open-loop control demonstration of micro-electro-mechanical-system mems deformable mirror

Open-loop control demonstration ofMicro-Electro-Mechanical-System

MEMS Deformable Mirror

Celia Blain1*, Rodolphe Conan1, Colin Bradley1, and Olivier Guyon2

1University of Victoria, Department of Mechanical Engineering, PO Box 3055, Stn. CSC,Victoria, BC, V8W 3P6, Canada

2Subaru Telescope NAOJ, 650 N. A’Ohoku Place, 96720 Hilo, HI, USA

*[email protected]

Abstract: New astronomical challenges revolve around the observationof faint galaxies, nearby star-forming regions and the direct imaging ofexoplanets. The technologies required to progress in these fields of researchrely on the development of custom Adaptive Optics (AO) instruments suchas Multi-Object AO (MOAO) or Extreme AO (ExAO). Many obstaclesremain in the development of these new technologies. A major barrier to theimplementation of MOAO is the utilisation of deformable mirrors (DMs) inan open-loop control system. Micro-Electro-Mechanical-System (MEMS)DMs show promise for application in both MOAO and ExAO. Despiterecent encouraging laboratory results, it remains an immature technologywhich has yet to be demonstrated on a fully operational on-sky AO system.Much of the research in this area focuses on the development of an accuratemodel of the MEMS DMs. In this paper, a thorough characterization processof a MEMS DM is performed, with the goal of developing an open-loopcontrol strategy free of computationally heavy modelling (such as the useof plate equations). Instead, a simpler approach, based on the additivity ofthe influence functions, is chosen. The actuator stroke-voltage relationshipand the actuator influence functions are carefully calibrated. For 100 initialphase screens with a mean rms of 97 nm (computer generated followinga Von Karman statistic), the resulting mean residual open-loop rms erroris 16.5 nm, the mean fitting error rms is 13.3 nm and the mean DM errorrms is 10.8 nm (error reflecting the performances of the model under testin this paper). This corresponds to 11% of residual DM error.

© 2010 Optical Society of America

OCIS codes: (010.1080) Active or adaptive optics; (230.4040) Mirrors;(120.0280) Remotesensing and sensors.

References and links1. J.A. Perreault, T. Bifano, B. M. Levine, and M. Harenstein, “Adaptive optic correction using microelectromech-

nical deformable mirrors,” Opt. Eng. 41(3), 561–566 (2002).2. M. N. Horenstein, T. Bifano, R. Krishnamoorthy, and N. Vandelli, “Electrostatic effects in micromachined actu-

ators for adaptive optics,” J. Electrostatics 42, 69–81 (1997).3. B. P. Wallace, P. Hampton, C. Bradley, and R. Conan, “Evaluation of a MEMS deformable mirror for an adaptive

optics test bench,” Opt. Express 14(22), 10132 (2006).4. K. M. Morzinski, D. T. Gavel, A. P. Norton, D. R. Dillon, and M. R. Reinig, “Characterizing MEMS deformable

mirrors for open-loop operation: High-resolution measurements of thin-plate behavior,” Proc. SPIE MEMSAdaptive Optics II 6888, 68880S (2008).

#120000 - $15.00 USD Received 16 Nov 2009; revised 30 Jan 2010; accepted 5 Feb 2010; published 2 Mar 2010(C) 2010 OSA 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 5433

5. T. G. Bifano, R. K. Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon,“Continuous-membrane surface-micromachined silicon deformable mirror,” Opt. Eng. 36(5), 1354–1360 (1997).

6. K. M. Morzinski, J. W. Evans, S. Severson, B. Macintosh, D. Dillon, D. Gavel, C. Max, and D. Palmer, “Charac-terizing the potential of MEMS deformable mirrors for astronomical adaptive optics,” Proc. SPIE Advances inAdaptive Optics II 6272, 627221 (2006).

7. C. Blain, R. Conan, O. Guyon, C. Bradley, and C. Vogel, “Characterization of influence function non-additivitiesfor a 1024-actuator MEMS DM,” in press (2009).

8. S. A. Cornelissen, P. A. Bierden, and T.G. Bifano, “Development of a 4096 element MEMS continuous mem-brane deformable mirror for high contrast astronomical imaging,” Proc. SPIE Advanced wavefront control:methods, devices and applications IV 6306, 630606 (2006).

