Vol. 3, No. 11/November 1986/J. Opt. Soc. Am. A 1833
Configuring an electrostatic membrane mirror by least-squares fitting with analytically derived influence functions
E. Scott Claflin and Noah Bareket
Lockheed Palo Alto Research Laboratory, 3251 Hanover Street, Palo Alto, California 94304
Received December 9, 1985; accepted June 26, 1986
Electrostatic membrane mirrors are potentially useful as aberration generators. To realize this capability, it isnecessary to know the influence functions (each giving mirror response to a single electrode) and then to use a fittingprocedure to determine the optimal voltage settings for reproducing a desired surface shape. We show that anapproximate analytical solution of Poisson's equation exists that can generate the influence functions for a circularelectrostatic mirror. We also show, by demonstration, that it is computationally feasible to calculate theseinfluence functions and to use them with a fitting procedure to fit the surface of a 109-electrode mirror to a desiredshape. Our methods allow one to test the theoretical performance of the 109-electrode mirror; we find that good fitsare obtainable for Zernike polynomials of up to degree 6.
1. INTRODUCTION
Deformable mirrors, which modulate the phase of an opticalwave front, find application as the active elements of adap-tive optical systems and as aberration generators. For thelatter application it is often desirable to operate the mirrorin an open-loop mode, i.e., without feedback. A standardapproach exists for mirror configuration in open-loop opera-tion. The surface is represented by a linear combination ofinfluence functions (the deformations produced by individ-ual actuators1 ) and then is least-squares fitted to the desiredsurface.2 The accuracy of this approach depends on theaccuracy of the calculated (or measured) influence func-tions, on the method used to fit them to the desired deforma-tion, and on the spatial resolution and linearity of the mir-ror.
We apply this approach to a circular, electrostatic mem-brane mirror. The influence functions are derived from anapproximate analytical solution of Poisson's equation, thedifferential equation governing small-amplitude displace-ment of a membrane. Pressures are computed by using theinfluence functions to least-squares fit a desired surface at aselected set of surface matching points. Finally, voltagesare obtained from pressures by using the standard equationfor the force per unit area between plates of a capacitor.
These computational techniques are applied to the config-uration of a mirror constructed at the University of Heidel-berg.3 Since properties of the mirror affect several aspectsof our calculations, we describe the mirror's design in somedetail. Figure 1 shows the important components of themirror assembly. A thin membrane of aluminized polypro-pylene, the deformable mirror, is placed halfway between anarray of control electrodes and a transparent bias electrodecoated onto one side of a glass window. Membrane thick-ness is 0.5 ,um, electrode-to-mirror spacing is 75 am, andmirror radius is 2.5 cm. The function of the bias electrode isto provide a restoring force to permit mirror deflections inboth directions. Control electrodes are biased to the samepotential as the bias electrode, and the membrane is ground-ed. Figure 2 displays the pattern of the 109 hexagonal
control electrodes that lie just beneath the mirror and deter-mine its spatial resolving power.
A wide variety of desired surface functions might conceiv-ably be chosen for demonstrating the mirror configurationmethod. We wish also, however, to show the quality of fitthat is theoretically possible with our particular mirror whenreproducing the optically useful Zernike polynomials. Forthis reason the Zernike polynomials are selected as the de-sired surface forms.
In the following sections we present the theoretical basisof our method and application to the electrostatic mirror,including illustrative examples using the Zernike polynomi-als. Accuracy and limits of applicability of the method arealso discussed.
