Ultralightweight deformable mirrors · 2017. 2. 27. · relevant mirror technologies. Section 3 describes various aspects of the proposed mirror concept. Section 4 describes the mirror

Ultralightweight deformable mirrors

Keith Patterson* and Sergio PellegrinoGraduate Aerospace Laboratories, California Institute of Technology, 1200 E California Blvd.,

Pasadena, California 91125, USA

*Corresponding author: [email protected]

Received 25 April 2013; revised 27 June 2013; accepted 1 July 2013;posted 1 July 2013 (Doc. ID 189333); published 22 July 2013

This paper presents a concept for ultralightweight deformable mirrors, based on a thin substrate ofoptical surface quality, coated with continuous active layers that provide separate modes of actuationat different length scales. This concept eliminates any kind of stiff backing structure for the mirrorsurface and exploits microfabrication technologies to provide tight integration of the active materialsinto the mirror structure, to avoid actuator print-through effects. Proof-of-concept, 10 cm diametermirrors with an areal density of 0.6 kg∕m2 have been designed, built, and tested to measure theirshape-correction performance and verify the finite-element models used for design. The low-cost manu-facturing scheme involves low-temperature processing steps (below 140°C) to minimize residual stresses,does not require precision photolithography, and is therefore scalable to larger diameters depending onapplication requirements. © 2013 Optical Society of AmericaOCIS codes: (220.1000) Aberration compensation; (220.1080) Active or adaptive optics; (220.4610)

Optical fabrication; (230.2090) Electro-optical devices; (230.4040) Mirrors; (230.4170) Multilayers.http://dx.doi.org/10.1364/AO.52.005327

1. Introduction

Recent advances in mirror technologies have starteda transformation in the architecture of space-basedtelescopes. Compare, for example, the monolithicprimary mirror of the Hubble Space Telescope, witha diameter of 2.4 m and an areal density of≈183 kg∕m2 [1], to the segmented aperture of theJames Webb Space Telescope, with an overall diam-eter of 6.6 m and consisting of 18 lightweight beryl-lium mirrors with an areal density of ≈20 kg∕m2,each mounted on a set of mechanical actuators thatprovide rigid-body pointing and a single curvatureadjustment capability [2]. Two key advances thatmade this larger and much lighter aperture possiblewere the use of a folding architecture and the useof wavefront sensing and control of the mirrorsurface error.

Further advances in technology are expected toenable even larger telescopes [3]. A recent studyof large space apertures, sponsored by the Keck

Institute of Space Studies [4], put forward the conceptof forming large mosaic mirrors through on-orbitself-assembly of identical active mirror segmentsmounted on modular, low-cost spacecraft (mirror-craft). Building on recent developments in autono-mous self-assembly in space [5–7], the mirrorcraftwould dock and become mechanically connected toone another, and the mirrors’ shapes would then beadjusted to formasingle coherent surface.Anon-orbitdemonstrationof identical, active lightweightmirrorsmounted on CubeSats is currently under develop-ment [8]. The current state of the art in primary mir-rors is the active hybridmirror technology [9], with anareal density of ≈10 kg∕m2 and a wavefront correc-tion capability of the order of 20 μm, which sets thestandard for further developments.

The approach chosen in the present research, firstpresented in [10,11], eliminates any kind of stiffbacking structure for the mirror surface and exploitsmicrofabrication technologies to provide tight inte-gration of the active materials into the mirror struc-ture, to avoid actuator print-through effects. As wellas decreasing the mass of the mirror, the use ofthinner structures results in a larger wavefront

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control authority than previous schemes, which canenable the modular telescope architectures envis-aged above. Opening the design space to flexible, ul-tralightweight mirrors also has the effect of makingthem more prone to dynamic excitation, which thenraises potential issues in relation to acoustic loadingand shock during launch. These issues will be sepa-rately addressed in a forthcoming mission by meansof new launch packaging solutions for the mirrors [8].

This paper presents the first step in such a scalableultralightweight mirror technology that has beendemonstrated at the 10 cm diameter scale. The pro-posed mirrors are based on an active laminate thatcontains several actuation layers able to provide sep-arate modes of actuation at the different lengthscales.

The paper is organized as follows: Section 2provides a review of the current state of the art ofrelevant mirror technologies. Section 3 describesvarious aspects of the proposed mirror concept.Section 4 describes the mirror shape control method-ology. Section 5 presents a set of experiments carriedout on a prototype 10 cm diameter mirror, andSection 6 concludes the paper.