9. B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, et al., “The Gemini Planet Imager,” Proc.SPIE Advances in Adaptive Optics II 6272, 62720L (2006).

10. F. Assemat, E. Gendron, and F. Hammer, “The FALCON concept: Multi-Object adaptive optics and atmospherictomography for integral field spectroscopy–principle and performance on a 8-m telescope,” MNRAS 376, 287–312 (2007).

11. O. Guyon, E. Pluzhnik, F. Martinache, J. Totems, S. Tanaka, T. Matsuo, C. Blain, and R. Belikov “High ContrastImaging and Wavefront Control with a PIAA Coronagraph: Laboratory System Validation,” in press (2009).

12. T. Fusco, G. Rousset, J.-F. Sauvage, C. Petit, J.-L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle,M. Kasper, P. Baudoz, and P. Puget, “High-order adaptive optics requirement for direct detection of extrasolarplanets: Application to the SPHERE Instrument,” Opt. Express 14(17), 7515–7534 (2006).

13. J. W. Evans, K. Morzinski, S. Severson, L. Poyneer, B. Macintosh, D. Dillon, L. Reza, D. Gavel, D. Palmer,S. Olivier, and P. Bierden, “Extreme Adaptive Optics testbed: performance and characterization of a 1024 de-formable mirror,” Proc. SPIE MEMS/MOEMS Components and their applications III 6113, 131-136 (2006).

14. L. A. Poyneer and D. Dillon, “MEMS adaptive optics for the Gemini Planet Imager: control methods and valida-tion,” Proc. SPIE Advances in Adaptive Optics II 6888, 68880H (2008).

15. E. A. Pluzhnik, O. Guyon, S. Ridgway, R. Woodruff, C. Blain, F. Martinache, and R. Galicher, “The PhaseInduced Aplitude Apodization Coronagraph: an overview of simulations and laboratory effort,” IAU, DirectImaging of Exoplanets: Science and Techniques 200, (2005).

16. F. Martinache, O. Guyon, J. Lozi, V. Garrel, C. Blain, and G. Sivo, “The Subaru Coronagraphic Extreme AOProject,” in press (2009).

17. D. Gavel, S. Severson, B. Bauman, D. Dillon, M. Reinig, C. Lockwood, D. Palmer, K. Morzinski, M. Ammons,E. Gates, and B. Grigsby,“Villages: An on-sky visible wavelength astronomy AO experiment using MEMS de-formable mirror,” Proc. SPIE Photonics West 3888-03, (2008).

18. C. Blain, O. Guyon, R. Conan, and C. Bradley,“Simple iterative method for open-loop control of MEMS de-formable mirrors,” Proc. SPIE Adaptive Optics Systems 7015, 701534 (2008).

19. K. Morzinski, K. B. Harpsoe, D. Gavel, and S. M. Ammons,“The open-loop control of MEMS: Modeling andexperimental results,” Proc. SPIE MEMS Adaptive Optics 6467, 64670G (2007).

20. J. B. Stewart, A. Diouf, Y. Zhou, and T. Bifano,“Open-Loop control of MEMS deformable mirror for large-amplitude wavefront control,” J. Opt. Soc. Am. 24(12), 3827–3833 (2007).

21. C. R. Vogel and Q. Yang,“Modeling, simulation, and open-loop control of a continuous facesheet MEMS de-formable mirror,” J. Opt. Soc. Am. A 23(5), 1074–1081 (2006).

22. M. C. Roggemann and B. Welsh, “Imaging through turbulence.”23. J. Nelson and G. H. Sanders, “The status of the Thirty Meter Telescope project,” Proc. SPIE Ground-based and

Airborne Telescopes II 7012, 70121A (2008).

1. Introduction

1.1. Background

1.1.1. MEMS deformable mirrors for astronomy

In the past decade, the astronomical community’s interest in Micro-Electro-Mechanical-System(MEMS) technologies has increased with the large number of MEMS studies conducted [1–3].These studies revealed the potential for astronomical adaptive optics (AO) applications. MEMSDeformable Mirrors (DM) [5] present the ability to organize individual micro actuators intoarrays that perform a macroscopic function. A flat, high quality mirror can be obtained byplacing a reflective membrane above the actuator array. A voltage applied between the actuatorand its ground pad creates an electrostatic force that deflects the actuator toward its pad, thusmoving the top membrane attached to the actuator.

