Electrostatic Membrane Deformable Mirror Wavefront Control Systems: Design and Analysis

Transcription

1 Electrostatic Membrane Deformable Mirror Wavefront Control Systems: Design and Analysis Keith Bush, Dee German, Beverly Klemme, Anthony Marrs, Michael Schoen Intellite, Inc Louisiana, NE Suite 202, Albuquerque, NM ABSTRACT Electrostatic Membrane Deformable Mirrors (DM) developed using silicon bulk micro-machining techniques offer the potential of providing low-cost, compact wavefront control systems for diverse optical system applications. The basic approach to electrostatic mirror construction, using bulk micro-machining, is relatively simple, allowing for custom designs to satisfy wavefront control requirements for most optical systems. An electrostatic DM consists of a thin membrane suspended over an actuator pad array that is connected to a high-voltage driver. Voltages applied to the array elements deflect the membrane to provide an optical surface capable of correcting for measured optical aberrations in a given system. The actuator voltages required to correct a given aberration are determined from wavefront sensor measurements and the mirror influence functions and/or through the minimization of measured error in the closed-loop control system. Electrostatic membrane DM designs are derived from well-known principles of membrane mechanics and electrostatics, the desired optical wavefront control requirements, and the current limitations of mirror fabrication and actuator drive electronics. In this paper, we discuss the electrostatic DM design process in some detail and present modeling results illustrating the performance of specific designs in terms of their ability to correct Zernike optical aberrations. Keywords: deformable mirror, electrostatic, membrane, wavefront control, MEMS 1.0 INTRODUCTION The concept of forming a Deformable Mirror (DM) based on electrostatic deflection of a thin membrane was first introduced in the literature in This work was based on well known concepts of membrane mechanics. 2 Basic theory and design considerations for developing easily manufactured and inexpensive membrane mirrors from titanium or nickel membrane materials was established. Later work 3, using this technology, further developed membrane DM design concepts and provided experimental results from a test system showing that mirror performance was predictable and desirable. With the advancement of silicon micro-machining techniques, the membrane DM concept was revived and is advancing to produce light-weight, low cost electrostatic membrane mirrors, based on this technology, for many different applications Intellite, Inc. has been developing electrostatic Micro-Electromechanical Systems (MEMS) DM technology for several years for both commercial and government applications. Our primary commercial products include 16 mm and 25 mm diameter DMs with 37 actuators and accompanying electronics and control system software. Much of our current research is focused on the development of large scale MEMS DMs (on the order of 100 mm diameter) with many actuators (up to 400) and large membrane throw to accommodate optical system designs for correcting Large Membrane Mirror (LMM) aberrations and atmospheric turbulence aberrations. The development of mirror designs for these applications is greatly enhanced with detailed modeling and analysis capabilities. We currently have modeling capabilities to predict the static response of various mirror designs to correcting arbitrary wavefront aberrations. The model is based on the membrane theory presented by Yellin 1 for membrane DMs and uses Finite Element Method (FEM) analysis to derive mirror actuator Influence Functions (IFs) for describing mirror surface displacement from applying actuator voltages. As for all DMs, the best method to approach wavefront control with a MEMS electrostatic DM depends on the given application. However, in our analysis we use a well-known zonal control method that provides a minimum least-squares solution to wavefront aberration compensation. Our analysis software predicts RMS Wavefront Error (WFE) and Strehl Intensity based on DM correction of wavefront aberrations

