## Abstract

A new adaptive optics (AO) system for controlling the mode profile of a diode-laser-pumped Nd:YAG solid laser has been set up in our laboratory. A 19-element piezoelectric deformable mirror (DM), which is used as the rear mirror of the solid-state laser, is controlled by a genetic algorithm (GA). To improve the system convergence rate, the GA optimizes the first 10 orders of Zernike mode coefficients rather than optimize 19 voltages on the DM. The transform matrix between the 19 voltages and the first 10 orders of Zernike mode coefficients is deduced. Comparative numerical results show that the convergence speed and the correction performance of the AO system based on optimizing Zernike mode coefficients is far better than that of optimizing voltages. Moreover, experimental results showed that this AO system could change TEM_{10}, TEM_{11}, and TEM_{20} transverse modes into a TEM_{00} mode successfully.

© 2007 Optical Society of America

## 1. Introduction

In recent years, solid-state lasers that can generate a high-quality fundamental-mode (TEM_{00}) output are indispensable in more and more scientific applications such as laser communication, high-precision laser machining, and so on. Therefore, there is an increasing demand for solid-state lasers to generate a TEM_{00} mode. Unfortunately, since both thermal lenses and thermally induced birefringence are the main thermal effects in solid-state laser resonators and can destroy the output laser beam quality greatly, the thermal effects must be first compensated for in order to obtain the fundamental mode output. Thermally induced birefringence may be eliminated successfully through the careful selection of a natural birefringent crystal. The spherical component of the thermal lens can be eliminated also by designing the resonator cavity well [1], whereas nonspherical aberrations cannot be compensated for in the same way [2]. Generally, one of the most conventional ways to make a solid-state laser generate a high-beam quality TEM_{00} mode output is to place a pinhole in its resonators. However, this way will abate the output power greatly and may cause the resonator to be misaligned once the pumping conditions change. Since thermal effects in the resonators change with varying pumping conditions [3], a promising way for compensating the thermally induced phase aberrations and thermal lenses is to adopt adaptive methods. It is known that the adaptive optics (AO) technique is a powerful technique that allows dynamic correction of phase aberrations. Although it was initially developed for astronomy, it can also be used in the field of lasers [4]. A 37-element membrane deformable mirror (DM) has been used successfully intracavity to optimize the laser modes [1]. However, the applied DM is not used as the rear intracavity mirror of the resonator. In this paper, we investigate the possibility of using a 19-element piezoelectric DM as the rear mirror of a laser resonator in conjunction with a genetic algorithm (GA) to control the beam mode of a diode-laser-pumped Nd:YAG solid laser [5]. The voltages applied on the 19 actuators of the DM are in the range of ±300 V and can vary with a step of 1 V; therefore, the possible number of DM surface shapes on the basis of voltage can reach 600^{19} [6]. Since the GA is used to find the optimum shape for optimizing the laser mode intracavity in such a vast DM surface shapes space, and thus to speed the system convergence, one promising approach is to decrease the search space. We take the first 10 orders of Zernike mode coefficients on actuators as the object function to optimize rather than to optimize voltages. The transform matrix between voltages and Zernike mode coefficients has been deduced, and both simulative and experimental results are presented and discussed. The paper is organized as follows. In Section 2 the matrix between Zernike mode coefficients and voltages on the actuators is established; in addition, the correction capability of the 19-element DM is demonstrated through numerical simulation. Section 3 briefly introduces the principle of the GA based on mode coefficients. Section 4 demonstrates the comparative results between the GA based on mode coefficients and based on voltages through numerical simulations. Section 5 describes the experimental setup of mode control system on a Nd:YAG solid-state laser as well as shows the mode optimization results.

## 2. The transform matrix between Zernike mode coefficients and voltages

Figure 1(a) shows the schematic configuration of the 19-element piezoelectric DM, and Fig. 1(b) corresponds to its photograph. The DM that was fabricated in our laboratory has a continuous faceplate with stacked PZT actuators [7] with an effective diameter of 32 mm, maximum deflection ±2 µ m [3], maximum voltage ±300 V, nonlinearity and hysteresis <±4%, and resonance frequency >10kHz. According to its principle, the DM can deform its surface shape by applying voltages on the actuators:

where v_{j} is the voltage applied on the jth actuator, V_{j}(x,y) is the influence function of the jth actuator on the wave-front, and n is the number of actuator.

The influence function of the DM can be written in following form:

where w is the coupling coefficient of DM and set to 0.082; (x_{j}, y_{j}) is the space position of the ith actuator; p is set to 2 and d is the distance between every two neighboring actuators; and x and y represent the value in the x-coordinate and the y-coordinate, respectively, of the orthogonal coordinate plane.