Compared to other DM technologies (e.g piezo-stack, magnetic) that exhibit bi-directional


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actuator motion, a MEMS DM actuator can only be “pulled” (through electrostatic force) inone direction. However, each actuator has low inertia and therefore, it can be positioned alongits total stroke with great accuracy and at high frequency (e.g in the kHz range). Further ben-eficial characteristics have also been demonstrated: a highly repeatable actuator position and anegligible hysteresis [3, 4].

A wide spectrum of AO instruments are currently under development and many are intendedfor use on the next generation of Extremely Large Telescopes or ELTs. Increasing the size ofthe primary mirror will increase the distortion of the image due to atmospheric turbulence.Combined with the needs of each specific AO instrument, this elevates the requirement for theDM specifications to very high levels. MEMS DMs fulfil many of the demanding requirementswhich accompany the next generation of AO systems; such as: speed, compact size, a largenumber of actuators with a relatively large stroke, no hysteresis and a reasonable fabricationcost. A 1024-actuator MEMS DM is commercially available and has been extensively testedby several research teams [6, 7]. The current state of the art is a 4096-actuator MEMS DM [8](with an active aperture of approximately 20 mm) designed to be implemented on the GeminiPlanet Imager [9].

1.1.2. New scientific goals and custom AO systems

The instrument specifications are defined by new scientific goals, such as the direct imaging ofexoplanets or the observation of faint galaxies and nearby star-forming regions. For the obser-vation of faint galaxies, the small number of photons received (long exposure time necessary foreach target) combined with the galaxy’s distribution [spread over a large Field of View (FOV)]make the classical Single Conjugate Adaptive Optics systems (SCAO) inefficient.

Multi-Object Spectroscopy (MOS) is a technique which relies on the insertion of severalapertures in a wide FOV (each one of them dedicated to a specific target) and the simultaneousmeasurement of the spectra of each object. To reach the angular resolution necessary to obtainrelevant information on each target, it needs to be coupled with a Multi-Object Adaptive Opticssystem (MOAO) [10]. An MOAO system is designed and optimised to provide a sharp correc-tion on several small areas spread over a large field of view (∼5-10’ ). Instead of correcting thewhole FOV, it consists of several sub-AO systems (with small FOV of approximately 2” each)working in parallel. However, in this configuration, the guide star light directed toward thewavefront sensor is independent of the science target light, directed toward the science camera.Therefore, close-loop control is not possible and open-loop control must be used.

The quest for the detection and the characterisation of exoplanets (from giant hot Jupitersto Earth-like planets) is at the center of many exciting research developments [9, 11, 12]. Tomeet the high performance coronagraphic needs, an AO system needs to be combined with thecoronagraph. Classical SCAO systems had to evolve toward the so-called Extreme AdaptiveOptics (ExAO) systems optimised to provide a high Strehl correction on axis over a small fieldof view. MEMS DMs have the ability to reach the desired ExAO performances [13, 14]. Toimprove the overall performance, new coronagraph designs [15, 16] also include MEMS DMsdirectly inside the coronagraphic path to correct for the residual aberrations due to alignmenterrors and to cancel speckles in the planet search area. In such cases, the choice of driving theDM with open-loop control is more appropriate.

For all these applications, it is likely that the DM will be driven using an open-loop (OL)control architecture. Because of the lack of feedback, an exact knowledge of the mirror shape inresponse to a set of control commands is needed to achieve the scientific requirement of image-sharpening (in the nm range). Much of the research in this area focuses on the development ofan accurate physical model [17–21].


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1.2. Goal of this paper

The work presented here was motivated by an attempt to find answers to the following ques-tions: Which types of AO systems require the use of a model for MEMS DMs? Which MEMSproperties need to be known to develop this model? What accuracy is needed for this model? Inorder to help the ongoing research focused on the development of MEMS models, a companionpaper [7] was focused on the characterisation of the actuator influence function non-additivities(non-linear coupling between neighbouring actuators) and stroke-voltage relationship (SVR).In this former study, it was observed that when neighbouring actuators are set in a push-pullconfiguration (one up, one down, relative to a bias half-stroke position), the influence functionnon-additivities are compensated. These results are recalled in Fig. 1.