2. Background

There are several approaches for deforming mirrorsby means of actuators. These approaches can bebroadly divided into three categories, depending onthe geometric arrangement of the actuators with re-spect to the mirror surface: (i) normal, (ii) parallel,and (iii) along the boundary of the mirror. In case(i), known as surface-normal actuation, an array ofstack or piston actuators push and pull on the mirrorsurface to produce local bumps and dips. In case (ii),known as surface-parallel actuation, actuators at-tached to a mirror facesheet bend the mirror. Case(iii), boundary actuation, applies forces and/or tor-ques on the mirror rim to produce distortions of themirror interior. We expect that the future implemen-tation of lighter and highly active space telescopesystems will restrict the adoption of surface-normalactuation to smaller mirrors, typically the secondaryor tertiary mirror of a telescope, whereas surface-parallel actuation will become established as thedominant lightweight solution for larger mirrorssuch as primary segments. Boundary controlled mir-rors are limited in the range of interior deformationmodes that can be activated by the actuators.

Microelectromechanical-systems (MEMS)-basedmirrors are well developed and are already used ex-tensively in adaptive optics for ground telescopes,where they are used for correcting atmospheric tur-bulence, laser beaming, and other applications. Bothsurface normal and surface parallel actuationschemes have been developed. Deformable mirrorswith diameters up to a few centimeters, hundreds tothousands of actuator channels, and actuationstrokes of the order of several micrometers basedon MEMS technology are marketed by Boston Micro-machine Corp. and OKO Technologies [12,13]; silicon

membrane mirrors actuated by PZT (lead zirconiumtitanate) have been fabricated with silicon technol-ogy [14]. These types of devices are quite limitedin the size of their clear apertures, and the requiredfabrication processes are not suitable for scalingthese designs up to larger diameters.

Larger surface-normal-actuationmirrors producedby Xinetics consist of lead magnesium niobate (PMN)stacks attached to a mirror facesheet. The Xineticsstandard deformable mirrors have a diameter up to200 mm, maximum deflection range of 3–8 μm, and941 actuators at a spacing of 7 mm [15]. The adaptivesecondary mirrors on multiple mirror telescope,Magellan Baade Telescope, and large binocular tele-scope use hundreds of surface-normal actuators. [16]

Boundary actuation has been investigated in largethin, lightweight shell mirrors [17] and also insecondary telescope mirrors where radial cantileverbeams attached to the rim of a glass mirror were de-flected by means of PZT actuators that impose long-range deformation while minimizing local dimplingeffects for low-order modes [18].

Surface-parallel-actuation schemes have beenstudied extensively. Examples include PZT actuatorslaminated to glass or silicon substrates [19], piezo-electric strips or sheets bonded to the back of a thinshell [20–22], and schemes involving surface-paralleltransducers integrated into the mirror structure[23]. Off-the-shelf deformable mirrors marketed byCilas include a range of flat bimorph mirrors basedon a symmetric arrangement of two glass plates en-closing two patterned piezoceramic plates; these mir-rors have diameters up to 100 mm and 188 actuatorsthat can provide a maximum curvature of 0.07 m−1.Cilas also markets a range of unimorphmirrors, withdiameters up to 115 mm and 63 actuators, which canprovide a maximum defocusing correction of 20 μmpeak to valley. For the active hybrid mirror technol-ogy, mentioned in Section 1, the reflective surface isprovided by a 10–100 μm thick nanolaminate foil,bonded to a lightweight silicon carbide structure.Hundreds of electrostrictive actuators are embeddedwithin the structure to make adjustments to the sur-face figure, in the form of surface parallel actuation.Diameters up to 1 m have been demonstrated [9].

Extremely lightweight concepts have also beenproposed for making mirrors from laminated poly-mer films [24,25]. A review of processes aimed atminimizing surface roughness, thickness variation,and thermal expansion has been compiled by deBlonk et al. [26].

3. Mirror Concept

The objective of this study is to develop deformablemirror designs that are lightweight, scalable, in-expensive, and with a sufficiently large shape correc-tion dynamic range to allow the same base design tobe used in many or all parts of a segmented asphericmirror and/or to compensate for thermally induceddistortion and long-term material effects such ascreep and aging. These requirements are to ensure

5328 APPLIED OPTICS / Vol. 52, No. 22 / 1 August 2013

economies of scale and simpler architectures infuture large telescopes.

For lightness, the approach of choice is surface-parallel actuation. Specifically, the proposedapproach is to develop designs and fabrication proc-esses for laminated shell mirrors consisting of a stiff,thin layer with an optically smooth surface, bondedto one or more layers of active material. The activelayers are coated with electrodes patterned in away that provides optimal control of the mirrorfigure. The most aggressive approach for lightnesswould be to make the whole mirror structure outof active materials. This alternative was pursuedat the early stages of the present study, but was laterabandoned due to the difficulty of fabricatingpiezopolymer thin shells by replication of an opticalquality surface.

