An innovative and efficient method to control the shape of push-pull membrane deformable mirror

We carry out performance characterisation of a commercial push and pull deformable mirror with 48 actuators (Adaptica Srl). We present a detailed description of the system as well as a statistical approach on the identification of the mirror influence function. A new efficient control algorithm to induce the desired wavefront shape is also developed and comparison with other control algorithms present in literature has been made to prove the efficiency of the new approach. © 2012 Optical Society of America OCIS codes: (110.1080) Active or adaptive optics; (230.3990) Micro-optical devices; (220.1000) Aberration compensation; (350.4600) Optical engineering; (150.5495) Process monitoring and control. References and links 1. N. Hubin and L. Noethe, “Active optics, adaptive optics, and laser guide stars,” Science 262, 1390–1394 (1993). 2. M. Rueckel, J. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing.” Proc. Nat. Acad. Sci. 103, 17137–17142 (2006). 3. E. Rossi, M. Chung, J. J. Hunter, W. H. Merigan, and D. R. Williams, “Imaging retinal mosaics in the living eye.” Eye (London, England) 25, 301–308 (2011). 4. R. K. Tyson,Adaptive Optics Engineering Handbook (CRC Press, 1999). 5. N. Ishii, T. Muroi, N. Kinoshita, K. Kamijo, and N. Shimidzu, “Wavefront compensation method using novel index in holographic data storage,” J. Europ. Opt. Soc. Rap. Public. 5, 10036 (2010). 6. A. Polo, V. Kutchoukov, F. Bociort, S. F. Pereira, and H. P. Urbach, “Determination of wavefront structure for a Hartmann wavefront sensor using a phase-retrieval method,” Opt. Express 20, 237–246 (2012). 7. F. Staals, A. Andryzhyieuskaya, H. Bakker, M. Beems, J. Finders, T. Hollink, J. Mulkens, A. Nachtwein, R. Willekers, P. Engblom, T. Gruner, and Y. Zhang, “Advanced wavefront engineering for improved imaging and overlay applications on a 1.35 NA immersion scanner,” Proc. SPIE 7973, 79731G–13 (2011). 8. G. Vdovin and P. M. Sarro, “Flexible mirror micromachined in silicon.” Appl. Opt. 34, 2968–2972 (1995). 9. E. Steinhaus and S. G. Lipson, “Bimorph piezoelectric flexible mirror,” J. Opt. Soc. Am. 69, 478–481 (1979). 10. J. Garcı́a-Márquez, J. E. A. Landgrave, N. Alcalá-Ochoa, and C. Pérez-Santos, “Recursive wavefront aberration correction method for LCoS spatial light modulators,” Opt. Lasers Eng. 49, 743–748 (2011). 11. G. D. Love, “Wave-front correction and production of Zernike modes with a liquid-crystal spatial light modulator,” Appl. Opt.36, 1517–1520 (1997). 12. S. Bonora and L. Poletto, “Push-pull membrane mirrors for adaptive optics.” Opt. Express 14, 11935–11944 (2006). 13. E. Fernandez and P. Artal, “Membrane deformable mirror for adaptive optics: performance limits in visual optics.” Opt. Express11, 1056–1069 (2003). 14. L. Zhu, P. C. Sun, D. U. Bartsch, W. R. Freeman, and Y. Fainman, “Wave-front generation of Zernike polynomial modes with a micromachined membrane deformable mirror.” Appl. Opt. 38, 6019–6026 (1999). 15. E. S. Claflin and N. Bareket, “Configuring an electrostatic membrane mirror by least-squares fitting with analytically derived influence functions,” J. Opt. Soc. Am. A 3, 1833–1839 (1986). #175322 $15.00 USD Received 6 Sep 2012; revised 4 Oct 2012; accepted 4 Oct 2012; published 30 Nov 2012 (C) 2012 OSA 3 December 2012 / Vol. 20, No. 25 / OPTICS EXPRESS 27922 16. Adaptica Srl, “Saturn user manual,” http://www.adaptica.com/site/en/pages/saturn. 17. C. Paterson, I. Munro, and J. C. Dainty, “A low cost adaptive optics system using a membrane mirror,” Opt. Express6, 175–185 (2000). 18. L. Jolissaint, “Synthetic modeling of astronomical closed loop adaptive optics,” J. Europ. Opt. Soc. Rap. Public. 5, 10055 (2010). 19. D. Malacara and W. T. Welford, Optical Shop Testing(John Wiley Sons, Inc., 2006). 20. A. Haber, R. Fraanje, and M. Verhaegen, “Linear computational complexity robust ilc for lifted systems,” Automatica48, 1102–1110 (2012). 21. D. A. Bristow, M. Tharayil, and A. G. Alleyne, “A survey of iterative learning control,” IEEE Control Systems 26, 96–114 (2006). 22. M. Verhaegen and V. Verdult,, Filtering and System Identification: A Least Square Approach (Cambridge University Press, New York, NY, USA, 2007). 23. I. G. Hughes and T. P. A. Hase, Measurements and Their Uncertainties: A Practical Guide to Modern Error Analysis(Oxford University Press, New York, NY, USA, 2011). 24. F. Roddier,Adaptive Optics in Astronomy (Cambridge University Press, 1999). 25. G. Brusa-Zappellini, A. Riccardi, V. Biliotti, C. Del Vecchio, P. Salinari, P. Stefanini, P. Mantegazza, R. Biasi, M. Andrighettoni, C. Franchini, and D. Gallieni, “Adaptive secondary mirror for the 6.5-m conversion of the multiple mirror telescope: first laboratory testing results,” Proc. SPIE 3726, 38–49 (1999).


