems might be too complex for certain applications where compactness and dedicated designs are in demand.

For the reasons above, an alternative method based on an optimization algorithm is provided and proved to be effective. In Ref. 7, a number of possible architectures are proposed to control multiple DMs with the stochastic parallel gradient decent (SPGD) algorithm. In the first architecture with independent controllers, DMs may be coupled when they are driven simultaneously. In the second architecture with single controller, two DMs are treated as one corrector whose actuator number is the summation of the two DMs, and the convergence rate would be governed by the slower DM.

These two methods all have limitations of convergence rate, the more DMs in the system the slower convergence rate is. In the third architecture with shared metric, bandpass filters are used to decouple DMs which make the system complicated.

To overcome the inconveniences above, a double-DM control method based on optimization algorithm is presented. Aberrations are decomposed and assigned to different DMs depending on their correction qualities. Each order or some orders of phase aberrations can be assigned to only one certain DM so the problem of coupling is solved well. The control method is proved to be effective by numerical simulations and experiments on slab laser beam cleanup with a double-DM AO system.

## 2. Principle of controlling two DMs with optimization algorithm

SPGD algorithm is an optimization algorithm which is proved to be effective in many laser systems [21, 22]. The principle of SPGD is described in details in Refs. 23-28. The control strategy of multi-DM based on SPGD algorithms is as follows: First an interferometers or other wave-front sensors are used to measure the response function of each actuator. ${R}_{j}^{n}(x,y)$ represents the *j*th actuator’s response function in the *n*th DM and ${u}_{j}^{n}$ is the voltage applied to this actuator. *Φ* is the aberration to be corrected, which can be expressed by orthogonal polynomials, for example Zernike polynomials which are orthogonal on a unit circle:

*δ*(

_{m}*x*,

*y*) denotes the

*m*

^{th}orthogonal polynomial with

*a*the coefficient and

_{m}*l*the total number of aberration orders. If the

*n*th DM is used to correct the

*m*

^{th}order Zernike aberration, the equation can be written as:where

*N*is the total number of actuators in the

*n*th DM. Equation (2) can be rewritten as:where ${u}_{m}^{n}$is the voltage vector for the

*m*

^{th}order Zernike aberration in the

*n*th DM.

*R*is the response function matrix of the

^{n}*n*th DM.

*δ*is the Zernike aberration’s matrix. For multiple DMs, Eq. (3) can be written as:

_{m}*t < m*, the wave-front distortion such as

*Φ*is corrected by the first DM in the system and so on.

_{1}*ε*is the residual wave-front.

For simplicity, two DMs are considered in the following section. In SPGD algorithm, the metric function can be written as *J*^{(}^{k}^{)} = *J*(*a _{1}^{k}*,

*a*,

_{2}^{k}*a*…,

_{3}^{k}*a*), where

_{l}^{k}*a*,

_{1}^{k}*a*,

_{2}^{k}*a*…,

_{3}^{k}*a*are coefficients generated by the controller at the

_{l}^{k}*k*th iteration and these coefficients can be transformed to voltage vectors accordingly. The iteration steps are as follows: firstly, a group of statistically independent random perturbations of coefficients

*δa*,

_{1}^{k}*δa*,

_{2}^{k}*δa*…,

_{3}^{k}*δa*is generated and then “positively” applied to the DM as (

_{l}^{k}*a*,

_{1}^{k}+ δa_{1}^{k}*a*,

_{2}^{k}+ δa_{2}^{k}*a*…,

_{3}^{k}+ δa_{3}^{k}*a*). The aberration

_{l}^{k}+ δa_{l}^{k}*Φ*is decomposed and assigned according to the correction qualities of the two DMs (tip and tilt are corrected by the tip/tilt mirror, so

*a*=

_{1}*a*= 0 and

_{2}*δa*=

_{1}^{k}*δa*= 0). If the third order of aberration is selected to be compensated by DM 1, let

_{2}^{k}*b*= (

_{1}*0*,

*0*,

*a*,

_{3}^{k}+ δa_{3}^{k}*0*, …,

*0*),

*b*= (

_{2}*0*,

*0*,

*0*,

*a*…,

_{4}^{k}+ δa_{4}^{k}*a*), then the voltages applied to DM 1 and DM 2 could be obtained from Eqs. (2) and (3). The metric function

_{l}^{k}+ δa_{l}^{k}*J*

_{+}

*is calculated after the voltages applied to DM 1 and DM 2. Similarly, apply a “negative” random perturbation array (*