2 represented by Zernike polynomial terms. These capabilities allow us to explore the significant design space available for MEMS DM fabrication and provide performance predictions that drive design optimization. The technical content of this paper is organized into 7 Sections. Section 2.0 summarizes the MEMS DM fabrication process used by Intellite. Electrostatic DM theory and our modeling approach is reviewed in Section 3.0. MEMS DM design concepts are discussed in Section 4.0. Section 5.0 describes the wavefront control approach used in our analysis. Modeling results presented in Section 6.0 provide performance predictions of various MEMS DM designs and illustrate design concepts discussed in earlier Sections. 2.0 MEMS DM FABRICATION OVERVIEW Intellite MEMS DMs are fabricated using bulk micro-machining techniques. The basic fabrication process is illustrated and described in Figure 1. We currently produce 10, 16 and 25 mm diameter membranes with good optical flatness from both 4 and 6 inch silicon wafers. We have also successfully produced 50, 75 and 125 mm diameter membranes. Some of these designs use a circuit board to support large actuator arrays rather than the silicon wafer approach depicted in the Figure. Membrane to pad array spacing is adjusted by selecting an appropriate spacer bead diameter. Large residual astigmatism on these mirrors has been an issue to date. However, we are currently investigating several promising approaches for reducing this artifact Start With Silicon Wafer Coated Both Sides With Low-Stress Silicon Nitride Remove Silicon Nitride in Membrane Area Using Photolithography & Plasma Etch Etch Away Bulk Silicon Wafer Material With KOH To Expose Silicon Nitride Membrane Deposit Reflective Coating On Front of Membrane & Conductive Coating On Back 5 Deposit Conductive Pad Array On Separate Silicon Substrate & Bond to Back Side of Mirror Substrate Using 50 µm Glass Spacer Beads Silicon Silicon Nitride Gold Aluminum Solder Glass Bead Figure 1. Five-step bulk micro-machining process used by Intellite to fabricate membrane mirrors. 3.0 ELECTROSTATIC MEMBRANE DM THEORY AND MODELING Steady-state deflection of a membrane mirror is governed by the Poisson equation given by: F( r) Z = (1) T where: Z is membrane deflection, T is membrane tension per unit length, and F is force on the membrane at a point r. For an electrostatic DM, the force of an actuator pad on the membrane is given by: 3 where: Fk - force on membrane due to actuator pad k ε - permitivity of free space F k ε Area V k 2 2D sep 2 k = (2)

3 Areak - area of actuator pad k V voltage applied to pad k Dsep - membrane to pad array separation The analytical solution to equation (1) given in Reference 3 describes the steady-state membrane deflection for an actuator in the center of the membrane. This solution is given by: Z( r, k) = Z( r, k) = Fk R 1 ln + 2πT S 2S 2 2 ( S r ) Fk R ln, S < r < R 2πT r 2, 0 < r < S (3) where: S - actuator pad radius Analytical DM actuator IFs are derived from equations (2) and (3) by successively applying the same voltage to each actuator pad and calculating the membrane deflection for each position (r) on the membrane surface. This provides an Influence Function (IF) with equal membrane displacements at each actuator location. In reality, the same force applied at each actuator location will cause unequal membrane displacement at each actuator due to the clamped edge of the membrane and the varying distance of each actuator from the edge. We use a Finite Element Method (FEM) correction to the analytical IF to correct the surface displacements due to this effect. The method converges on new surface displacements by balancing the membrane stress, the membrane to actuator pad separation, and the second derivative of the surface at each actuator location. Finally, the fundamental frequency of a membrane DM is important for determining the temporal response of a given mirror design. Neglecting friction and air damping the fundamental frequency of a membrane of diameter D, tension/meter T, and mass/area, σ, is given by: 1-3 f = T σ πd (5) Consideration of friction and damping effects of air on the mirror vibrational modes is particularly necessary when designing a mirror for high temporal bandwidth correction. This will be discussed in some detail in the following Section. However, in this work we are primarily concerned with static mirror control and spatial wavefront correction. 4.0 MEMS DM DESIGN The flexibility and relatively low cost of bulk micro-machining allows for designing MEMS DMs to meet the needs of many different applications. Parameters available for varying DM designs include the membrane stress, diameter and thickness, the number of actuator pads and their size and their locations, and the separation distance between the membrane surface and the actuator pad array. Intellite s design and modeling software (DMModel) provides a GUI for developing mirror designs. The design parameters are depicted in Figure 2 along with a representation of the DMModel GUI. Once design parameters are specified, DMModel can be used to generate actuator IFs using the approach outlined in Section 3.0. DMModel can then be used to evaluate the mirror response to correcting Zernike aberrations, to determine the effectiveness of a given design. More detailed design and wavefront control analysis is accomplished by porting mirror design data and IFs from DMModel to MatLab for use by a script written for this purpose.