Figure 1(a) presents the membrane deflection obtained in two cases. First, two neighbour-ing actuators are pulled consecutively to 0V, 50V and 100V while the rest of the actuators aremaintained at a bias voltage of 150V. The influence functions obtained for both actuators arenumerically summed for 0V, 50V, and 100V and a transversal cut of the result is plotted andtagged “sum” on the figure’s textbox. Then actuators are pulled simultaneously to 0V, 100Vand 150V while the other actuators are maintained at a bias voltage of 150V. These measure-ments (transversal cut) are tagged as “poke2” on the figure’s textbox. Figure 1(a) shows that asthe relative voltage (the difference between the neighbouring actuator’s voltage and the rest ofthe membrane voltage) increases, the membrane deflection created by simultaneously pullingboth actuators (tagged “poke2” in the textbox) separates from the numerical sum of their influ-ence functions (tagged “sum” on the textbox).

On Fig. 1(b), the red and green plots represent the transversal cut of the influence functionsof two neighbouring actuators. First, the measurement are taken consecutively, one actuator ispushed up, then the neighbouring actuator is pulled down (while the rest of the actuators aremaintained at a bias voltage of 100V). Then, the sum of both actuators influence function (darkblue plot tagged “sum” in the figure textbox), is plotted beside the measurement obtained whenboth actuators are set simultaneously on the push-pull configuration (light blue plot tagged“poke2” in the figure). It can be seen that in this particular configuration, the non-additivity ofthe two neighbouring actuator’s influence function becomes negligible (the dark and light blueplots are on top of each other).

From this previous observation emerged a new research path: what accuracy can be reached(best residual rms) using an open-loop control strategy based on the modelling of the influencefunctions additivity? Indeed, with a thorough characterisation process, the control command(voltage maps) can be computed by using the information obtained from both the actuatorstroke-voltage relationship and the actuator influence functions.

In this paper, we will rely on the following two assumptions: first, for a typical turbulentphase screen, there is more power at low spatial frequencies than at high spatial frequencies.Second, the statistical distribution of actuators compensating for a phase screen type Kol-mogorov is such that the number of actuators on the “up” position is roughly equal to thenumber of actuators on the “down” position. Thus, from the plot presented in Fig. 1(b), thenon-additivity of the influence functions are considered negligible.

We first present the experimental apparatus and the data collection process in Sec. 2. Theresults of the actuator SVR and influence function calibration are introduced in Sec. 3. Section 4focuses on the performance obtained when the DM is controlled in an open-loop fashion usingonly the knowledge acquired from Sec. 3. The residual rms is minimised through the optimisedutilisation of the calibration data. Finally, a statistical study is performed over 100 computergenerated phase screens simulating the atmospheric turbulence as seen by a 30 metre diametertelescope.


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2. Experimental setup

2.1. Experimental setup

The experimental setup is illustrated by the diagram in Fig. 2(a) and consists of a 1024-actuatorBoston Micromachines MEMS DM with 200 volt (14 bit resolution) electronics manufacturedby NASA JPL. The actuator pitch is 340 μm and the maximum stroke is given for 2.5 μm.A Zygo PTI 250 interferometer is positioned in front of the DM. The interferometer beampasses through a density filter to improve the fringe contrast. A mask can be set on top ofthe interferometer beam using the software provided with the interferometer. One computeris dedicated to the Zygo interface (metrology software, interferometer measurements and datatransfer to the laboratory data server), while a second computer controls the DM electronics andinitiates the interferometer measurements. The DM and interferometer are setup on a vibrationisolation optical table. The active area on the DM is a square of 18 by 18 actuators. Aligningthe pupil edges (the edges of the interferometer mask) to half of the edge actuators reduces thefitting error (this result will be detailed in Sec. 4.3). In this optimised configuration, the effectivearea of the DM will become a square of 17 by 17 actuators. For an optimised pupil size of 106by 106 pixels, the spatial resolution of the interferometer is 6.2 pixels per actuator for a 17 by17 actuator array. The repeatability of the Zygo measurement was verified and is less than twonm rms. Finally, all results presented along this paper are given in nm or μm “surface” withone nm “phase” equal to two times a nm “surface”.