The high-stiffness substrate included in thepresent scheme has the advantage of providinghigher bending stiffness that aids shape retentionwithout complex mounting fixtures and providesan initial shape for the mirror that is close to thedesired optical figure. However, a disadvantage ofthis approach is that it decreases the dynamic rangefor shape adjustment.

A. Active Materials

A list of active materials suitable for the actuationlayer of a deformable mirror based on the presentconcept, in order of increasing maximum actuationstrain, is presented in Table 1. The field that is usedto control the actuation strain—electric, magnetic ortemperature—is listed in the second column of thetable, and, since precise control can be achieved mosteasily for electric voltage, our choice is focused on pie-zoelectric (both ceramic and polymer) and electro-strictive materials, and also dielectric elastomers.Two parameters that are used for the selection of ac-tuators [27,28] are the maximum actuation strain,i.e., the strain achieved by raising the control fieldto its highest safe value, and the blocked stress,i.e., the stress required to hold the material at zerostrain overall while raising the control field to itshighest safe value. The values of these parametersare listed in columns three and four.

Electrostrictives and piezoceramics were notpursued in the present research, although they do

provide large actuation stresses with good strain lin-earity, and low hysteresis in the case of some electro-strictives. Their brittleness and high-temperatureprocessing made it more desirable to use piezopoly-mers instead. The issues related to brittleness havebeen alleviated in the currently available macro-fiber-composite actuators based on piezoceramic fi-bers bonded with epoxy and prestressed in a polymerpackage [29]; however, these are discrete devicesthat would lead to potential print-through problemsand would also be difficult to integrate into a mirrorconcept that requires a large number of independentactuators. Thin coatings of ceramic actuator materialdo have potential for certain mirror applications, andare currently being developed by others. [30]

Piezoelectric polymers are a reasonable compro-mise between performance and ease of processing;hence they were selected for the present study. Theirprimary drawbacks are relatively low blocked stressand maximum actuation strain, but these issues canbe addressed by designing a laminate with a suffi-ciently flexible passive layer.More specifically, the ac-tive material used in the present research was thecopolymer of vinylidene fluoride and trifluoroethy-lene, P(VDF-TrFE). This material is a semicrystal-line, electroactive, thermoplastic fluoropolymer,which can be made piezoelectric by poling at roomtemperature. This process involves placing a largeelectric field through the material thickness in orderto align the electric dipole domains within thematerial. Note that P(VDF-TrFE) is similar to thehomopolymer, PVDF, but does not requiremechanicalstretching to achieve a piezoelectric crystallinestate [31].

An extensive study has been done by Sandia Na-tional Laboratories [31] on the performance of PVDFand its copolymers under conditions similar to lowEarth orbit and including: temperature variation,UV exposure, atomic oxygen, and other effects.Subject to certain restrictions on the allowable tem-perature range to prevent loss of piezoelectricity, ap-proximately −80°C to 90°C, and the need to shieldagainst atomic oxygen, the study concluded that thispolymer is suitable for space applications. It shouldalso be noted that an instrument on the NewHorizons mission to Pluto utilizes films of PVDFto count dust particle impacts on its journey toward

Table 1. Candidate Active Materials

TypeControlField Examples

Free Strain(%)

Blocked Stress(MPa) Notes

Piezopolymers Voltage PVDF, P(VDF-TrFE) 0.1 5 Easily controlled; low-temperatureprocessing

Electrostrictives Voltage PMN, lead magnesiumniobate-lead titanate, leadlanthanum zirconium titanate

0.1 100 Low hysteresis; requires bias voltage

Magnetostrictives Magnetic Terfenol-D 0.2 70 Require large magnetic fieldsPiezoceramics Voltage PZT, BaTO3 0.2 100 Easily controlled; high-temperature

processingShape memory alloys Temperature Nitinol, CuAlNi 5 200 Large strains; requires heating


the outer solar system by measuring currents inthe film [32].

B. Substrate Selection and Active Laminate Construction

The substrate should have high extensional stiffness,and yet its bending stiffness should be low to maxi-mize the range of curvature changes that can beachieved for any given active material. Hence thematerial used for the substrate should be stiff, manu-facturable in small thicknesses, and with low surfaceroughness. Alternately, it should be a material thatcan be polished to an optical quality finish and thatwill retain this finish in a space environment. Ideally,it would also have low density to reduce the overallmass of the mirror, have high thermal conductivity toprevent thermal gradients from distorting the shape,and be durable enough for handling and processingpurposes. The surface roughness desired for opticalwavelength mirrors is of the order of a few nano-meters or less. Table 2 provides a list of severalpotentially suitable materials.