Introduction
Reaching a diffraction-limited image in an optical system can be done by means of adaptive optics (AO) techniques.A typical AO systems consists of a wavefront sensor, a deformable mirror (or some other active optical element) and a controller [1].For many years devices capable to control and modify the wavefront shape have been developed especially in the astronomic and military field.Recently, due to the cost reduction of these devices, wavefront correction systems have widened their application to other research areas such as microscopy [2], ophthalmology [3], optical communication [4,5] and even lithography steppers [6,7].One of the reasons of the cost reduction is the development of new tools that are able to modify the wavefront of an optical wave, such as electrostatic membrane mirror [8], bimorph piezoelectric mirror [9] and spatial light modulator (SLM) [10,11].In AO systems, the most used wavefront correction device is still a deformable mirror (DM).Compared to SLM, DM offers more advantages: efficient reflection, achromaticism and large dynamics.Among the DMs, membrane electrostatic mirrors have low hysteresis, low power consumption and low cost.The first mirror of this kind proposed by Vdovin and Sarro [8], consisted in a conductive silicon-nitride membrane aluminium coated, placed on top of an array of electrostatic actuators that pull the membrane in one direction.Although this mirror showed relatively good results in correcting aberrations, it showed some difficulties to obtain high stroke deformations and high order aberrations because of the unidirectional motion of the membrane.To overcome this problem, push and pull membrane deformable mirrors have been developed with additional set of transparent actuators above the membrane.In this way a motion of the membrane in both directions is possible [12].
Several control strategies for inducing the desired wavefront have been proposed in the literature [13,14].However, controlling the DM using these methods might result in a non optimal performance of the overall AO system.Namely, these control strategies might result in a low convergence of the wavefront error and a significant steady state residual wavefront.Furthermore, in some cases, saturation of the actuators (i.e.voltages close to the maximum allowed value) could occur with these methods [12].
In this paper we analyse the performance of a commercial push-pull membrane deformable mirror and we show how to boost its performance in terms of the generation of wavefront shape with this device.This is done by choosing an appropriate working point and introducing an innovative method to control efficiently the membrane surface deformation.
The outline of the paper is as follows.In Section 2 the design of the deformable mirror and the experimental setup used to characterise are given.In Section 3, the mathematical model of the DM and the algorithms used to control it are proposed.In Section 4 the experimental measurements of the characteristics of the mirror are shown.Moreover the procedure for a statistical identification of the influence function is given as well as the performance comparison with different control algorithms.Finally, the conclusions are presented in Section 5.

