## Abstract

A simple but effective decoupling control algorithm based on Zernike mode decomposition for adaptive optics systems with dual deformable mirrors is proposed. One of the two deformable mirrors is characterized with a large stroke (woofer) and the other with high spatial resolutions (tweeter). The algorithm works as follows: wavefront gradient vector is decoupled using the Zernike modes at first, and then the control vector for the woofer is generated with low order Zernike coefficients to eliminate high order modes. At the same time the control vector for the tweeter is reset by a constraint matrix in order to avoid coupling error accumulation. Simulation indicates the algorithm could get better performance compared with traditional Zernike mode decomposition control algorithms. Experiments demonstrate that this algorithm can effectively compensate for phase distortions and significantly suppress the coupling between the woofer and tweeter.

© 2013 Optical Society of America

## 1. Introduction

Wavefront correctors are key components in adaptive optics systems, and the performance of adaptive optics systems are strongly related to them. In many applications [1, 2], both large strokes and high spatial resolutions are demanded, which could not be practically fulfilled by a single wavefront corrector. During the past decades, researchers have noticed this problem, and dual deformable mirrors systems are proposed as a solution, which generally employ two deformable mirrors: one with high stroke and low spatial resolution (woofer), the other with low stroke and high spatial resolutions (tweeter). A typical dual deformable mirrors adaptive optics system is shown in Fig. 1 [1, 3]. Control algorithms are critical for such systems. As reconstruction algorithms and detection errors always induce coupling between wavefront correctors, if the coupling error is not well suppressed, the deformable mirrors would generate opposite surface shapes and gradually become saturated over time [4], leading to severe waste of strokes, and the predicted high correction qualities would never be achieved.

Several control algorithms for dual deformable mirrors systems are proposed and proved effective, including the 2-step control method [2,5,6], Zernike mode decomposition [1,7,8], Fourier mode decomposition [9], wavelet mode decomposition [10], distributed mode decomposition [11], zonal reconstruction [12–15], and so on. Because of Zernike modes are widely used for aberration descriptions in laser beam clean-up, retina imaging and astronomy, decoupling control methods based on Zernike mode decompositions are very suitable for these applications.

However, traditional Zernike mode decomposition decoupling is based on separating modal coefficient correction or confinement correction [1], the separating modal coefficient correction algorithm could not fully make use of the correction abilities of the deformable mirror (DM), and the confinement correction algorithm are difficult to eliminate the coupling accumulation.

In this paper, a simple but effective algorithm with global constraint-based strategy is presented. Instead of focusing on decoupling the gradients of the residual wavefront, we concentrate on constraining the control vectors of both woofer and tweeter. In this algorithm, the control vector of woofer is generated with low order mode coefficients to eliminate the higher order mode accumulations. At the same time, the control vector of tweeter is reset by a constraint matrix to avoid low order mode accumulation. With this method, coupling errors could be effectively removed. As a result, the strokes and spatial resolutions of double deformable mirrors systems could be sufficiently utilized. To prove the effectiveness of our algorithm, direction cosine between the phase compensation of the woofer and tweeter is used as coupling coefficient to quantify the coupling. Then a numerical model is set up to compare the control algorithm proposed in this paper with these traditional Zernike mode decomposition control algorithms. At last an experiment system is used to validate the performance of our algorithm.

## 2. Principle of the Zernike decomposition decoupling control algorithm

In our algorithm, the dual deformable mirrors adaptive optics system employs a Shack-Hartmann wavefront sensor to measure the residual wavefront errors. The slope of wavefront *g* could be written as:

*Z*is the slope response matrix from Zernike modes to the Shack-Hartmann wavefront sensor, and

*a*is the coefficient vector of the Zernike modes.

*a*could be calculated as:where

*Z*is the pseudo-inverse matrix of

^{+}*Z*. The woofer only corrects the lower order modes, and the remaining modes are assigned to the tweeter. Then the lower order mode coefficients

*a*could be written as:where

_{w}*I*is a p × p diagonal matrix, p is the number of Zernike modes used for reconstruction. If the woofer is assigned to correct the i

_{w}^{th}mode,

*I*is one, and the other elements of

_{w}(i,i)*I*are zeros. For example, if the woofer is used to correct defocus and astigmatism,

_{w}*I*= diag(0 0 1 1 1 0 … 0).