It is also known that any given wavefront in a unit circle can be described by a series of the Zernike polynomials:

where *b*0 is the piston coefficient; *Z _{k}*(

*x,y*) and

*b*(k=1, 2… m) are the kth Zernike polynomials and their corresponding coefficients, respectively; and m is the Zernike order.

_{k}Consider that whether optimizing the voltages or the Zernike mode coefficients by GA, the ultimate aim is to obtain the needed DM surface shapes. Moreover, as to our DM, voltages are the key parameters that affect its surface shape directly, and therefore we have to change the Zernike mode coefficients into voltages after the optimization sequence. As a result, the relationship between the mode coefficients and the voltages should be ascertained. According to Eq. (1) and Eq. (3), we can obtain:

In most applications the piston coefficient is often omitted, and then Eq. (4) can be rewritten as:

Now our object is to build the transform matrix between the Zernike coefficients and the actuator voltages, which can be calculated according to:

Equation (6) can be rewritten:

In order to deduce the proper transform matrix, *U*, we investigate the correction capability of the 19-element DM through simulation first.

Figure 2 indicates the correct capability of this DM, where the red line represents the RMS value of the original wavefront, the green curve stands for the RMS value of the wavefront generated by the DM (which is used to correct the original wavefronts that correspond to each Zernike order), and the blue curve represents the RMS value of the residual wavefront. These results are obtained through 35 simulative correction calculations. What we know from Fig. 2 is that the DM employed is very suitable for correcting the first 10 orders of Zernike aberrations, although it can also correct the 11^{th} to the 14^{th} and the 18^{th} to 20^{th} orders of Zernike aberrations, to some extent. As a result, choosing the first 10 orders of Zernike polynomial coefficients as the basis to deduce the matrix *U* is suitable.

Limited by the DM deflection range (±2µ m), we set the value of each Zernike coefficient to 0.1. The simulative target grid is set to 100×100, and then the size of each influence function matrix (j=1, 2…19) is also 100×100. For calculation convenience, we transform the form of 19 influence function matrices into a 10000 x 19 matrix and use V (x,y) to describe it. We use the first 10 orders of Zernike polynomials and their corresponding mode coefficients (each is set to 0.1) in sequence to produce a surface shape matrix *ψ*(*x,y*) with a size of 10000×1, and *u* represents the voltage vector. According to Eq. (3) and Eq. (5) we can obtain:

System stability and the capability of resistance to interference is evaluated by the condition number of matrix V (x,y) :

where δ_{max} and δ _{min} are the maximum and minimum singular values, respectively. In fact, the condition number is the measurement of the ill-condition of the matrix. Traditionally, this value should be smaller than 20. Since the condition number is 4.47 calculated by Eq. (9), the capability of V (x,y) is very good. Once *ψ* (*x,y*) and V (x,y) are ascertained, the transform vector between the Zernike mode coefficients and the actuator voltages can be calculated:

where V (x,y)^{+} is the generalized inverse matrix of V (x,y). Finally, the transform matrix *U* between the first 10 orders of Zernike mode coefficients and 19 actuator voltages on the DM can be obtained by calculating Eq. (10) ten times:

where *u _{i}* (i=1, 2…10)is a 19×1 vector.

When *U* has been set up, the GA will optimize the first 10 orders of Zernike mode coefficients rather than the 19 voltages on the actuators. It should be noticed that since each coefficient is set to 0.1, the voltages calculated by (6) can not be applied directly on actuators before they are multiplied by a factor of 10.

In order to evaluate the effectiveness of the transform matrix, we use the first 10 orders of Zernike polynomial coefficients first to produce a wavefront, and then we use the transform matrix to figure out 19 voltages that will drive the DM to produce another wavefront. To obtain the residual wavefront, we compare the similarity of the two wavefronts and then subtract the Zernike-generating wavefront from the DM-generating wavefront at their corresponding positions.

What we know from Fig. 3 is that, although there is little difference in the peak-to-valley (PV) between the Zernike-generating wavefront and that of DM-generating wavefront, their RMS values approach each other, and their shapes are also similar. Because the RMS value is conventionally regarded as one of the most important parameters with which to evaluate the whole surface quality of a wavefront in applications, we think the results are able to testify to the availability of the transform matrix because the RMS of the residual wavefront is small; as a result, the transform matrix can be used to describe the relationship of the first 10 orders of Zernike mode coefficients and 19 voltages.