Push-Pull membrane deformable mirror design and experimental setup
The deformable mirror examined in this paper is a commercial device supplied by Adaptica Srl (Pavia, Italy).It consists of a 5 µm thick, silver coated nitrocellulose conductive membrane placed between two set of actuators structures.The optical useful area diameter is 11 mm with a maximum achievable stroke claimed by the company of ∼10µm.A sketch of the mirror is shown in Fig. 1.The first set of 32 actuators are installed below the membrane.They are electrodes connected to an external amplifier which allow, by applying a voltage distribution V (x, y) (between 0 and 250 Volts) to control, by a pull action, the membrane shape S(x, y) that are solution of the Poisson equation [15].By placing an additional set of 16 transparent electrodes above the membrane push and pull capability is obtained.The transparency and the conductivity of the latter electrodes is provided by a indium-tin-oxide (ITO) coated glass on which the electrodes are installed.According to the factory specifications the transmission of the ITO glass is 89% [16].
The experimental setup built to characterise the mirror performance is shown in Fig. 2.
A semiconductor intensity stabilised laser (wavelength λ =633nm) coupled by a single mode fiber is first collimated by a 100 mm focal length lens L1.Then the beam is folded by a beam splitter (BS) and brought to illuminate uniformly the surface of the mirror.An iris of diameter 9 mm is placed in front of the mirror in order to use the optimal area of the membrane [12,17].A Shack-Harmann wavefront sensor (S-H WFS) (Thorlabs WFS S300-14AR, 1.3 Mpixel, λ /50 rms accuracy) is used to measure the absolute wavefront shape when voltages are applied to the actuators.The S-H sensor is conjugate with the DM by a relay system consisting of the lens L2 ( f = 250 mm) and L3 ( f = 100 mm).The relay system has also the function of decreasing the beam size to be fitted in the S-H sensor (clear aperture 4.8 mm).Both S-H sensor and DM are connected to each other in a close-loop [18] by a LabView ® interface to evaluate the generation of desired wavefront with an iterative routine as explained below.

Mathematical model and control algorithm
In this section we will present a DM model that is suitable for control.In the literature the DM is modelled as a nonlinear function of inputs [12,13].However, for simplicity, in this paper we are going to use a linear model for the DM.Such a model will enable us to develop an efficient and simple control strategy.All the model uncertainties will be handled by the control strategy that is based on the feedback principle.Under the assumption of linearity the DM model is given by the following equation: where W ∈ R 36 is the membrane shape expressed as Zernike polynomials expansion up to the 36 th terms (the Malacara notation is used through this paper [19]), M ∈ R 36×48 is the influence matrix of the DM and V ∈ R 48 is a vector of voltages applied to the DM.
The influence function M is usually measured by sequentially applying voltages to each channel following by measuring the corresponding response of the system.For example, the first column of the matrix M, denoted by m 1 , is obtained by applying the voltages to the first actuator and measuring the response.
In order to produce the wavefront of the desired shape, we need to solve the system of equations (Eq.( 1)), where V are the unknowns.However, the system is underdetermined, hence in the general case there are multiple solutions.In [12], the solution of the system of Eq. ( 1) have been determined by solving the following optimisation problem: where W d is the desired shape, • 2 2 is a 2-norm [22] and Z is the vector of square voltages.In particular, the solution has been found using a non-negative least square algorithm (NNLS) without applying an initial value for Z to the mirror (i.e., a bias voltage) and introducing an iterative correction algorithm for the saturated channel.
Even though this method achieves good results thanks to the push-pull mirror design, we will show in Sec. 4 that such an algorithm reduces the domain of solutions of Eq. ( 2) and force some of the channels to go into saturation.This could create a strong mechanical stress on the membrane.Furthermore the convergence of the residual wavefront is slow.
To overcome these issues we propose a novel control strategy that is based on the Iterative Learning Control (ILC) algorithm [20] .Let n ∈ N be a real discrete-time instant.The wavefront produced by the DM at the time instant n will be denoted by W n .The wavefront error e n is defined as follows: where V n are the voltages applied to the DM at the time instant n.The error at the n + 1 time instant is given by e From Eq. ( 4) and Eq. ( 1) we have: where ∆V n = V n+1 −V n is a control update.ILC finds the control update by solving the following optimisation problem min Here β is a positive regularisation parameter between 0 and 1 used to penalise ∆V n .In our setup β = 0.1.The solution of Eq. ( 6) is: From Eq. ( 7) we obtain the control law Throughout this paper we will refer to this law as ILC.The influence of β on the overall performance of the system has been analysed in [20,21].By means of this approach we can study the dynamics behaviour of the system.In particular by choosing large β we force the voltages to be close to the bias, while on the other hand the convergence of the tracking error decreases.On the contrary by decreasing β we increase the convergence speed of the algorithm while the voltages will assume larger value.In that way, we prevent the actuators to go into saturation.
For comparison purpose, we will also use the following control algorithm: where α is parameter varying from 0 to 1 accounting for the convergence speed and stability of the algorithm.A suitable value for our setup is 0.4.The control law Eq.( 9) has the form of the steepest descent method (SD) [22] and it also belongs to the class of ILC algorithms.
In order to apply the algorithms described by Eq. ( 8) and Eq. ( 9) we need to set the initial value of V n .This initial value, denoted by V 0 , is set to the 50% of the maximum voltage (bias voltage).With parameter β we ensure that ∆V n will not be too large and close to V 0 .We would like to emphasize here that in the case of NNLS solution, V n+1 is found iteratively (of the order of 20 iterations) for each cycle of measurement at the time instant n.On the contrary with the control law described by Eq. ( 8) and Eq. ( 9) we attempt to find a solution at the time instant n and then correct it in the n + 1 iteration making use of the knowledge of the measurement W n .The typical time step t for our setup is of about 1.5s due to the slow response of the sensing hardware.Within that time step the DM has reached its steady state since the typical response of this type of devices is of the order of a few hundreds Hertz [8].