^{k}*a*,

_{1}^{k}- δa_{1}^{k}*a*,

_{2}^{k}- δa_{2}^{k}*a*…,

_{3}^{k}- δa_{3}^{k}*a*) and calculate the metric function

_{l}^{k}- δa_{l}^{k}*J*

_{-}

*. Update coefficients as:*

^{k}*a*

_{i}^{k}^{+}

*=*

^{1}*a*+

_{i}^{k}*γδa*(

_{i}^{k}*J*

_{+}

*-*

^{k}*J*

_{-}

*),*

^{k}*i*= 1 to

*l*, where

*γ*is the gain coefficient. The values of

*γ*and

*δa*depend on experimental systems. In numerical simulations and experiments, the initial value of vector

_{i}*a*at

_{i}*k*= 0 are all zeros. Because of the properties of the SPGD algorithm, the convergence rate of the algorithm depends not only on the initial

*a*but also on

_{i}*δa*,

_{i}*γ*and the wave-front aberrations to be corrected. The metric function is calculated from the focal spot. Encircle energy, mean radius and Strehl ratio are usually chosen as the metric functions. The iterations run continuously until a manual stop or some promising results are achieved.

With the recent development of the SPGD algorithms, the convergence speed of optimization process can be significantly increased. Though the convergence speed of SPGD algorithm is faster than that of ant colony algorithm [29] and genetic algorithm [7], the convergence time is increased along with the increase of the control channels of the adaptive optics system. Even with very favorable estimation, the convergence time increases especially in a large number of control channels, at least as a factor of *N _{c}^{1/2}* [23], where

*N*is the number of control channels of the adaptive optics system.

_{c}## 3. Numerical simulation of controlling two DMs

#### 3.1 Correction quality of the DM and the principle of distributing the aberrations to DMs

How to decompose the aberration into orthogonal orders and assign them to different DMs is very important in multi-DMs adaptive beam cleanup systems. One reasonable way is calculating the correction qualities of every DM and then the orthogonal aberrations are assigned to different DMs accordingly. Two 39-actuator rectangular DMs are used in the following simulation. The actuator arrangement of the rectangular piezoelectric DM employed in our simulations and the experiment is shown in Fig. 1 . The parameters of this DM are as follows: effective area 40mm × 40mm, range of stroke ± 3μm, maximum voltage ± 450V. The distance between two neighboring actuators is 10 mm.

The length of square beam is chosen as 80% length of DM’s. The Zernike polynomials used to represent the aberrations are orthogonal and normalized in unit square area [30]. The response function of the *j*th actuator of the DM can be expressed as [31]:

*d*is the distance between every two neighboring actuators and equals to 10mm in our case;

*α*is Gaussian coefficient and sets to 2.14;

*ω*is the coupling coefficients of the DM and sets to 0.18; (

*x*,

_{j}*y*) is the position of the

_{j}*j*th actuator. RMS of all incident aberrations is 1μm. The correction quality of the 39-actuator DM for the first 35 orders polynomials aberration is shown in Fig. 2 . The first 16 orders of polynomial aberrations are compensated well and the RMS of residual wave-front is less than 0.2μm.

#### 3.2 Comparison of correction qualities between single ideal stroke DM and two limited stroke DMs

Usually, the correction ability of a single DM is limited by the strokes and spatial frequencies. In double-DM AO system, one DM with large stroke compensates for the large-scale low-order aberrations and the other one with high spatial frequency compensates for the small-scale high-order aberrations.

In numerical simulation, the wave-front before correction in Fig. 3(a) is made of a large-scale defocus and small-scale high order aberrations. The PV and RMS of the wave-front aberration before correction are 9.03μm and 1.55μm, respectively, while the stroke of the actuator is limited to ± 3μm. So a limited stroke 39-actuator DM could not compensate for the wave-front aberration completely. The focal spot with normalized intensity before correction is shown in Fig. 3(b). The Strehl ratio is about 0.04. Comparisons of correction qualities between the double-DM system based on the above control strategies and a single DM are illustrated in Figs. 4 –6. The residual wave-front after corrected by one limited stroke 39-actuator DM is shown in Fig. 4(a) with PV 2.99μm and RMS 0.24μm, in comparison with the residual wave-front after corrected by one ideal stroke 39-actuator DM shown in Fig. 4(b) with PV 0.95μm and RMS 0.07μm. PV and RMS can reach 0.84μm and 0.08μm in residual wave-front after compensation by a limited stroke double-DM system as is shown in Fig. 5(c) . In Figs. 5(a) and 5(b), the wave-front generated by DM 1 and DM 2 are shown. Defocus is generated solely by DM 1 and other orders of aberrations are corrected by DM 2. The focal spots are compared in Fig. 6 when the wave-front aberration is corrected by a single ideal stroke DM, a single limited stroke DM and double limited stroke DMs respectively. The Strehl ratio after corrected by single limited stroke DM is about 0.74. The correction performance of two DMs with limited stroke is better than that of a single ideal stroke DM and a limited stroke DM. The Strehl ratio can reach about 0.95. The Strehl ratio corrected by single ideal stroke DM is about 0.83 which is better than that of a limited stroke DM.