2. THEORY
Small displacements of a membrane under tension are gov-erned by Poisson's equation.4 5 Solutions of Poisson's equa-tion are linear in electrode pressures6 and may, therefore, bewritten in the form
Ne
(1)Zi = E A 'jx
j=1
where Aqj are coefficients derived from solutions of Pois-son's equation, zi is the displacement of the ith surface posi-tion, Pj is the pressure exerted over the surface of the jthelectrode, and Ne is the total number of electrodes. Dis-placement along the boundary is taken to be zero. Thecolumn vector obtained from .Ajj by fixing j and varying i iscalled an influence function,' or, to be more exact, the valuesof an influence function at surface positions labeled i = 1,... , N, where N, is the number of surface points. Eachinfluence function is the surface displacement caused byunit pressure from a single electrode and, according to Eq.(1), combines linearly with the other influence functions togive total surface displacement zi. Let A1 be the matrix withelements Ai. We call A the influence function matrix inrecognition of the role of its columns.
1834 J. Opt. Soc. Am. A/Vol. 3, No. 11/November 1986
GLASS WINDOW CONDUCT IN G A. Determination of the Influence Function MatrixLAYER For small displacements, the relation between surface dis-
IN SULATOR placement z and pressure P is described by Poisson's equa-MEMBRANE tion 45 :
v 2z =-PIT, (5)
where T is surface tension (force/distance) of the membrane.A displacement is small if the angle between a line tangent tothe membrane surface and the z = 0 plane is small and if thetension T is unaffected by surface deformation. The size ofthe surface displacement that violates the constant-tension
condition depends on the elastic properties of the mem-brane.
For convenience of calculation, the membrane radius isnormalized to unity and the boundary condition is taken to
SPACING ELECTRODE be z = 0 at r = 1, where r is radial distance. The solution toRI N GS I N SU LAT ION Poisson's equation in polar coordinates (r, 0) is then 8
where P(r, O) is the pressure applied to the membrane atCONNECTOR position (r, 0) and a is the actual membrane radius.
To permit integration in Eq. (6), each hexagonal controlelectrode can be approximated by an electrode of the same
5 CM 1.0
Fig. 1. Cross section of deformable membrane mirror. 3
The central problem of mirror configuration is to find thepressures Pj that best fit the membrane surface z to a desiredsurface z' at a selected set of surface-matching points. The 0.5quantity to be minimized is
Np
S =E (Zi - )2 . (2)i=1 Y 0.0
Equation (1) may be written in matrix form
Z=.P,(3)
where column vector z has components zi and column vector -0.5
P has components Pj. Solving Eq. (3) for P in the least-squares sense of minimizing S in Eq. (2), one finds thesolution 7
p = ( )t ,)-' At z' (4) -1.0-1.0 -0.5 0.0 0.5 1.0
where the superscript t denotes transpose. XThe remainder of this section deals with methods of deter- Fig. 2. Electrode configuration of deformable mirror. The actual
mining the matrix elements Aij from a solution of Poisson's mirror has narrow insulating gaps between electrodes. The gapsequation and with conversion of pressures to voltages. have been ignored in this simulation.
E. S. Claflin and N. Bareket
Vol. 3, No. 11/November 1986/J. Opt. Soc. Am. A 1835
1.0
0.5
Y 0.0
-0.5
-1.0-1.0 -0.5 0.0 0.5
X
1.0
Fig. 3. Approximated electrode shapes chosen to permit analyticalintegration of matrix elements.
area but with boundaries defined by lines of constant r and0. Each approximated electrode has a shape and positiondetermined by the ad hoc requirements that its radial extentbe equal to the length of the arc at its midradius and themidway point of its radial bisector be at the center of theoriginal hexagon that it replaces. Note that these simpleshape and positioning choices result in a small outward radi-al displacement of the electrode's centroid. The approxi-mated electrode shapes are shown in Fig. 3. Equation (6)can then be rewritten as
In writing Eq. (7) it is assumed that the pressure associat-ed with a given electrode position is caused only by thatelectrode, that is, that edge effects are negligible. Thisassumption is valid if the ratio of electrode width to elec-trode-mirror spacing is much larger than 1. For the mirrordescribed in this paper, the ratio is 100, which satisfies thecondition. It is also assumed that the pressure on the mem-brane area above the electrode does not vary as a function ofdisplacement. This assumption holds for our mirror, sincedisplacement variations over an electrode are much smallerthan the electrode-mirror spacing of 75 Aim.