4 D mem Mirror t mem D sep Bond N act Pads a) b) Figure 2. a) Illustration of a MEMS DM cross-section depicting physical mirror parameters that can be adjusted to meet design requirements. b) DMModel screen shot showing the design for Intellite s 25 mm diameter mirror with 37 actuators. The number of actuator pads selected and their size and placement in an actuator array is dependent on the spatial frequency of the aberrations that require correction, 12 the size of the mirror being developed, and the physical limits of the technology. Our modeling has shown that close packing of the actuator pads provides the best mirror performance in terms of wavefront correction and it maximizes the force applied to the membrane according to Equation (2). However, large numbers of actuators present technical challenges. The presence of high electric fields could cause arcing between actuators in the case of densely packed arrays. Dense actuator packing also limits space available to bring out lead wires to the electronics. For designs with large numbers of actuators a through-hole process can be used, whereby the actuator leads are brought out through holes etched through the silicon wafer and electrically isolated using a dielectric or through holes in a circuit board actuator pad array. According to Eq.2, the force on the membrane goes inversely as the separation between the actuator array and mirror membrane squared. However, there is a lower bound to the practically usable separation because of membrane snapdown. Snap-down occurs when, as a result of actuation, the membrane to actuator distance is reduced to about one third of the total, ambient separation. To compound the problem, we can only pull the mirror in one direction using the actuators, so a certain amount of focus must always be added to produce a given higher-order aberration. Not only does this reduce the total stroke available for the desired aberration, but it requires additional optics to compensate for the focus added by the mirror. The membrane tension has implications for both the total mirror stroke and the frequency response. To increase stroke, one needs a low-tension membrane according to equation (1). However, there is an important design trade-off involving membrane tension between stroke and fundamental frequency as given by Equation (5). It is important to notice that if the equation is written in terms of the fundamental material properties; density (ρ = σ/t) and stress (S=T/t), the membrane thickness cancels out. Thus, the fundamental frequency (f 0 ) is independent of membrane thickness. Figure 3 shows curves of f 0 versus membrane diameter (D) as a function of the membrane tension (T) as calculated using Equation (5). These curves clearly show that as membrane diameter is increased the membrane tension must also be increased to maintain mirror frequency response. Thus, when designing large format mirrors for large stoke, mirror frequency response may be sacrificed. Air resistance will tend to increase the fundamental frequency, but this effect may not be significant enough to effect a mirror design. We plan to study the vibrational modes of the membrane and the effects of air resistance in our future work.

5 Figure 3. Effect of membrane tension and mirror diameter on fundamental resonance frequency Cross-sections of IFs calculated for two different mirror designs using DMModel are shown in Figure 4. The plot labeled a) shows IFs of adjacent actuators from the middle to the edge of a 37-actuator hexagonal array. IFs for a 632- actuator mirror with a 75 mm diameter are shown in the plot labeled b). In each case, the center IF is perfectly symmetric and the adjacent actuators have IFs skewed based on their location in the array. a) b) Figure 4. Actuator Influence functions for a) 25 mm diameter DM model with 37 actuators in hexagonal array b) 75 mm diameter DM model with 632 actuators in rectangular array. Figure 5 provides a comparison of mirror surfaces modeled using DMModel IFs with a Zygo interferometer measurement of the actual mirror surface for the same applied voltages. The mirror surfaces shown represent Zernike trefoil and astimatism aberrations. Assuming the actuator forces add linearly, the modeled surface is obtained by summing the IF from each actuator weighted by the individual actuator voltages. The modeled surfaces in Figure 5 show reasonably good agreement with the measurements, indicating that our modeled IFs and the linear sum of IFs provides a good approximation to the membrane surface deformation for these aberrations. We plan to do similar comparisons for large diameter mirrors in future research efforts.