Assuming that the actuation layer is much thinnerthan the substrate, the curvature change, κ, can beestimated with Stoney’s formula [33],

κ � 6εaMatat2sMs

� 6σatat2sMs

; (1)

where ϵa is the free strain and ts, ta and Ms, Ma arethe thicknesses and the biaxial moduli of the sub-strate and actuator, respectively. For an isotropicmaterial, the biaxial modulus is E∕1 − ν, where Eis Young’s modulus and ν is Poisson’s ratio. The (biax-ial) blocked stress, σa � ϵaMa, may be used in place ofthe actuation strain, if preferred.

In order to increase κ without changing the blockedstress, the substrate thickness should be reduced or asofter substrate be chosen. However, there are somepractical limits. First, when it was attempted to con-struct whole mirrors out of layers of P(VDF-TrFE), itwas found that the mirror would wrinkle (buckle)very easily unless it was held under a state of preten-sion. Second, high-modulus substrates that are toothin tend to bend due to gravity effects and becomedistorted due to fabrication stresses. Last, packagingsuch low-stiffness mirrors for launch is morechallenging.

All of these effects can be analyzed with standardtechniques, but a specific effect related to fabrication

should be mentioned. In cases where the activematerial is poled after deposition onto the substrate,a residual poling strain, ϵp, will remain due to thepermanent reorientation of the dipole domains. Ifthe poling strain is too large, this residual straincan cause the substrate to buckle into a cylindricalmode. The minimum (critical) thickness at the onsetof buckling of a circular plate of radius R is givenby [34,35]

tcrit ≈�1.05εpMataR2

Ms

�1∕3: (2)

Note that there could be other sources of residualstrain beyond poling, for example, coating processesand thermal cycling. All such effects can be treated ina similar manner.

Because of their ready availability, high thermalconductivity, and excellent surface quality (rough-ness of 1–5 nm), the substrates utilized to build mir-ror prototypes during the present study consisted ofthin (100–200 μm thick), 100 mm diameter single-crystal (1-0-0 orientation) silicon wafers, polishedon both sides. The piezopolymer layer was spin-coated onto the wafer. An organic solvent was usedto dissolve the P(VDF-TrFE) copolymer in powderform and create a resin with reasonably high viscos-ity. If the viscosity is too high, it is difficult to spin-coat evenly, whereas a low viscosity produces layersthat are too thin. This resin was poured onto the wa-fer surface, and the wafer was then rotated on a vac-uum chuck for a set time and spin rate to produce anearly uniform coating. The sample was then bakedon a hot plate to boil off the solvent and anneal thethermoplastic polymer. This process was repeatedmultiple times to build up a film layer of the requiredthickness. The reason for building each active layerout of several thin coats is to lessen the probability offorming pinhole defects, as well as to attain thickenough layers for actuation purposes.

Each piezoelectric layer was coated on either sidewith a conductive film formed by vacuum sputteringa metal such as aluminum, copper, or gold. As analternative to sputtering, thermal evaporation wasalso investigated, but the latter process may causethe active layer to heat up, which could introducedamage. Sputtering is preferable because it is aroom temperature process. To avoid oxidation, the

Table 2. Candidate Materials for Mirror Substrates

Material Type Notes

Si (single crystal) Ceramic Benefits from existing fabrication technologies; readily available;limited diameters but increasing over time

SiC Ceramic Very stiff; can be made to any sizeGlass (FS, BK7, borosilicate, Zerodur, etc.) Ceramic Traditional mirror material; new flexible electronics

display glasses could be suitableCarbon fiber composites Polymer Low thermal expansion; scalable using tape dispensing techniques;

surface roughness is open concernAl, Be Metal Ductile and easy to machine; low thermal stability; beryllium is toxic.


best metal is gold, but an intermediate layer of tita-nium or chromium is required in order to promoteadhesion to the adjacent layers. Much care wasneeded to develop a successful process and to avoiddefects such as pinholes, cracking, and delamination.Patterning of themetal film was done by covering thefilm with a shadow mask during sputtering; thisworks for simply connected geometries, but photoli-thographic techniques could be used for more compli-cated patterns.