Performing the integrations in Eq. (8) yields analyticalexpressions for the coefficients Aij. Different expressionsare needed depending on the values and relative sizes ofradial position ri of the ith surface point and radial limits rjand r2j of the jth electrode. Five cases arise:
Case : r = 0, rj = 0.
A = - Ai\.roj(rI - ln r2 )2
Case II: r = 0, rj > 0.
A = I Aj[r 2( - ln r21)
(9)
(10)
Case III: r2j < r.
Ai = 1 I Aj[-In ri][r 2 -r 2]/2 - 2 [r?- ]
7r (j1 Z1+ ,n 2(n + 2) [r-1
[hf+2 - hn+2J
2 N"Zi = T E Aij P,
j=1(7)
where zi is surface displacement at position (ri, Xi), P ispressure exerted by the jth electrode, and (a2/T) Aij = ij ofEq. (1). The unitless coefficients Aij are given by
Aij 0j[rn(11rj) r'dr' + r'dr'ln(1/r')]
-J r'dr' 2 [(rir')n - (r'/ri)n]n=1n
X [sin n(02j - 0j) - sin n(01 j - )]
r'dr' 2 2 [(rir')n - (ri/r')nJn=1n
X [sin n(02j - 0j) - sin n(lj - )]j (8)
where 4lj and 02j are the lower and upper limits, respective-ly, on the angular coordinate for the jth electrode andwhere Aqoj = 2j - 0j. All integrands are zero except whenthe variable of integration lies between the lower and upperradial limits of thejth electrode, i.e., when r1j <r' <r2j. Theinfinite sums in Eq. (8) are a generalized form of Clausen'sintegral and hence cannot be further simplified.
X [sin n(02j - 0j) - sin n(0Plj-hi)]
Case IV: ri < rj.
A = I {A j{r2j(1 - In r21 )
- r2 - n rij)]/2
Z +1 [ri (rn+2 -rnl+2)/(n + 2) + a]n=
(11)
X [sin n(02j - h) - sin n(0j - 0i)]} (12)
where
a = -ri(r2j - r1j) when n = 1,
a = -r?(ln r2j - In r1j) when n = 2,
a = r [(i n - ( i 2 /(n - 2) when n > 3.
Case V: r <ri <r 2j. Substitute ri for r2j in Eq. (11) andri for r1j in Eq. (12). Add the results.
E. S. Claflin and N. Bareket
- r2 1 -In rij1j( 2
1836 J. Opt. Soc. Am. A/Vol. 3, No. 11/November 1986 E. S. Claflin and N. Bareket
1.0B. Conversion of Pressure to VoltageThe conducting elements of a membrane mirror attract oneanother like the plates of an ideal capacitor, since, as men-tioned earlier, edge effects and displacement variations overan individual electrode are negligible. The resulting up-ward pressure P on the surface at a given point iS
4
20 V2 _ I2 d2 d2}
0.5
(13)
where EO is the permittivity of free space, d2 the distancefrom the membrane to the transparent bias electrode, V2 thevoltage on the transparent bias electrode, d the distancefrom the membrane to a control electrode, and V1 = V1B +AV the voltage on a control electrode, where V1B is the biasvoltage on all control electrodes and AŽV is the adjustablevoltage on a single control electrode. It follows from Eq.(13) that
y 0.0
-0.5
-1.0
(14)-1.0 -0.5 0.0 0.5 1.0
X
Distances di and d2 may be taken to be constant as long asmembrane displacements are small by comparison.
Fig. 4. Points used for least-squares fitting of surface.
2.0
3. APPLICATION OF THE METHOD
The mirror configuration method is illustrated by fitting themirror surface to selected functional forms. We choose Zer-nike polynomials9 "10 because they represent optical aberra-tions useful in evaluating the performance of an electrostaticmirror. These polynomials will take the place of the desiredsurface deformations z' in Eqs. (2) and (4).