#### 3.3 Effect of unmatched correction orders and aberration orders

Because there isn’t any wave-front sensor in the system to measure the wave-front distortions, then the order of the orthogonal polynomials in wave-front distortions couldn’t be estimated accurately. In the algorithm, the correction orders of orthogonal modes are hard to be just equal to the order of aberration modes to be corrected. In general, three different situations are considered: the number of correction orders is larger than, equal to or smaller than the number of real aberration orders. In the simulations below, the wave-front aberration to be corrected is generated by the 3rd-10th Zernike polynomial within a unit square area (the 1st and 2nd order are tip/tilt in two orthogonal directions corrected by a tip/tilt mirror). The wave-front distortion before correction showed in Fig. 7(a) whose PV and RMS are 4.71μm and 0.73μm, respectively. The corrected results in the three different situations are shown in Fig. 8 . The 3rd −15th orders, 3rd −10th orders and 3rd −7th orders are chosen to correct the aberration in these three situations respectively. The incident wave-front aberration has a large defocus and is beyond the correction capability of a single DM. Then DM 1 is chosen to correct defocus and encircled energy is the metric function. The residual wave-front when correction orders are 3rd to 15th orders which are larger than aberration orders is shown in Fig. 8(a). The PV value of the residual wave-front is 0.76μm and RMS is 0.07μm. The 3rd to 10th orders’ coefficients of the residual wave-front are smaller than that before correction. However, the 11th to 15th orders of aberrations are brought into the residual wave-front as is shown in Fig. 8(b). These orders of aberrations are considered in the algorithm though do not exist in the wave-front aberration before correction. Mode confusion takes place between the 11th to 15th orders of aberration and the 3rd to 10th orders of aberration. In Fig. 8(c), the PV value of the residual wave-front is 0.36μm and RMS is 0.03μm when the number of correction orders is equal to that of the wave-front aberration before correction. 3rd to 10th orders of the aberration is compensated well, as is shown in Fig. 8(d). When the number of correction order is smaller than that of the wave-front aberration before correction, the PV is 0.94μm and RMS is 0.13μm after correction, as is shown in Fig. 8(e). The 3rd to 10th orders of aberrations are not completely compensated because the correction orders are 3rd to 7th in control strategy. Mode coupling exists between correction orders and aberration orders. The 8th to 10th orders of aberrations are not affected, as is shown in Fig. 8(f). The reason is that these orders of aberrations are not corrected in the algorithm.

Large numbers of simulations are performed and the average convergence curves are shown in Fig. 9 . The Strehl ratio of the focal spot increases with the iteration process in three situations but the results are different. The Strehl ratio can reach about 0.95 when the number of correction orders is equal to that of aberration orders. But in fact, it is very difficult to make them equal without wave-front detection. How to select the correction orders must be considered in the control strategy. When the number of selected correction orders is larger than that of the aberration orders, the Strehl ratio can reach about 0.84, larger than 0.60 obtained when the number of selected correction orders is smaller than that of the aberration orders. For the above reasons, the number of correction orders should be selected larger than the number of aberration orders based on the correction quality of the DMs.

## 4. Experiment of beam cleanup of slab lasers using two DMs

To evaluate the control strategy, a series of experiments are carried out. The optical layout of the double-DM beam cleanup system is shown in Fig. 10 . It mainly consists of a tip/tilt mirror, two DMs, a relay system and a controller implementing the control algorithm. The response function of each actuator is measured by an interferometer and the relation matrix of response functions and orthogonal polynomials are calculated before the experiment which is shown in Fig. 11 . The wavelength of slab laser system is 1.064μm. A cylindrical expander system is added after the slab laser system for reshaping the laser beam into about 40mm × 40mm in order to match the aperture size of DMs. The laser beam is first reflected by the tip/tilt mirror and then the wave-front aberration is corrected by DM 1 and DM 2. A relay system is employed here to make DM 2 conjugate with DM 1. A camera is placed at the focal plane of the lens. The controller implements the control algorithm and sends out signals to the tip/tilt mirror and DMs. The focal length of the lens before the camera is 1.4m.