Performance
A characterisation of the mirror needs to be carry out in order to have an accurate control of the mirror.Since the DM model (Eq.( 1)) is assumed to be linear, a measurement of this characteristic gives information on how realistic is this assumption.Figure 3 shows the normalised displacement of the membrane when a linear increment of voltages is applied to some actuators above and below the membrane.These measurements show that the actuators taken in con-Fig.3. Linearity behaviour of the mirror on some channels: channels 9 and 24 are situated on the center of the membrane while the others are on the edge.
sideration have approximately the same behaviour with a Pearson correlation coefficient [23] around 0.96.
The second step in the characterisation involves the measurement of the influence function M (Fig. 4).The matrix M is measured as described in Section 3 after subtracting the shape of the membrane when no voltage is applied to it.In this way the contribution of the initial non-flatness of the mirror is filtered-out.The influence function obtained using this procedure will be referred as the 'standard influence function' M. The maximum displacement occurs for the central actuators below and above the membrane, that is, 2.3µm and 1.2µm.We would like to stress that this way of identification does not take fully into account the cross talk action between adjacent actuators.Furthermore, the measurement errors are directly included in the matrix M.These factors degrade the performance of the AO system as we experimentally confirm in Section 4.2.In order to include these factors into the model and therefore improve the performance of the device, a statistical approach for the identification of the influence function is developed and is described in Sec.4.1.To make a distinction with the previous one we will call it the 'Statistical Influence Function' M stat .

Statistical identification of the influence function
In order to improve the performance of the system we will identify a new influence function M stat using the following procedure [22].
Let us consider the measurements illustrated in Fig. 3. From these data we can assume that the linearity behaviour of the mirror is justified in the range from 30% to 70% of the maximum voltage where the Pearson correlation coefficient is around 0.99.Taking that into account, at each time instant n a Gaussian random distribution voltage V n varying between the linearity bounds is applied to the actuators and a correspondent measured W n is taken.By grouping the applied voltages and measured responses, we define the following matrices: The number of measurements N should be a relative large number [25].In this paper we choose N = 200.The influence matrix M stat is identified by solving the following least-squares optimisation problem: min where • F denotes the Frobenius norm [22,24].The random voltages ensure that V has full row rank so, the solution of the optimisation problem 11, is then given by: where V † = V T (VV T ) −1 denotes the matrix pseudo-inverse.We will show in the next section that this identification approach leads to the improvement of the performance of the overall AO system.Compared to the standard way of identification of the influence function, this procedure takes into account the mechanical cross-talk between the response of the actuators and filters out the noise due to the measurement equipment.