6 Figure 5. Modeled and measured mirror responses for Zernike Trefoil and Astimatism aberrations. 5.0 MEMS DM WAVEFRONT CONTROL Wavefront control has been accomplished with MEMS electrostatic DMs using varied approaches. 5-11,13-17 Primarily they either depend on WaveFront Sensor (WFS) measurements and least-squares correction estimates for the mirror surface 6-11,13-16 or on the optimization of an appropriate metric using a stochastic method 5,17 or dithering technique to determine appropriate actuator commands. The least-squares techniques generally use either zonal control (individual actuator influence functions) or modal control (Zernike influence functions) schemes. Intellite has built systems using both least-squares correction (modal control) and metric optimization techniques. Our metric optimization control system is depicted in Figure 6. Video Camera ND Filter* Outgoing Beam Beam Splitter* Sampled Beam 37-Actuator Deformable Mirror Focusing Lens* Ribbon Cables To Frame Grabber Turning Mirror* Clarifi Mirror Control Software Laser System* High Voltage Interface Box Clarifi Computer System * These items are provided by the user To Parallel Port D40DI 40-Channel High Voltage Mirror Driver Figure 6. Clarifi system configured to optimize far field spot metrics

7 Our wavefront control system modeling is accomplished using zonal control with the individual actuator IFs described in Sections 3 and 4. Slope IFs are produced for each actuator by calculating slopes (x and y) from the mirror surface IF for each actuator at specified WFS measurement locations as mapped to the mirror surface. Singular Value Decomposition (SVD) 16 is then performed on the Slope IF to determine the relative magnitude of the mirror spatial modes. Low magnitude modes may then be removed to improve mirror control. As we show in Section 6, the number of modes removed for some designs has a strong effect on mirror performance. Inversion of the slope IFs provides a mirror slope control matrix that calculates mirror actuator commands from wavefront slope measurements. Zernike polynomial terms are used to generate wavefront aberrations for correction. Wavefront slopes are calculated from this aberration at the WFS measurement locations across the mirror surface to provide a signal for calculating the mirror actuator commands using the slope control matrix. Actuator commands calculated in this manner will contain negative as well as positive values. Since electrostatic actuator pads can only pull on the membrane, these commands (voltages) must be biased to provide a minimum of zero volts in the actuator commands. This bias adds induced focus to the correction. For static correction, the actuator commands are applied to the mirror surface (OPD) IF to provide the mirror correction. Residual RMS WFE is minimized in a closed loop by adjusting an actuator gain factor. This process is illustrated by the matrix equation representations shown in Equations (6). C1 S1 n x m slope... = control matrix... C n S m S Slope measurements ( m) i C DM electrostatic pad commands ( n) i OPD1 C1 N x n OPD... = influence function. + Bias.. OPD N C n OPD Optical Phase Difference from DM i (6) Results from performing SVD on slope IFs for 37-actuator and 632-actuator mirror designs are shown in Figure 7. The large 75 mm mirror with 632 actuators shows a significant number of modes that will present control problems. a) b) Figure 7. Results of SVD on slope IFs for a) 25 mm DM with 37 actuators, b) 75 mm DM with 632 actuators