_{w}The low order mode coefficients are regarded as intermediate control vector, so the total low order mode coefficient vector *A _{w}* could be generated using the well-know digital PI controller as:

*pid_a*and

*pid_b*are the coefficients manually tuned for specific adaptive optics systems. Then we can use the transition matrix

*T*from mode coefficient vector to the control vector of woofer to get the control vector

*V*:and the transition matrix

_{w}*T*could be deducted as follow: here

*R*is the pseudo-inverse of the woofer’s response matrix

_{w}^{+}*R*.

_{w}*v*is the control vector of the woofer to correct the low order mode parts of the residual wavefront.

_{w}The tweeter is controlled in another way: the slope vector of higher order aberrations could be solved as:

where*I*is a m × m identity matrix, and m is the dimension of the slope vector

*g*, thus the control vector for correcting the higher order parts of the residual wavefront ishere

*R*is the pseudo-inverse of the tweeter’s response matrix

_{t}^{+}*R*. Then the control vector

_{t}*V*is generated as:

_{t}’*V*consists of two parts, one is

_{t}’*V*which would make tweeter correct high order modes, the other is

_{t}*∆V*which make tweeter correct low order modes inducing coupling.

*V*is generated by the high order mode slope, so

_{t}’*∆V*is often far smaller than

*V*. Then we have:where

_{t}*I*is a n × n identity matrix, and n is the number of the tweeter’s actuators. As described in Ref. 1, we can get,

*R*is the j

_{m}(i,j)^{th}order Zernike mode coefficient of the i

^{th}woofer’s influence function

*V*,

_{i}(x,y)*k*is an empirical parameter. A larger

*k*will enhance the elimination of low order components in the surface shape of the tweeter, but the correction qualities of higher order phase aberrations will be decreased. Thus

*k*should be carefully chosen to make a better compromise.

*Z*is the Zernike modes assigned to be corrected by the woofer.

_{j}Combing Eqs. (12) and (13), the control vector of the tweeter could be written as:

*V*could be calculated from

_{t}*V*using Eq. (17), and low order Zernike modes would be confined in the resultant surface shape.

_{t}’After obtaining *V _{w}* and

*V*, decoupling control between woofer and tweeter is realized. Different from the traditional Zernike mode decomposition algorithms concentrating on decoupling the gradients of the residual wavefront,

_{t}*V*is constrained by the matrix

_{w}*T*which converts mode coefficient vectors to the control vectors of woofer, and

*V*is reset by the constraint matrix

_{t}*C*in each control iteration, so the coupling error is suppressed and the problem of coupling accumulation could be well solved.

## 3. Numerical simulation

An adaptive optics system with dual deformable mirrors is simulated to compare the algorithm with the traditional Zernike mode decomposition algorithms [1]. In the numerical model, the woofer is a DM with 19-actuators which is used to correct the defocus, and the tweeter is a DM with 61-actuators which is used to correct the other aberrations, the influence function both of woofer and tweeter is described as a Gaussian function:

*(x*is the position of the i

_{i}, y_{i})^{th}actuator, α is the Gaussian coefficient, and

*d*is the distance between every two neighboring actuators. We set ω to 0.1 and α to 2.35 in the numerical model.

The configurations of the Hartmann-Shack wavefront sensor’s sub-apertures and the DM’s actuators are shown in Fig. 2. The aberrations of the input wavefront are made up of the first 35 Zernike modes in this simulation, as is shown in Fig. 3. The tip/tilt are removed completely in the simulation because of they are often corrected by the tip-tilt mirror and the woofer and tweeter could suppress the tip/tilt by Eqs. (5) and (17).