Zernike generation
The performance evaluation concerns the ability of the mirror to reproduce a desired shape as well as the speed of the used algorithm to converge to the closest solution.We use five control algorithm combinations.The steepest descent and the ILC algorithms are compared together with the definition of M and M stat .Additionally, for completeness, a NNLS algorithm making use of the definition of the influence function M has been used to control the DM.We first evaluate the capability of the mirror to correct for its own non-flat surface.Then, the first 4 orders of the Zernike polynomials (8 modes) are evaluated.To assess the quality of the generated shape that we can obtain by the different methods, we have followed the strategy From these results we can state that the definition of M stat in conjunction with the ILC algorithm gives the highest performance for the considered aberration.In particular the RMS deviation and the Purity values achieved with the latter control algorithm are always better compared to the same values achieved by the other algorithms.The improvement is more evident for the higher order aberrations (e.g. for spherical aberration we obtain a P ILC Mstat =0.76 compared to P NNLS =0.69 and a RMS deviation of 0.064 and 0.093 respectively).
Figure 5 shows also that this combination converges fastest and smoothest among the other algorithms, with a maximum number of iterations that varies from 5 to 10 for high order aberrations.

Voltages
Finally, an analysis on the final voltages that are applied to the mirror to generate the desired shape has been carried out.This parameter is rarely taken into account in the performance analysis and gives information about the mechanical stress of the membrane.A value of some voltages close to 100% means that the mirror is working in the limits of the safe region which can lead to a negative effect in the elastic property of the membrane in a long term use. Figure 6 shows the mirror's average voltages V and standard deviation σ V for every membrane shape we considered previously for the different algorithms.The highest standard deviation in the voltages referred to the NNLS algorithm (σ V ≃ 40%) means that there are channels with values close to the saturation limit.These channels are strongly pushing (or pulling) the membrane to generate the desired shape inducing strong stress to it.On the contrary the final voltages referred to the other algorithms are more stable with a lower standard deviation (σ V ≃ 10%) and they do not deviate significantly from the applied bias voltage (50% of the maximum value).Therefore, in these cases a fine tuning of the mirror's voltages is sufficient to generate the desired shape without inducing strong stress to the membrane.Mathematically this means that the voltage optimisation problem converges to the closest solution in a very smooth way without jumping from a local minimum to another.

Conclusion
We have carried out a characterisation of a commercial push-pull deformable mirror with 48 actuators.We have developed an easy iterative linear-based control model that can be used to generate a desired wavefront shape.We also suggested a statistical method to identify the influence matrix of the mirror that boosts the performance of the device.Comparison of the new control algorithm with other methods presented in the literature has been made and it was shown that a substantial improvement of the performance in terms of quality of the shape generation as well as faster convergence is obtained.An analysis on the voltage required to the mirror to generate a desired shape has been carried out for all described algorithms.Also in this case we showed an improvement compared to the methods that are present in literature.

Fig. 1 .
Fig. 1.Sketch of the Saturn mirror supplied by Adaptica Srl: (a) 3D representation; (b) planar representation of both the upper and lower electrodes

Fig. 2 .
Fig. 2. Experimental setup for the performance testing of the push-pull membrane deformable mirror

Fig. 4 .
Fig. 4. Measured Influence Function: each figure is not at the same scale with the others for an easier visualisation.Each plot refers to the membrane response to a single actuator.Red colour means positive displacement (push action), while the blu colour mean a negative displacement (pull action)

Fig. 5 .
Fig. 5. Convergence of the RMS wavefront deviation for different control laws for different Zernike polynomial generation.(a) flat wavefront; (b) tilt; (c) defocus; (d) astigmatism; (e) trefoil; (f) coma; (g) tetrafoil; (h) 2 nd order astigmatism; (i) spherical aberration.The ILC control law that is based on M stat gives in all the cases the best performance.

Fig. 6 .
Fig. 6.Final average voltage applied to the mirror and relative standard deviation (error bars) for different Zernike shapes resulting from different algorithms: (a) Non-Negative least square (NNLS); (b) steepest descent solution with standard influence function (SD M); (c) steepest descent solution with statistical influence function (SD M stat ); (d) Iterative Learning control with standard influence function (ILC M); (e) Iterative Learning control with statistical influence function (ILC M stat ).