After completing themirror stack, it was necessaryto change the state of the active material from para-electric into piezoelectric. The method that wasimplemented consisted in directly applying a highvoltage potential across the top and bottom electro-des of the whole stack, although corona poling meth-ods could be used as an alternative [31]. At fieldlevels of 50–100 MV∕m, the polymer used for thepresent study undergoes a transformation as localelectric dipoles within the material align themselvesto the external field. After removing the high voltage,the polymer is poled and thus ready to be actuatedwith lower voltages (approximately 25 MV∕m). Ifthere are pinhole or crack defects present in the lam-inate, the high voltage may cause arcing across thedefect and hence permanent damage. Typically, themetallization around the defect becomes vaporizedas a result of local heating at the arc location, andthe momentary electrical short is then reopened. Ac-tually, the laminate is still largely functional afterthis occurrence, albeit with a small defective spot.

Measurements taken during an example polingprocess on a laminate consisting of a single 20 μm

thick P(VDF-TrFE) copolymer layer over a 200 μmthick, 100 mm diameter wafer are shown in Fig. 1.The electrode covered only the central 80 mm diam-eter of the mirror, to prevent electrical arcing aroundthe edge. A digital image correlation setup was usedto measure the deflection of the center of the mirrorrelative to its edge as well as the average curvatureacross the mirror. In Fig. 1(a) note the transitionfrom a horizontal slope around 0 V to a strongly pos-itive slope at the end of the poling process. Also notethe significant residual strain, ϵp, and overall deflec-tion due to poling; this residual strain is larger thanthe typical linear actuation strain and must be ac-counted for in themirror design. Luckily, the residualstrain is nearly uniform, and hence results in anearly axially symmetric deformation (defocus) ofthe laminate. Because P(VDF-TrFE) is a thermoplas-tic, one possible way of countering the poling stress isto anneal this layer. Figure 2 shows the behavior ofthe same laminate after it had been poled. Cycles of−500 to �500 V were applied and resulted in anearly linear behavior with some observable hystere-sis. The measured cycles were fairly repeatable,although the relatively low resolution of the mea-surements prevents us from reaching quantitativeconclusions about potential accuracy of this laminateused as part of a deformable mirror.

Estimates of the curvatures that can be achievedby using a uniformly actuated, single layer of P(VDF-TrFE) with a thickness of 20 μm on substratesof arbitrary thickness have been obtained by usingEq. (1); the results are plotted in Fig. 3. The theoreti-cal lower limit imposed by buckling into a cylindrical

0 500 1000 1500 2000 25000

0.05

0.1

0.15

0.2

Voltage (V)

Dis

plac

emen

t (m

m)

0 500 1000 1500 2000 25000

1

2

3

x 10−3

Voltage (V)

Pol

ymer

Str

ain

(a) (b)

Fig. 1. Deformation of uniformly coated disk consisting of 20 μm P(VDF-TrFE) copolymer on a 200 μm thick silicon wafer, during polingcycle. (a) Peak-to-valley deflection of mirror and (b) estimated piezoelectric strain in the polymer layer fromEq. (1), showing residual polingstrain.

−500 0 500−0.01

−0.005

0

0.005

0.01

Voltage (V)

Dis

plac

emen

t (m

m)

−500 0 500

−200

0

200

Voltage (V)

Pol

ymer

Str

ain

(µst

rain

)

(a) (b)

Fig. 2. Cyclical actuation of uniformly coated disk consisting of 20 μmP(VDF-TrFE) copolymer on 200 μm thick silicon wafer, after polingprocess. (a) Peak-to-valley deflection and (b) estimated piezostrain in the polymer layer, from Eq. (1).


mode, predicted by Eq. (2), bounds the practical de-sign space. Using this type of plot, an appropriatesubstrate material and thickness could be chosenbased on a required actuation curvature and for achosen active material.

Note that the poling fields have to be appliedacross the entire active layer, not only the patternedelectrode regions, to prevent print-through of theelectrode pattern into the mirror figure, due to polingstresses. In the present case this was achieved by de-positing a thin, uniform coating of metal across theentire surface of the laminate, on top of the patternedelectrodes, to act as a temporary poling electrode.A quick plasma etch after poling was then used toremove the uniform metal surface while leaving be-hind only the thicker, patterned metal underneath. Ifthere are multiple active layers, the entire stack canbe poled at once.

A summary of the layer thicknesses and processingsteps for a complete mirror laminate is presented inTable 3, where layers 5–8 can be repeated to providegreater shape control; see Subsection 3.C. Note thatfor curved mirrors the substrate requires additionalprocessing (see Subsection 3.D), and also note thatthe reflective layer thickness can be increased toachieve thermal balance (see Subsection 3.E).