A comparison between the algorithm proposed in this paper and traditional Zernike mode decomposition algorithms is made by adding the same phase aberrations, and the PI controller parameters as Eq. (4) are 0.998 and 0.1. The numerical system is controlled by one of the following algorithms: (a) the decoupling algorithm proposed in this paper, (b) separating modal coefficient correction algorithm [1], (c) confinement correction algorithm [1]. And to qualify the coupling, the coupling coefficient is defined as follow:

*S*is the correction of woofer and

_{w}*S*is the correction of tweeter. In mathematics, the coefficient

_{t}*r*is the direction cosine between

*S*and

_{w}*S*, when it is zero that means the

_{t}*S*and

_{w}*S*are orthogonal with each other, and when it equal to one that means the S

_{t}_{w}and S

_{t}are linearly dependent. So the smaller

*r*indicates the smaller coupling between phase compensation of woofer and tweeter.

Figure 4(a) shows the RMS of the residual wavefront error reduction curves during the correction. After correction by algorithm (a) and (c), the RMS of the residual wavefront both decreased to 0.05λ, and when the algorithm (b) is used to correct, the RMS of the residual wavefront decreased to 0.11λ. It means that the algorithm (a) and (c) get almost the same correction performance, and the algorithm (b) could not efficiently use the correction ability of the dual deformable mirrors system. Figure 4(b) shows the coupling coefficient during corrections. When then algorithm (a) be used, the coefficient is less than 0.01, and when the algorithm (b) be used, the coefficient is about 0.04, both of them could be keep stable when the correction has converged. But when algorithm (c) is used, the coefficient would increase during the correction, it means the coupling could not be well suppressed. The simulation indicates that the algorithm proposed in this paper could get the best correction performance and least coupling compared with the traditional Zernike mode decomposition algorithms.

## 4. Experiment

An experimental adaptive optics system with dual deformable mirrors as Fig. 5 is built to evaluate the effectiveness of the algorithm. In this experiment system, the woofer is a bimorph DM with 19-actuators, and the tweeter is a bimorph DM with 37-actuators. Figures 6(a) and 6(b) show the actuator geometry and the clear aperture of the DMs. Besides, as this algorithm does not set any fundamental limitations on actuator numbers, it could be easily transferred to a dual-DM system with more actuators.

Three experiments are carried out to test the effectiveness of our algorithm. In first experiment, the woofer is used to correct defocus (the third Zernike mode), in the second experiment the woofer is used correct 0° astigmatism (the 4th Zernike mode), and in the third experiment, the woofer is used correct defocus, 0° astigmatism and 45° astigmatism (the third, 4th and 5th Zernike mode). The tweeter is used to correct the other aberrations. In each experiment, the aberrations are corrected by three methods, one is woofer only worked with our algorithm to correct defocus or astigmatisms, the other is tweeter only worked to correct aberrations except which is corrected by the woofer, and the last one is dual deformable mirrors both worked.

#### 4.1 Woofer is used to correct defocus

The initial aberration is shown in Fig. 8(a). Before correction, RMS of the wavefront is 0.665λ (λ = 650nm). The system parameter *pid_a* is 0.998, *pid_b* is 0.1, and *k* is 10000. In the experiment, after corrected only by the woofer to compensate defocus, RMS of the residual wavefront decreased to 0.373λ, as is shown in Fig. 8(b). After corrected only by the tweeter to compensate the aberrations except defocus, RMS of the residual wavefront decreased to 0.568λ, as is shown in Fig. 8(c). At last, the dual deformable mirrors both joined the correction controlled by the proposed decoupling algorithm, and RMS of the residual wavefront is further lowered to 0.021λ, as is shown in Fig. 8(d). Figure 7(a) shows the RMS of the residual wavefront error reduction curves during the correction, and Fig. 7(b) illustrates the composition of the wavefront errors quantified by Zernike mode coefficients. From Fig. 7(b), it is easy to find that the aberrations are split exactly to defocus and high order parts. The woofer almost deals with defocus only, while the other parts of aberration are mainly corrected by the tweeter as expected.