## 3. The principle of the GA based on coefficients

A GA is a stochastic parallel algorithm based on natural selection and biological genetics. Using one parameter as the object function to optimize, it adopts the concept of survival of the fittest in evolution to find the best solution to some multivariable problem [8]. As for our DM, each actuator voltage represents one independently adjustable variable. We use a Charge Coupled Device (CCD) camera on the focal plane to record the far-field light spot intensity within a selected aperture, and then the intensity is used as the fitness parameter of the GA. The GA first generates a population of individuals represented by chromosomes. In the GA adopted in this paper, each individual corresponds to a DM surface shape, which is also a possible solution of the system. In order to reduce the search space, each individual has 10 chromosomes (Zernike polynomial coefficients) rather than 19 voltages on the actuators. The fitness parameter of our GA is also regarded as the object function to optimize. After the initial population is created according to the roulette selection principle, we select excellent individuals from the population and then create new individuals by randomly crossing the chromosomes of the old individuals with a probability of P_{C}. Finally, some chromosome positions of individuals are mutated randomly with a mutation rate of P_{m} for introducing a new individual and avoiding the convergence to a local maximum. By going through this process, the GA will gradually find the optimum mirror shape that can yield the best fitness parameter [9–11]. The flowchart of the GA is illustrated by Fig. 4.

## 4. Comparative simulations

To compare the performance of our AO system based on optimizing voltages and the Zernike mode coefficients, we have accomplished some comparative numerical simulations. On two bases, the initial populations both consist of 20 individuals (DM surface shape). The far-field light spots are obtained through executing the Fast Fourier Transformation (FFT) algorithm. The initial phase aberrations of both cases are generated randomly by a group of Zernike polynomial coefficients. For briefness and convenience, we just use one example to illustrate the comparative effects.

Figure 5 shows the comparative fitness function curves; the y-coordinate represents the fitness value, which is normalized to 1, whereas the x-coordinate shows the number of iterative generations. What we know is that the GA based on optimizing the first 10 orders of Zernike mode coefficients has nearly converged after 200 generations, whereas the GA based on voltages is far from reaching convergence. Furthermore, after about 20 iterative generations, the fitness value of the GA based on coefficients can reach 0.5, which will take the GA based on voltages about 200 generations to reach.

Figure 6(I) is the near-field wavefront before and after the phase aberrations are corrected by DM on the basis of optimizing the voltages. Fig. 6(I)A and Fig. 6(I)B are the planar and three-dimensional wavefronts, respectively, before phase aberrations are corrected, whereas Fig. 6(I)C and Fig. 6(I)D correspond to the corrected cases. We know that the PV value and the RMS value of the wavefront are reduced to 1.45λ (λ=1064nm) and 0.30λ from 3.38λ and 0.64λ, respectively, even if after 400 times of iterative calculation. Thus, we cannot say the correction effect is promising. Similarly, Fig. 6(II) is the near-field wavefront before and after the phase aberrations are corrected by DM on the basis of optimizing the Zernike mode coefficients. Fig. 6(II) is the plot according to the optimizing case of the mode coefficients. Fig. 6(II)A and Fig. 6(II)B show the planar and three-dimensional wavefronts, respectively, before phase aberrations are corrected, whereas Fig. 6(II)C and Fig. 6(II)D correspond the corrected cases. Fig. 6(II) samples are corrugated to some extent after correction. There are two possible reasons to explain this: one is owing to the reduced basis, and another is owing to the deformable mirror (DM) that was employed. There is probably no DM that can generate a completely precise surface shape needed to compensate for the phase aberrations generated and represented by Zernike polynomials. However, we know that the PV value and RMS value are reduced to 0.52λ and 0.06λ from 3.38λ and 0.64λ, respectively, after 200 generations of iterative calculations. Therefore, the correction ability based on optimizing the Zernike polynomial coefficients is powerful and the wavefront improvement is obvious.

Figure 7(I) and Fig. 7(II) are the far-field focal spot distributions for optimizing the voltages and mode coefficients, respectively. Fig. 7(I) demonstrates that most of the energy of the far-field focal spot converges to the center after phase aberrations are corrected, and the Strehl ratio is raised from 0.06 to 0.64. Compared with Fig. 7(I), after correction, more energy is converged towards the spot center in Fig. 7(II), and the Strehl ratio is increased from 0.06 to 0.95.

## 5. Mode control experiments

We have known that most of the phase aberrations in the Nd:YAG solid-state lasers are low-order aberrations (such as defocus, astigmatism, and coma) and can be described well by the first 10 orders of the Zernike polynomial [3]. Consequently, a GA based on optimizing the first 10 orders of Zernike mode coefficients is suitable for application in solid-state lasers.