C. Electrode Patterns

To minimize print-through effects on the front (mir-ror) surface of the laminate, the active material layeris coated uniformly over the entire substrate, andthen patterned electrodes are deposited on top. Thisprovides a range of individually addressable actuatorregions. See Fig. 4(a) for an example design with aglass substrate and annular electrode pattern. Com-plementary sets of actuation modes can be created bystacking multiple active layers within the laminate,each with its own set of unique electrode patterns.For example, the design in Fig. 4(b) shows large an-nular electrodes to provide broad, low-order correc-tions with relatively large stroke capability, whilea high-density lattice of smaller electrodes in anupper active layer provides localized corrections atshorter length scales, albeit with lower strokecapability. See Fig. 5 for example influence functionsgenerated by selected electrodes from the low- andhigh-density patterns of Fig. 4(b).

The topology and geometry of the high-densityelectrode pattern were selected from a trade studythat considered several different patterns. Figure 6shows six designs grouped into two families of pat-terns with varying actuator densities. The substrateis 200 μm thick silicon, and the actuation layer is acontinuous 20 μm layer of P(VDF-TrFE). The firstpattern type, Figs. 6(a)–6(c), consists of rectangularstrips arranged in a triangular lattice similar tothat shown in Fig. 4. The second pattern type,Figs. 6(d)–6(f), is a hexagonal tessellation.

A finite-element model was set up for each of thesepatterns to compare their performance in making ar-bitrary corrections. The model was constructed in thesoftware package Abaqus using S3T and S4T ther-moelastic shell elements. Thermal expansion wasused to simulate the piezoelectric effect, with thetemperature field used as a substitute for the electricfield, and the thermal expansion coefficient replacingthe d31 piezoelectric coefficient. The value of d31 inthe model was scaled so as to make a temperaturechange of 1 K equivalent to the application of 1 V.This model was used to calculate the influence modesof each individual actuator (in units of m/V), by com-puting the mirror deflections due to a unit tempera-ture change. The analysis assumed small deflections,and the material properties of the substrate and ac-tive layer were as given in Table 4. The results were

Fig. 3. Estimated actuation capability of 20 μm thick P(VDF-TrFE) coating as a function of substrate biaxial modulus and thick-ness. A practical lower limit is imposed by substrate buckling dueto residual poling strain.

Table 3. Summary of Mirror Layers

Position Mirror Layer Material Thickness (μm) Fabrication Order Fabrication Method

1 Reflective surface Al or Au 0.1 2 Sputtering/evaporation2 Substrate Si or Glass 200 1 COTS wafers3 Adhesion layer Ti 0.01 3 Sputtering/evaporation4 Ground electrode Au 0.1 4 Sputtering/evaporation5 Adhesion layer Ti 0.01 5 Sputtering/evaporation6 Piezo layer P(VDF-TrFE) 20 6 Spin coating7 Adhesion layer Ti 0.01 7 Sputtering/evaporation8 Electrodes Au 0.1 8 Blanket sputtering � ion mill pattern etch back


then postprocessed in MATLAB to determine theperformance of the mirror.

Shape errors in circular apertures are analyzed interms of Zernike modes [36], i.e., a set of orthogonalpolynomials defined over the unit disk. For eachZernike mode, all mirror designs were evaluatedfor how well they can correct an error in the shapeof the chosen mode; the maximum RMS amplitudethat can be corrected prior to saturating any of theactuators was also evaluated. The mathematical de-tails are presented in Subsection 4.A, but the mainresults of this study are presented here.

Figure 7 shows the correctability for each of thefirst 30 Zernike modes, for the 6 high-density elec-trode patterns presented in Fig. 6. Here, correctabil-ity is defined as the ratio of input RMS error to theoutput (corrected) residual RMS error, without con-sidering the effects of actuator saturation.

For each Zernike mode, as the amplitude of theerror increases, more andmore of the actuation chan-nels reach saturation at 500 V. Hence, the relation-ship between the amplitude of the input RMS errorin the chosen mode and the output (corrected)residual RMS error, shown schematically in Fig. 8,is nonlinear as more and more actuators become sa-turated. This becomes linear again with a slope equalto 1 when all actuators are saturated. The actualpredictions for three important low-order modes (de-focus, astigmatism, and coma) are shown in Fig. 9.Note that after partial saturation the Hex43 residualRMS is lower for coma than the Hex91, owing to theorientation of the mode with respect to the pattern ofthe electrodes.

The general trend is that, for any chosen pattern,increasing the actuator density can improve both theshape correction accuracy and available stroke, butthis comes at the cost of increasing the complexity

Electrodepattern

Glasssubstrate

P(VDF-TrFE)active layer

Reflectivecoating

Groundplane

(a)

Fineelectrodes

Coarseelectrodes

Siliconsubstrate

First activelayer

Secondactivelayer

Reflectivecoating

Groundplane

(b)

Fig. 4. Exploded views of example mirror layers: (a) single activelayer and (b) double active layer.