2.2. Data collection

There are only two DM properties which need to be evaluated to complete the characterisationprocess necessary for the control strategy presented in this paper: the actuator stroke-voltagerelationship (SVR) and the actuator influence function.

The actuator SVR is the relationship between the membrane deflection and the correspondingapplied voltage. To determine this relationship, a set of voltages were applied to the actuators,starting at the lowest voltage that can be sent (0 Volts) to the highest voltage that can be sup-ported by the DM actuators. This highest voltage is determined by the DM physical limit (size


of the actuator gap) and by the electronics. The DM physical limit is 250V. If an actuator isexposed to a higher voltage, the electrostatic force generated between the actuator plate andthe actuator base would bring the actuator plate all the way down to the actuator base. Theelectronics used for this experiment can provide a maximum voltage of 200V.

The actuator influence function is the characteristic shape of the mirror response to the actionof a single actuator. The DM influence functions are measured by sequentially pushing the DMactuators and measuring the resulting shape over the whole membrane.

The MEMS DM being tested has an array of 32 by 32 actuators. However, a large numberof actuators on the right side of the mirror are dead or malfunctioning; therefore, the maximumsize of the array available for this test was 18 by 18 (a total of 324 actuators). This 18 by 18array will be subsequently referred to as the DM’s active area. The position of this array withrespect to the DM membrane is presented in Fig. 2(b). The actuator situated at the bottom leftcorner of the array is coupled with an actuator located outside of the array and has a truncatedmaximum stroke.

To ensure the reliability of the calibration data, a new set of data should to be taken if theexperimental setup is modified (optical re-alignement, change in ambient temperature..). Thiswill guarantee up-to-date DM properties necessary to achieve an accurate control of the DM.

3. Deformable mirror characterisation

3.1. Measurement of the actuator stroke-voltage relationship

The precise calibration of each actuator stroke-voltage relationship is a critical step toward theaccurate open-loop control of the deformable mirror. The plots for each actuator of the 18 by18 array are generated by driving the actuator from 0 Volts to 200 Volts in steps of 20 Voltswhile the rest of the actuators are set to a voltage bias of 140V (this bias voltage correspondsto the mid-stroke, directly measured from an actuator’s stroke-voltage plot).

The stroke-voltage relationship is quadratic and for each actuator k can be stated as:

stroke(k) = gain(k)· V (k)2 + o f f set(k) (1)

where the “gain(k)” refers to the slope of the SVR for the actuator k, the “offset(k)” refers tothe 0 position offset and “V” corresponds to the voltage applied to the actuator.

The maximum stroke observed is approximately 800 nm. However, the coupled actuatorin the bottom left corner, described in Sec. 2.2, has a maximum stroke of only 400 nm. TheSVR plots obtained for the 324 actuators are presented on Fig. 3. The x axis represents thesquared voltage sent to the actuator and the y axis the corresponding actuator stroke (in nm).The standard deviation of the measurements presented in Fig. 3 is 23.4 nm rms.

3.2. Measurement of the actuator influence function

The influence functions are measured by releasing, one at a time, the actuators at 0 Volts whilethe rest of the membrane is pushed at the bias voltage of 140 Volts. For each actuator, the phaseis measured by the interferometer and normalised. To normalise the influence function, eachmeasured phase map is divided by the absolute value of the maximum phase point in the phasemap. The normalised influence functions are non-unit phase maps with the scale ranging fromzero to one. The normalised influence function for actuator # 171 is presented in Fig. 4.

Note that because the actuators are released one at a time, the influence function measure-ments do not include any non-linear information related to the mechanical coupling betweenneighbouring actuators.


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Fig. 5: (a) original sample phase screen ϕ generated with Matlab, (b) corresponding fitted phaseϕ obtained by the multiplication of the influence function F and the stroke coefficients ak, (c)stroke map (ak coefficients obtained by projection of the original phase ϕ onto the influencefunctions F), (d) voltage map (vertical scale in Volt) obtained with Eq. (6), (e) projection of theoriginal phase screen ϕ onto the DM (phase ϕm measured by the interferometer) and (f) “open-loop” error or “measurement” error = ϕ - ϕm. This error map incorporates both the fitting errorand the DM error (non-linear effects such as the mechanical coupling between neighbouringactuators). Vertical scales for (a), (b), (c), (e), and (f) are in nm.