The real Zernike polynomials of degree n and azimuthalfrequency m are defined by 9
Indices n and m are positive integers satisfying m < n and n- m even. The radial functions Rn (r) are normalized suchthat R'( = 1.
To fit the membrane surface to the Zernike polynomials,we choose to use approximately twice as many surface-matching points as electrodes. This choice gives good spa-tial resolution without being too burdensome for the com-puter. Figure 4 shows a set of 217 matching points chosenfor the 109-electrode mirror by an algorithm that takes asinput the approximate number of points desired and distrib-utes them with constant areal density. The outer edge ofthe area containing the matching points, the optically activeregion, is at radius 0.72. Choice of an optically active regionwith radius less than 1.0 is necessary to accommodate theZernike polynomials, which generally have nonzero or spa-tially varying values at their outer edge. The outer edge ofthe optically active region should be roughly one electrodewidth inside the boundary of the membrane. Placing it
I-
LU
LU
-Ja-IA
0
1.0
0.0
-1.00.0 0.5 1.0 1.5
RADIAL DISTANCE (CM)2.0 2.5
Fig. 5. Displacement of the membrane surface along a radial line at= 0 for Zernike polynomial U0, with an amplitude of 1 im.
2.0
2
I-
zLi
LU
-J
IA
1.0
0.0
-1.00.0 0.5 1.0 1.5
RADIAL DISTANCE (CM)2.0 2.5
Fig. 6. Displacement of the membrane surface along a radial line at0 = 0 for Zernike polynomial U0, with an amplitude of 0.109 m.
XXXXXXX XX
XX XX X XX X
X XX x X XXXX X XX X X X XXX XX
X X X X X X X X XX X X XXX X X XXX X X X XXXX X X X X X X X X X X X X X X
X X X X X X X X X X X X X X XX X X X X X XX X X X X X X X XX XX XXX X X X X X X X X
X X X X X X XX XX X X X XX XX X X X X X X X X X /
X XXX X X X X /X X XXX X
X XXXXX XX XX XXX XX
LEGEND ' 'ZERNIKE POLYNOMIAL N
< ZERNIKE BOUNDARY X
SIMULATED SURFACE X/
I -
/ \~~~~~~~~~~~~~~~~~
\~~~~~~I
LEGENDZERNIKE POLYNOMIAL
, ZERNIKE BOUNDARY _ESIMULATED SURFACE / N
/ I/I
I
. .
V22 2 1/2AV=dl - _ - P - V1B-2 fo
d2
Vol. 3, No. 11/November 1986/J. Opt. Soc. Am. A 1837
nearer the boundary causes a poor fit at the edge of theoptically active region and excessive voltages, while placingit farther from the boundary causes loss of resolution withinthe optically active region and large displacements outsidethat region.
The coefficients Aij, defined by Eqs. (9)-(12), were calcu-lated for the 217 surface-matching points and 109 electrodesby using a lower limit on series term size (before multiplica-tion by sine term) of 1 X 10-7. Values of the coefficients arein the range 10-2 to 10-5. Maximum number of terms re-quired for convergence is 174. CPU time on a VAX 11/780 is23 min. If one took advantage of symmetry both of theelectrode pattern and of the surface-matching points aboutthe x and y axes, then coefficients for only one mirror quad-rant would need to be calculated directly. It should beemphasized that the coefficients depend only on the elec-trode pattern and the set of surface-matching points and,therefore, do not need to be recalculated for each new sur-face function.
Electrode pressures and voltages were calculated for Zer-nike polynomials up to degree n = 6, and the rms error andpeak error of the fitted surfaces were calculated at thematching points. The fits were excellent but, as expected,the errors increase with n. Maximum error occurs for U6;rms error is 4% and peak error is 16% of the total displace-ment range of the Zernike surface.