0 0.04 -0.001(a) (b)

0.003

Fig. 5. Finite-element predictions of the influence of functionshapes for selected electrodes from (a) coarse and (b) fine patternsof Fig. 4. Units are μm/V.

(a)

(d)

(c)(b)

(e) (f)

Fig. 6. Electrode design patterns: (a)–(c) triangular lattices with42, 90, and 156 actuators; (d)–(f) hexagonal patch tessellationswith 43, 91, and 151 actuators.

Table 4. Material Properties

Material Property Value

Biaxial modulus M 180 GPaSi Coefficient of thermal

expansion α2.6 ppm∕K

Elastic modulus E 1.5 GPaPoisson’s ratio ν 0.34

P(VDF-TRFE) Biaxial modulus M 2.3 GPaCoefficient of thermalexpansion α

200 ppm∕K (est.)

Piezoelectric coefficient d31 16 pm∕VBiaxial modulus M 120 GPa

Al Coefficient of thermalexpansion α

23 ppm∕K


of the driving electronics. The choice of the electrodepattern is also important. Figure 6 shows that thetriangular lattice pattern significantly outperformsthe hexagonal pattern in terms of modal correctabil-ity, which can be explained by noting that placing theactuating strips in different orientations providesbetter control on the bending of the substrate in ar-bitrary directions. This aids in correcting shapes thatrequire nonaxisymmetric deformations. Figure 9shows that the hexagonal pattern is able to achievelarger strokes before saturation, which can be ex-plained by noting that the hexagonal pattern pro-vides a higher coverage of the surface and hencethe total available actuation moment is higher. Itwould appear that a combination of these two pat-terns, where the unutilized area in the lattice pat-terns is filled in with triangular and hexagonalpatches, may provide a good compromise betweenstroke and correctability. Clearly there is scope forfuture optimization of the electrode pattern.

D. Curved Mirrors

Glass or silicon wafers are excellent substrates forthe mirrors; however they are usually manufacturedto be nominally flat. In order to use deformable

mirrors in a primary aperture, it is desirable to intro-duce a base curvature to decrease the demand on theactuators. In the case of a low-cost telescope with asegmented primary mirror, identical spherical activesegments with curvature equal to the average overallcurvature of the aperture would be required [10].

One way to introduce a permanent curvature intoa nominally flat substrate is by applying a stressedcoating to the substrate, e.g., by depositing coatingswhile either cooling or heating the substrate. Thisworks well up to a limit defined by Eq. (2), after

101

102

Cor

rect

abili

ty

Zernike Mode Indices

0 0

−1 1

1 1

0 2

−2 2

2 2

−1 3

1 3

−3 3

3 3

0 4

−2 4

2 4

−4 4

4 4

−1 5

1 5

−3 5

3 5

−5 5

5 5

0 6

−2 6

2 6

−4 6

4 6

−6 6

6 6

−1 7

1 7

Tri lattice, 40Tri lattice, 90Tri lattice, 156Hex patches, 41Hex patches, 91Hex patches, 151

Azimuthal:Radial:

Fig. 7. Correctabilities of six actuation patterns in Fig. 6, for first 30 Zernike modes.

0 1 2 30

0.5

1

1.5

2

2.5

3

Input Modal RMS

Res

idua

l RM

S

No co

rrecti

on (i

nput

= o

utpu

t)

Unsaturated Partly saturated

Fully s

atur

ated

Fig. 8. General relationship between residual and input error inany chosen mode.

0 2 4 6 8 100

0.5

1

Res

idua

l RM

S (

wav

es)

0 2 4 6 8 100

0.5

1

Res

idua

l RM

S (

wav

es)

0 2 4 6 8 100

0.5

1

Input RMS (waves)

Res

idua

l RM

S (

wav

es)

Tri42Tri90Tri156Hex43Hex91Hex151

(a)

(b)

(c)

Fig. 9. Variation of residual RMS error with RMS error in threechosen modes for the six actuation patterns in Fig. 6. (a) ModeZ02 defocus, (b) mode Z−2

2 astimagtism, and (c) mode Z−13 coma.

The reference wavelength is 633 nm.


which the substrate will buckle from a spherical intoa cylindrical shape [34]. To go beyond this limit, aboundary constraint could be used to hold the axi-symmetric shape at higher levels of stress; howeverthis comes at the cost of a reduced actuation range.

An alternative approach, suitable for glass sub-strates, is to support the substrate on a mold andraise the temperature to the glass transition temper-ature of the glass. This process, known as slumping,will cause the substrate to take the curvature of themold (without sacrificing the optically smooth sur-face finish of the original wafer). It was used forthe mirrors of the NuSTAR x-ray telescope [37]. Ithas been demonstrated that under carefully con-trolled conditions silicon can be slumped as well,at around 900°C [38].