In the experiment, DM 1 is set to correct defocus and DM 2 is set to correct other aberrations among the first 15 orders except tip/tilt which is corrected by the tip/tilt mirror. The process of control is shown in Fig. 12 . The mean Strehl ratio is 0.25 before correction and increases to 0.48 after compensation by DM 1, then DM2 is turned on to make a further correction. Though heat is produced as the laser operation which would have negative effect on the laser beam quality, the mean Strehl ratio reaches nearly 0.65 after 1500 iterations. As when DM 1 is turned on, the mean Strehl ratio increases considerably, we could figure out that defocus occupies a large part of the wave-front aberrations. The iterative number is about 270 per second with closed-loop AO system operation in the experiment, so the convergence time is not long.

The dithering amplitude of iteration curve is analyzed. Before correction, the dithering of Strehl ratio is caused by the laser system. The standard deviation of the iteration curve in this section is 0.0156. When DM 1 is working, the dithering amplitude of the iteration curve increases in this section. The standard deviation of the iteration curve in this section increases correspondingly and reached 0.0403, which is 1.58 times larger than that before correction. DM 1’s surface shape continuously changes during the optimization process, and is reflected in the iteration curve. After DM 2 begins to work, the Strehl ratio increases because of wave-front distortions are further compensated. The standard deviation of iteration curve is 0.0535 in this section which is 1.33 times larger than that when DM 1 solely operated. The reason is that DM 2’s surface shape continuously changed during the control process and is reflected in the iteration curve as well as the thermal effects produced in the laser system.

The experiment results are shown in Fig. 13 and Fig. 14 . Long time exposure far-field spots are shown in Fig. 13(a), 13(b) and 13(c). The fraction of energy within the white circle is 84% in total. Figure 13(d) is the PIB (Power in the bucket) curve for far-field of the laser beam. A PIB curve, i.e. a plot of fractional power within a given beam diameter or beam width versus the diameter or width [32]. β, the times of diffraction limited, is used to represent the beam quality [33]. Before correction, β = 6.7. After compensation by DM 1, β = 3.7. With DM 1 and DM 2 on, β = 3.0. The focal spot converged after correction by the double-DM beam cleanup system and the peak intensity increases obviously. The horizontal and vertical cut of the focal plane images are shown in Fig. 14.

In simulations, nearly no coupling occurred between two DMs. The correction capabilities of the two DMs are fully used. The surface shape of these two DMs in the experiments after correction are calculated and decomposed into orthogonal polynomials, as shown in Fig. 15 . The coefficients of orthogonal polynomials are compared. The results indicate that small amount of coupling occurred between two DMs, including the 3rd, 10th, and 13rd order. In Fig. 16 , the voltages applied to half of the DM’s actuators reach the predefined limit, indicating that DM 1’s correction capability is nearly fully used for defocus correction in the experiment. From the above results, the way that aberrations assigned to different DMs for correction is reflected by the simulations.

## 5. Conclusion

A multi-DM control method based on SPGD algorithm has been presented. The correction performances among a single limited stroke 39-actuator DM, double limited stroke 39-actuator DMs and a single ideal stroke 39-actuator DM have been compared through numerical simulations. The results indicated that for large-scale aberration compensation beyond a single DM’s correction ability, the correction performance of two limited stroke 39-actuator DMs is better than a single limited stroke 39-actuator DM. The Strehl ratio can achieve 0.95 with double-DM system but only 0.74 by a single limited stroke 39-actuator DM system in our simulations. Because there is not a wave-front sensor to detect the distortion, the number of aberration orders can’t be accurately determined. It is better to select almost all orders of aberrations that the DM can reach a good correction performance. The wave-front of a slab laser has been corrected by the double-DM beam cleanup system. Before correction, β = 6.7 and β = 3.0 with DM 1 and DM 2 on. The double-DM experiment has proven the validity of the multi-DM control method.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) and Preeminent Youth Fund of Sichuan Province under grants 10974202, 60978049 and 2012JQ0012.

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