4. Performance evaluation in open-loop control

This section describes the experimental protocol that was followed, starting from the generationof the 100 test phase screens to their final projection onto the DM. To illustrate the results ob-tained, a sample phase screen among the 100 generated was selected. Figure 5 presents imagescorresponding to its original phase, its projection onto the influence functions and onto the DMas well as the matching voltage map.

Figure 6 illustrates the open-loop control process and the various errors introduced in thefollowing sections.

4.1. Generation of phase-screens

100 phase screens ϕ are generated using Matlab. To create phase screens that match the dy-namical range of the MEMS DM, the generated phase screens are first numerically rescaled.They are generated using turbulence parameters, which follow a Von Karman statistic. TheFried parameter, r0, is set to 15 cm. The Fried parameter determines the seeing cell size (thiscorresponds to the aperture size beyond which increases in diameter provide no further increasein resolution) [22]. The outer scale, L0, is set to 60 m and corresponds to the size of the largestturbulence cell. Finally, the pupil diameter, D, is set to 30 m. The characterisation of the actua-tor SVR presented in Sec. 3.1 shows that the DM being tested has a maximum stroke of about800 nm. With a bias voltage of 140V, the DM stroke varies from -400 nm to +400nm. However,the scale of the raw phase screens generated through Matlab is around 15000 nm. In order toavoid DM stroke saturation, the phase screens are scaled to 88% of the DM maximum stroke.The piston is also removed in order to match the phase screen dynamic range to the actuatordynamic range (±400nm) with no bias voltage.

Figure 5(a), shows a representative phase screen ϕ . The phase varies approximately from250 nm to -300 nm.

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Fig. 6: (a) Diagram of the open-loop control process and (b) error estimation. The original phasescreens ϕ are generated using Matlab. The influence functions F and the SVRs are measuredduring the DM calibration. The stroke maps (ak coefficients) are computed using Eq. (4). Thevoltage maps [obtained using Eq. (6)] are sent to the DM and the interferometer measures theDM membrane deflection (named the measurement phase screen ϕm). ϕm is the projection ofthe original phase ϕ onto the DM.


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Fig. 7: (a) Fitting error versus the size of the interferometer mask. When the size of the mask isdecreased, the phase screen is rescaled to match the interferometer mask size. As the mask getsmaller, the number of actuators available to reproduce the phase screen decreases, and the fit-ting error increases as a result. The fitting error is minimum when the mask edges are positionedat the center of the edge actuators of the DM’s active area (b) Diagram of the interferometermask size relative to the first three outer actuator coronas.

4.2. Projection on influence function: stroke map computation

For a pixel of coordinate (x, y), the phase for the MEMS DM at this pixel is given by:

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where ak represents the unknown stroke coefficients and Fk represents the normalised influencefunctions, measured and described in Sec. 3.2 (one ak coefficient per influence function andone influence function per actuator).

Using matrix representation, it is re-written as

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The evaluation of the ak coefficients is done by performing a least square fit (LSF) projectionof the phase screen onto the influence functions which corresponds to the minimisation of theFrobenius norm of ϕ −F· a with respect to a.

The minimisation of the norm corresponds to the following matrix operation:

a = F†· ϕ with F† = (FT F)−1FT (4)

In the following, the ak coefficients related to a particular phase screen are called the strokemap. Once the stroke maps are obtained , the fitted phase ϕ can be reconstructed through thefollowing matrix multiplication,

F · a = ϕ (5)

The process described through Eq. (3) to Eq. (5) is commonly called the projection of thephase screen onto the influence functions. Figure 5(b) presents the fitted phase ϕ obtained bythe multiplication of the stroke maps and the influence functions F. Figure 5(c) shows the corre-sponding stroke map, results of the projection of the phase screen onto the influence functions.

Because the phase screens ϕ have been scaled to fit the DM’s maximum stroke capability,the stroke coefficients ak also match the DM’s stroke capability. Phase screens ϕ and influencefunction phase maps are 106 by 106 pixels. To ensure that the piston was removed properly,the mean value of both the generated phase screens ϕ and the stroke maps were checked to beequal to 0. In Sec. 4.4, the stroke coefficients will be utilized to generate the voltage maps.