Two Zernike polynomials were selected to illustrate thefitting procedure. They are
US = r 3 cos 3, (18)
UL = 20r6 -30r4 + 12r2 -1. (19)
Computed surface displacements for Zernike polynomialsU3 and U°6 along a radial line at X = 0 are shown in Figs. 5 and6. The difference between the desired and computed sur-face displacements is caused by the limitations in spatialresolution of both the finite number of electrodes and thefinite number of surface-matching points. In practice therewould be additional discrepancies, principally due to vari-able membrane tension and approximation of electrodeshapes, as discussed in Section 4.
Figure 7 shows the electrode pressure distribution re-quired for Zernike polynomial 0J3. Note that nonzero pres-sures are needed only around the periphery of the mem-brane. The electrode pressure distribution for US is shownin Fig. 8. The structure in the central region reflects thecomplex radial dependence of Eq. (19).
Figures 9 and 10 show the voltage distributions AV forZernike polynomials U andV. Parameters for these exam-ples have the values V1B = V2 = 100 V, d, = d2 = 75 gm, T =15 N/m, and a = 2.5 cm. The noticeable asymmetry aboutAV = 0 in Fig. 9 is caused by the nonlinearity of Eq. (14).
4. ACCURACY OF THE CONFIGUREDMIRROR SURFACE
Two general categories of error sources affect the accuracy ofthe configured mirror surface: (1) mirror resolution and (2)modeling and fitting error. Mirror resolution is determinedby the fixed electrode pattern of a mirror assembly. Theeffect of an electrode pattern on resolution can be seen in theinfluence functions for the 109-electrode mirror. Figure 11
Fig. 7. Membrane pressure distribution for the Zernike polynomi-al U3 (1-pm amplitude).
Z'I
U'
Fig. 8. Membrane pressure distribution for the Zernike polynomi-al U (0.109-jum amplitude).
shows influence functions along a radial line at = 0 result-ing from the individual actuation of six electrodes under thissame radial line. Surface functions requiring a higher spa-tial resolution cannot be accurately reproduced by this mir-ror.
Modeling error also affects surface accuracy. In practice,the single most important source of modeling error is expect-ed to be spatial variation and directional asymmetry inmembrane tension (T). The present model can compensatefor spatial variation, if known, by adjustment of the elec-trode pressures but has no provision for either directionalasymmetry or shear stress. Generalizing Eq. (5) for a non-uniform stress state is possible,"",12 but to utilize the equa-tions it would be necessary to devise experimental methodsfor measuring the stress state.
Error incurred in approximating electrode boundaries aslines of constant r and 0 may also be significant. One way ofmeasuring this error is to compare the modeled surface witha known surface deformation, in this case the deformationcaused by constant electrode pressure over the entire mem-brane. The exact known solution is compared in Fig. 12with the surface simulated by putting the same electrodepressure on all 109 electrodes but with the approximateelectrode shapes shown in Fig. 3. It can be seen that thediscrepancy is largest near the membrane center, where theelectrode shapes and centroids are least accurate.
Accuracy of the modeling could be improved by precisepositioning of centroids of approximated electrodes on hexa-gon centers or by breaking up each electrode into a numberof smaller regions. The latter procedure would better ap-proximate the actual electrode shape.
E. S. Claflin and N. Bareket
1838 J. Opt. Soc. Am. A/Vol. 3, No. 11/November 1986
31 0 -45 points of comparison for the least-squares fit may be chosen
to occupy only a portion of the membrane surface area. The32 7 0 0 -6 -48 shape of the remaining membrane surface is not known a
31 7 0 0 0 0 0 -6 -45 priori but will be determined by the fitting procedure. Byavoiding the mirror area near the outer rim, one can config-
0 0 0 0 0 0 0 0 0 0 ure the mirror without being restricted by the fixed bound-
ary condition inherent in a stretched membrane. This free--45 0 0 0 0 0 0 0 0 0 31 dom in fitting also implies that the fitted portion need not
-6 0 0 0 0 0 0 0 0 7 necessarily be circular but could be rectangular, for example.Another significant aspect of the method is that the coeffi-
-48. 0 0 0 0 0 0 0 0 0 32 cients Aij are computed from readily evaluated analyticalexpressions, permitting Eq. (7) to be used as a general-
-6 0 0 0 0 0 0 0 0 7 purpose simulation tool. More specifically, one can use Eq.