#### 4.2 Woofer is used to correct 0° astigmatism

Before correction, RMS of the wavefront is 0.672λ, as is shown in Fig. 10(a). The system parameter *pid_a* is 0.998, *pid_b* is 0.1, and *k* is 10000. In the experiment, after corrected only by the woofer to compensate 0° astigmatism, RMS of the residual wavefront decreased to 0.618λ, as is shown in Fig. 10(b). After corrected only by the tweeter to compensate the aberrations except 0° astigmatism, RMS of the residual wavefront decreased to 0.305λ, as is shown in Fig. 10(c). At last, the dual deformable mirrors both joined the correction controlled by the proposed decoupling algorithm, and RMS of the residual wavefront is further lowered to 0.019λ, as is shown in Fig. 10(d). Figure 9(a) shows the RMS of the residual wavefront error reduction curves during the correction, and Fig. 9(b) illustrates the composition of the wavefront errors quantified by Zernike mode coefficients. From Fig. 9(b), it is easy to find that the aberrations are split exactly to 0° astigmatism and other parts. The woofer almost deals with 0° astigmatism only, while the other parts of aberration are mainly corrected by the tweeter as expected.

#### 4.3 Woofer is used to correct defocus and astigmatisms

Before correction, RMS of the wavefront is 0.673λ, as is shown in Fig. 12(a). The system parameter *pid_a* is 0.998, *pid_b* is 0.1, and *k* is 20000. In the experiment, after corrected only by the woofer to compensate defocus and astigmatisms, RMS of the residual wavefront decreased to 0.665λ, as is shown in Fig. 12(b). After corrected only by the tweeter to compensate the aberrations except defocus and astigmatisms, RMS of the residual wavefront decreased to 0.120λ, as is shown in Fig. 12(c). At last, the dual deformable mirrors both joined the correction controlled by the proposed decoupling algorithm, and RMS of the residual wavefront is further lowered to 0.031λ, as is shown in Fig. 12(d). Figure 11(a) shows the RMS of the residual wavefront error reduction curves during the correction, and Fig. 11(b) illustrates the composition of the wavefront errors quantified by Zernike mode coefficients. From Fig. 7(b), it is easy to find that the aberrations are split exactly to defocus, astigmatisms and other parts. The woofer almost deals with defocus and astigmatisms only, while the other parts of aberration are mainly corrected by the tweeter as expected.

#### 4.4 Coupling test

In the dual deformable mirrors system, if the coupling error accumulation could not be efficiently suppressed, the two deformable mirrors would start to oppose each other and become saturated over time. To evaluate the effectiveness of suppressing coupling error accumulation of the algorithm, in each of the above experiments, the woofer-tweeter system ran about 2000 seconds, and 5 samples per second were recorded to allocate the less storage. The phase compensation of woofer and tweeter are calculated by their voltage data, and then the coupling coefficient could be generated by Eq. (19). Figure 13 shows the coupling coefficient during 2000 seconds. It is clear that the coupling coefficient is always smaller than 0.01 and almost stable, it indicates that the algorithm presented in this paper could suppress the coupling error efficiently.

## 5. Conclusion

In summary, we have proposed and discussed a simple but effective algorithm based on Zernike mode decomposition to control dual deformable mirrors adaptive optics system. This algorithm could be applied to dual deformable mirrors with different spatial resolutions, and could also eliminate the coupling error accumulation between them. Simulation result means that the algorithm could get better performance than traditional Zernike mode decomposition control algorithms. A woofer-tweeter AO system is built to prove the effectiveness of the algorithm. The experimental results show that the dual deformable mirrors system controlled by the proposed algorithm could combine the advantages of woofer and tweeter for aberration correction. The effectiveness of suppressing coupling of the two deformable mirrors is also confirmed by the experiments, thus longtime stable operations could be conveniently achieved. This algorithm could be used in the laser beam clean-up, large telescope and other situations for corrections of large-scale and high-spatial resolution aberrations.

## Acknowledgments

This work was supported by the Preeminent Youth Fund of Sichuan Province under grant 2012JQ0012.

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