4.3. Fitting error minimisation

The multiplication of the stroke maps by the normalised influence functions gives the fittedphase screens ϕ . The fitting error is the result of the difference between the original phasescreens ϕ and the fitted phase screens ϕ (see Fig. 6). The fitting error gives an estimation ofthe limits of the DM’s performance due to the limited DM spatial resolution (limited numberof actuators available to reproduce a given phase screen). This is the DM’s sampling error. Thisshows how well the mirror can reproduce a specific phase screen when the only limiting factoris the number of actuators. The fitting error decreases as the number of actuators increases.Non-linear effects (such as the mechanical coupling between neighbouring actuators) are nottaken into account in the estimation of the fitting error. This is due to the fact that the influencefunctions described in Sec. 3.2 are obtained by releasing one actuator at a time, thus the effectsdue to coupling between neighbouring actuators are not present.

The fitting error can be minimised by carefully choosing the size and position of the pupilprojected onto the DM. In this experiment, the pupil is the interferometer mask. Figure 7(a)shows the variation of the fitting error for various interferometer mask sizes. The initial size is110 by 110 pixels which corresponds to the whole 18 by 18 array of actuators [see also Fig. 7(b)


for an illustration of the size and position of the mask onto the DM’s active area]. The masksize is decreased two pixels at a time with the phase screen being rescaled to maintain the initialphase property. The phase screen is not truncated but rescaled over a smaller number of pixelsthrough an interpolation process. The D and r0 are identical as well as the variance of the phasescreen before and after rescaling, only the resolution varies. When the mask size decreases, thenumber of actuators available to reproduce the phase screen also decreases, and the fitting errorincreases as a result.

The smallest fitting error is reached when the edges of the mask are positioned at the centersof the actuators located at the edge of the 18 by 18 array [called the first outer actuator corona inFig. 7(a)]. In this experiment, this corresponds to a mask of 106 by 106 pixels, represented bythe red dashed line in Fig. 7(b). Because half of the actuators located on the first outer actuatorcorona are outside the pupil, the DM area used to correct the turbulent phase screen is now only17 by 17 actuators.

The minimisation of the fitting error occurs by optimising the size of the projection of theentrance pupil on the DM’s active area and is critical for the design of all optical elementslocated upstream (between the entrance pupil and the DM).

Note that in Fig. 7(a), it appears that aligning the pupil to have approximately three quarterof the edge actuators in the pupil would generate the smaller fitting error while maximising thesize of the pupil (which is desirous in order to maximise the DM’s spatial resolution).

4.4. Generation of the voltage command maps to be sent to the DM

Section 3.1 presents the SVR for each actuator and shows that it follows a quadratic pattern.Each actuator stroke-voltage plot can be fitted to a second order polynomial. Each polynomialfit provides two coefficients corresponding to the “offset” and the “gain” introduced in Eq. (1).To generate the voltage maps, we insert these coefficients into Eq. (1). This equation is theninverted and, for an actuator k, becomes:

V (k) =

√ak −o f f set(k)

gain(k)(6)

where the stroke maps (thus the coefficients ak) are estimated from the phase screens projectiononto the normalised influence functions.

From this point forward, the voltage maps can be obtained following two slightly differentpaths. First, option A, the mean polynomial coefficients (for the offset and gain) from Eq. (1)are inserted in Eq. (6). Second, option B, each actuator’s individual polynomial coefficient isinserted in Eq. (6). The benefit (improvement in open-loop performance) of employing optionB over option A is evaluated in the following sections.

The comparison of rms performance obtained with these two options will be presented inthe following sections. Note that for a given phase screen, the variations between the voltagemaps obtained with option A and option B are in the mV. Figure 5(d), presents the voltage mapobtained with option A.

4.5. Statistical analysis of DM errors

Once the voltage maps are computed, the voltage commands are sent to the DM. The measuredmembrane deflections corresponding to the projection of the phase screens onto the mirror aredenoted by ϕm. Two sets of data are taken, one with option A and one with option B. Figure 5(e),shows ϕm obtained with option A. Figure 5(f) shows the corresponding measurement or open-loop error.