To investigate the availability of the 19-element DM as the rear mirror to optimize the Nd:YAG solid laser, a Nd:YAG laser resonator in Fig. 8 was configured. A 5x telescope was used intracavity to expand the laser beam (6.2 mm) to about 30 mm in diameter so that the beam could match the DM aperture (32 mm) and cover as many actuators as possible. In this way the resolution of the beam/mirror interaction could be enhanced and allow a greater degree of optimization [1]. The active medium was an 85 mm-long Nd:YAG rod with a diameter of 7 mm. The Nd^{3+} doping concentration of the crystal was 0.8%, and the rod was surrounded by an antireflection-coated cooling sleeve. The largest repetition rates, the pumping currents of the pump head, were 100 Hz and 70 A, respectively. A 70% reflective mirror was used as the output coupler (OC), and the output power could be adjusted from 0 to 50 W. After being attenuated by an attenuator, the output beam first was reflected by a beam splitter (BS1) and then passed through a 1064 nm narrow filter before it was focused by a lens onto an infrared CCD camera. The intensity information of the focus light spot was acquired with a frequency of 25 Hz by a frame grabber. Using one part of the intensity information as the object function to maximize (within a selected center area), the industrial computer calculated the 10-element Zernike mode coefficients, which finally could be transformed into 19 voltages by the transform matrix *U* obtained in Section 2. Finally, these voltages were amplified by a high-voltage amplifier (HVA) before being applied to DM actuators. A monitor that was placed behind another beam splitter (BS2) was used to watch the laser mode profile. A power meter was also employed to detect the output laser power in real time.

We first tested the optimization performance of the solid-state laser at a pump current of 40 A. Before optimization, the output power was 5.6 W; the initial transverse mode shown in Fig. 9(a) was a TEM_{10} mode. During the course of optimization, we found that the TEM_{10} mode converged toward a fundamental TEM_{00} mode, shown in Fig. 9(b), after about 1 minute. This phenomenon may be explained as follows: as the DM changed its surface shape in the course of optimization, its curvature radius also changed, which resulted in a change of the resonator configuration and thereby established conditions for generating the TEM_{00} mode more efficiently and the suppression of the higher order modes, to a great extent. The output power was reduced to 5.3 W after mode optimization was accomplished. A simple video of the TEM_{10} optimization procedure is displayed in Fig. 10.

We gradually increased the pump current and simultaneously observed the output laser mode. It was found that the beam mode structure became more and more complex as the pumping current increased. Experimental results showed that it was impossible for the DM to select the TEM_{00} mode successfully when the mode structure became too complex. In order to control the mode structure efficiently in a relatively high-pumping current status, a variable-size pinhole was placed near to the OC of the resonator to restrict the complex high-order modes roughly.

Figure 11 shows the far-field beam profiles from the Nd:YAG laser recorded at various intervals during an optimization sequence when the pump current was 50 A with a 3 mm pinhole in the resonator. In this case, the aperture of the laser beam on the DM was restricted to about 15 mm; however, experimental results demonstrated that it was still effective to optimize the mode in spite of only the central part of the DM being covered by the laser. Figure 11 shows that the TEM_{20} mode distribution was changed into a TEM_{00} mode after about 85 seconds. The relative power in the selective region of the CCD camera was increased from 1 to 5.6. The selective region could be altered by the program built into the computer; in this case, the size of the selected region was as large as one diffraction limit. As a consequence of optimization, the output power was increased from 3.2 W to 4.7 W.

Figure 12 shows the far-field beam profiles from the Nd:YAG laser, which were recorded at various intervals during another optimization sequence, where the pump current was 59 A with a 2.5 mm pinhole in the resonator. Figure 12 shows that the TEM_{11} mode distribution was also changed to the TEM_{00} mode after optimization was finished, and the relative power in the selective region of the CCD camera was increased from 1 to 5.2. In this case, the size of the selected region was also set as large as one diffraction limit. The output power was brought down from 7.2 W to 5.8 W during this course. It took about 140 seconds to finish the optimization. A video about the TEM_{11} mode optimization performance before and after the AO system is on is given in Fig. 13.

## 6. Conclusions

An intracavity transverse mode for controlling AO systems has been presented. A 19-element piezoelectric DM is introduced as the rear intracavity mirror of a Nd:YAG solid-state laser to control the output laser modes. A GA is adopted to optimize the first 10 orders of Zernike mode coefficients rather than the actuator voltages on the DM. The transform matrix between the first 10 orders of Zernike mode coefficients and the 19 voltages on the actuators has been established. Numerous simulations show that the performance of the system based on optimizing the Zernike coefficients is much better than that of optimizing the voltages. It has also been demonstrated experimentally that the capability of our AO system for intracavity mode control is very promising. It can change TEM_{10}, TEM_{20}, and TEM_{11} modes into TEM_{00} mode within 1 to 3 minutes, based on an industrial computer with a 400 MB CPU and 256 MB RAM. We believe that the speed of our system greatly depends on the industrial computer, and if the speed of the industrial computer is increased, the time of mode optimization procedure can be reduced to only a few seconds.

## Acknowledgments

We express our thanks for the support of the National High Technology Project of China. We also acknowledge Professor Xiao Yuan for his generous advice and help.

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