8 6.0 MEMS DM DESIGN PERFORMANCE EVALUATION DM designs developed using DMModel are evaluated in MatLab using a DM performance evaluation script known as DMZernError. Wavefront aberrations are formed from Zernike Polynomial terms as given by Noll. 20 DM design data and IFs are read from a MatLab format file and applied in a static correction loop to determine the performance of a DM design in correcting specific aberrations. Performance is evaluated by calculating residual RMS Phase error and far field Strehl intensity. The software also calculates two physical parameters that place limits on design performance. These include the force ratio that detects mirror snap-down and the maximum voltage limit imposed by the mirror electronics design. Total mirror Peak-Valley (PV) throw and the induced focus from correcting an aberration are also calculated. The force criterion for limiting snap-down with equal voltages on all actuator pads is given by: where: IFOPD - OPD influence function F lim = < all act all act sep Fk such that Max IFOPDFk (7) k k 3 The force ratio (F ratio ) is the ratio of the sum of all actuator forces applied for a correction to the value of F lim to determine if snap-down has been violated for a given correction. Figure 8 shows Zernike correction results in terms of residual RMS WFE and Strehl Intensity using four different mirror designs with varying numbers of actuator pads. The initial WFE for each Zernike (terms 5 through 22) was set to 0.25 microns RMS (1 2 microns PV) across the active portion of each mirror design (roughly 60% of the membrane diameter). Performance generally improves with increasing numbers of actuators (particularly for the higher order Zernike terms) up to the case with 384 actuator pads. Performance degrades for some of the Zernikes when using the 632-actuator design. In this case, the actuator pads are small enough to require large voltages for correction. Also, an insufficient number of mirror modes (see Figure 7. b)) were subtracted from the control matrix. This also increases voltage requirements and results in poor mirror control, particularly for higher order aberrations. D

9 a) b) c) d) Figure 8. Zernike correction performance with varying actuator density a) 37 hexagonal actuators b) 112 rectangular actuator pads c) 384 rectangular actuator pads d) 632 rectangular actuator pads Modeling results shown in Figure 9 illustrate the performance improvements that can be obtained by appropriate SVD mode subtraction in forming the mirror control matrix. Corrected Strehl Intensity data is plotted in Figure 10 a) as a function of the number of actuator pads from correcting a Zernike composite aberration typical of Large Membrane Mirrors (LMM) (x and y astigmatism, coma, and 3 rd and 5 th order spherical aberration). All of the mirror designs used a 120-micron membrane-to-actuator pad array separation to allow for the correction of large aberrations. The trend indicates improved performance with number of actuators. However, results vary for each actuator case. The spread of Strehl values obtained using 316, 384, and 632 actuators was obtained by varying the number of SVD modes subtracted for the mirror control. In this case, subtracting too few or too many modes can significantly reduce the Strehl values. The optimum number of removed modes for the 384 and 632 actuator designs was 50 and 100, respectively.. An optimum of 25 was found for the 316-actuator case. The images in Figure 10 b) illustrate the best Strehl correction obtained for the 632-actuator design when subtracting 100 SVD modes. This performance is near diffraction-limited for imaging applications. The images in Figure 10 c) and d) show the wavefront correction differences for 112 and 632 actuator designs.

10 a) b) c) d) Figure 9. a) Strehl variation with actuator density, aberration strength, and subtraction of SVD modes, b) Strehl improvement using 632 actuator mirror, c) Initial LMM WFE, DM correction and residual phase for a 112 actuator mirror without SVD mode subtraction, d) same data for the 632-actuator mirror when subtracting 100 SVD modes. 7.0 SUMMARY AND CONCLUSIONS Intellite, Inc. designs and fabricates MEMS electrostatic DMs and control hardware and software and conducts MEMS DM research for commercial and government applications. This work summarizes our approach to MEMS DM fabrication, electro-static DM theory and modeling, MEMS DM design concepts, and the wavefront control approach we use to analyze MEMS designs. Modeling and analysis results are presented that illustrate MEMS DM performance for different design and mirror control conditions. The analysis results illustrate the importance of subtracting low-magnitude mirror modes from the actuator influence functions prior to constructing a MEMS DM control matrix used for zonal least-squares mirror control. Some MEMS DM designs have many low-magnitude modes that will severely limit mirror performance in terms of requiring excessive actuator voltages to correct for large magnitude aberrations. Our analysis approach allows the designer to optimize the number of mirror modes to obtain the best aberration correction with a given mirror design. 8.0 ACKNOWLEDGEMENTS The work required for this manuscript was supported by the Air Force Research Laboratory s Directed Energy Directorate (under contracts F D-0238/D03 and FA C-0024) at Kirtland AFB, New Mexico.

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