(7) to find displacements of any set of points on the mem-brane surface for any set of electrode pressures.
o 0 0 0 0 0 0 0 0 0
0 0 0 -6 -45
32 7 0 0 -6 -48
31 0 -45
Fig. 9. Electrode voltage distribution required to generate the dis-placements of Fig. 5 and the pressure distribution of Fig. 7. Unitsare volts. Values are rounded to the nearest volt.
0.14
38 12 38
0.1237 -2 -2 -2 -2 37
38 -2 -8 -3 -3 -3 -8 -2 38 a
12 -2 -3 0 1 1 0 -3 -2 12 ZLi
38 -2 -3 1 1 1 1 1 -3 -2 38 0-Jn-
-2 -3 1 1 0 0 1 1 -3 -2 0
037 -8 0 1 0 -1 0 1 0 -8 37
6. CONCLUSIONS
A method based on the use of analytically derived influencefunctions and least-squares fitting has been developed foropen-loop configuration of a circular electrostatic mem-brane mirror. The method allows surface functions havingvariable displacement at their outer edge to be fitted to an
0.10
0.06
0.06
0.04
0.02
-2 -3 1 1 0 0 1 1 -3 -2
38 -2 -3 1 1 1 1 1 -3 -2 38
12 -2 -3 0 1 1 0 -3 -2 12
38 -2 -8 -3 -3 -3 -8 -2 38
0.000.0 0.5 1.0 1.5 2.0 2.5
RADIAL DISTANCE (CM)Fig. 11. Surface response to actuation of a single electrode with apressure of 0.24 N/M2 . Curves show surface displacement fromeach of six electrodes along a radial line at = 0.
37 -2 -2 -2 -2 37
38 12 38
Fig. 10. Electrode voltage distribution required to generate thedisplacements of Fig. 6 and the pressure distribution of Fig. 8.Units are volts. Values are rounded to the nearest volt.
5. SUMMARY AND COMMENTS
The process of finding the best set of electrode voltages toconfigure a membrane mirror to a desired shape requiresfour steps: (1) Choose a set of points at which the surfacewill be matched, in the least-squares sense, with the desiredsurface; (2) use an analytical solution of Poisson's equationto find the influence function matrix; (3) solve the least-squares problem to find the optimal electrode pressures tocreate a desired surface function; and (4) convert pressuresto voltages.
An especially significant feature of this method is that the
2.5
2.0
zLi
-J0L(aa
1.5
1.0
0.5
0.00.0 0.5 1.0 1.5
RADIAL DISTANCE (CM)2.0 2.5
Fig. 12. Effect of approximated electrode shapes on surface shape.Curves show surface displacement along a radial line at 0 = 0 due toa constant pressure of 0.24 N/M2 applied by all electrodes.
31 7 0 0
E. S. Claflin and N. Bareket
Vol. 3, No. 11/November 1986/J. Opt. Soc. Am. A 1839
interior portion of the membrane, thus avoiding conflictwith the membrane's fixed outer boundary. The methodhas been successfully applied to optimize electrode pres-sures and voltages to fit Zernike polynomials and can beapplied to any other function, provided that the function'sspatial Variation is compatible with the resolution of theelectrode pattern. Use of the method to perform a simula-tion test on the University of Heidelberg mirror demon-strates that that mirror is capable of reproducing Zernikepolynomials up to n = 6 with good accuracy.
ACKNOWLEDGMENTS
We thank T. V. Huynh for many useful suggestions duringthe writing of this paper. We also thank C. A. Felippa, L. S.Weisstein, and D. L. Flaggs for helpful discussions concern-ing membrane mechanics.
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