60 80 100 120 140 1600

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rms distribution: original phase screen ϕ

(a)

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rms distribution: stroke map (ak)

(b)

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rms distribution: open-loop error - option A

(c)

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rms distribution: open-loop error - option B

(d)

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25rms distribution: DM error - option A

(e)

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25rms distribution: DM error - option B

(f)

Fig. 8: Histogram representation of the statistical study over 100 generated phase screens. (a)distribution of the original phase screen rms (vary from 56 nm to 155 nm), (b) distribution of thestroke map or ak coefficients rms, (c), (d), (e) and (f) illustrate the variation between option Aand B for the open-loop error rms and the DM error rms. “x-axis” represents the rms wavefronterror (nm surface), “y-axis” represents the number of corresponding phase screens.


The open-loop error corresponds to the difference between the original phase screen ϕ andthe measurement phase screen ϕm, which is the projection of the original phase screen ontothe DM. The DM error, corresponding to the difference between the fitted phase ϕ and themeasurement phase screen ϕm, gives an estimate of the error due to the DM non-linear effects(inter-actuator mechanical coupling). See illustration Fig. 6.

For an initial mean rms phase screen of approximately 97 nm, the mean open-loop error is16.5 nm rms, the mean fitting error is 13.3 nm rms and the mean DM error is 10.8 nm rms. TheDM error is a critical value because it reflects the ability of this modelling approach to predictthe shape of the DM. Note that for the 100 phase screens tested, option A and option B giveapproximately the same performance. Because option A is computationally less expensive thatoption B, it appears to be a more effective approach.

Figure 8 shows a histogram representation of the rms distribution over the 100 phases testedfor the original phase screens, the stroke maps, the DM errors and the open-loop errors. Thelack of significant improvement between option B and A is clearly visible.

An overview of this study is presented in Table 1 and Table 2. The values tabulated corre-spond to the mean value over the 100 phase screens generated and are given in nm rms.

Table 1: Mean and standard deviation rms of the fitting errors. All values are given in nm.

phase scr. stroke map fitting error

97 ± 20 96 ± 20 13 ± 3

Table 2: Mean and standard deviation rms of the measurement errors. All numbers are given innm except for the ratio values given in %.

option open-loop DM openloopphasescr.

DMphasescr.

A 17 ± 4 11 ± 3 17% 11%B 17 ± 4 11 ± 3 17% 11%

5. Conclusion

In this paper, a low computational cost approach is used to control a MEMS DM in open-loop. This control strategy assumes that the non-additivites of influence functions, studied ina companion paper [7], are negligible in the case of a DM who’s shape matches a randomKolmogorov turbulence phase screen. This approach appears as a possible solution for immi-nent R&D in MOAO system design because of its promising results, its simplicity and its lowcomputational cost.

By driving the DM in stroke command instead of voltage command, the relationship betweenthe membrane deflection and the applied voltage, commonly assumed to be linear, is replacedby the calibrated quadratic relationship.

Promising results of low open-loop residual rms errors obtained with a 1024-actuator MEMSdeformable mirror are shown. The characterisation process relies on both the thorough stroke-voltage relationship calibration and the actuator influence function measurements.


A statistical study over 100 phase screens is performed. With an optimised characterisationprocess, the residual open-loop error obtained is 17% and the residual DM error obtained is 11%(of the original phase screen rms). The test phase screens have a mean rms of approximately97 nm, the mean open-loop error is 16.5 nm rms. With a mean fitting error of 13.3 nm rms, thisbrings the DM error to 10.8 nm rms. The DM error shows that the modelling approach used inthis paper is good to 10.8 nm within the spatial frequency accessible to the DM.

The fitting error is also highlighted as a critical parameter for reducing the residual open-loop rms error. The projection of the entrance pupil onto the DM’s active area will impact theoptical design upstream and need to be taken into account in the very early phase of the design.Section 4.3 experimentally shows that a minimised fitting error is obtained when the pupil edgesare aligned with the centers of the actuators located at the edges of the DM’s active area.

The University of Victoria AO Lab is currently implementing the next step of this experimentwhich consists of running the MEMS DM in open-loop in a real-time control setup. The woofer-tweeter AO test bed currently in function was modified. It is now an hybrid AO bench with thetip-tilt and the woofer mirrors controlled in closed-loop while the MEMS DM is controlled inopen-loop. The details and results of this experiment will be presented in an upcoming paper.


open-loop control demonstration of micro-electro-mechanical-system mems deformable mirror

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