E. Reflective Coating

Polished silicon is reflective in the visible spectrumand its reflectivity is adequate for laboratory testing,but mirrors with high-throughput requirementsneed to be vacuum coated with a thin aluminumor silver film for visible light, or gold for infrared.A layer of a transparent ceramic oxide can be coatedover the reflective film in order to protect these softmetals from damage and to prevent oxidation in thecase of aluminum or silver. Depending on the coatingthicknesses, care must be taken that residual coatingstresses do not affect the thin substrate’s figure, or doso in a desired, predetermined manner.

F. Thermal Balancing

A laminate made of layers of different materials will,in general, bend when it is subjected to bulk temper-ature changes. These effects can be analyzed andcounteracted as discussed in this section.

Consider a laminate consisting of a substrate andany number of layers. Let αs and αi be the coefficientsof thermal expansion of the substrate and the addi-tional layers, respectively. It is assumed that all ofthe layers are much thinner than the substrate,and hence a simple estimate for the bending curva-ture resulting from a temperature change, ΔT, canbe constructed by substituting into Eq. (1) the ther-mal strain (relative to the substrate thermal strain),ϵi − ϵs � �αi − αs�ΔT. Allowing for layers attached toboth the top and the bottom of the substrate, theoverall curvature can be obtained by superimposingtheir individual effects:

κ �Xi

6si�εi − εs�Mitit2sMs

�Xi

6si�αi − αs�ΔTMitit2sMs

� 6ΔTt2sMs

Xi

si�αi − αs�Miti; (3)

where si � �1 for a layer on top of the substrate andsi � −1 for a layer on the bottom.

Thermal bending can be prevented by means of ad-ditional coating layers that balance the laminatethermal stresses. To do this, set the curvature to zero

and add additional layers (either on top or bottom)with appropriate thicknesses until the overall ther-mal curvature is zero:

0 �Xi

si�αi − αs�Miti: (4)

For example, a 100 μm thick silicon substratecoated with a 20 μm thick layer of PVDF can be bal-anced by means of a 3 μm thick aluminum coatinglayer on the front, and buckling can be avoided bycontrolling the substrate temperature during deposi-tion, to minimize the coating stress. The additionalcoating thickness required is small because of thehigher stiffness of the metal coating relative to thepolymer, and hence there is only a small increasein the bending stiffness of the mirror. Thereforethe corresponding reduction in actuation capabilityis small. Since a metallic coating is required anywayfor reflectivity, then the thermal balance issue cansimply be addressed by ensuring that the reflectivecoating has the thickness required to balance out theactuation layers.

G. Kinematic Mounting

The surface parallel actuation scheme adopted in thepresent study is well suited for a kinematic mountingscheme, which mechanically constrains the six de-grees of freedom (three translations, three rotations)of the mirror without adding any additional con-straints that may put the mirror in an unintendedstate of self-stress. This approach allows the flexibleoptic to deform itself into any desired shape and tohold its shape without being affected by the behaviorof the mount. However, great care is needed in de-signing a kinematic mount for an ultrathin plateor shell. For example, if mechanical flexures wereemployed to hold the mirror, any deformation intro-duced into the mirror by the flexures should have anamplitude of the order of optical wavelengths. Hence,the flexures would need to be designed to be so soft asto be impractical.

A simple and effective scheme to hold the mirrorhas been devised that takes advantage of the largethrough-thickness stiffness of the mirror. Note thatthe mirror can be pinched without significant deflec-tions by applying equal and opposite forces on thefront and back surfaces. The pinching forces needto be as close as possible to collinear, as any misalign-ment introduces shear and bending loads, whichwould induce significant deflections. Coalignmentcould be achieved with machined parts that havetight tolerances, but a simpler approach is to useself-aligning magnetic forces.

Figure 10 shows an implementation of thisscheme. The mirror is attached to a printed circuitboard (PCB) at three points, and at each of thesepoints two polished, magnetized steel spheres pinchthe mirror and hold it in place. The top sphere is un-constrained, and hence the magnetic field automati-cally aligns the upper and lower contact points.


Additional constraints would need to be added to re-strain the remaining three in-plane degrees of free-dom (two translations, one rotation), but, for thepurpose of laboratory testing, friction at the pinchedpoints is sufficient to hold the mirror in place.

Once the mirror is mounted on the PCB in thisway, electrical connections are then made betweenthe PCB and the mirror using very thin, prebuckledwires made of soft, ductile metal such as gold or alu-minum, bonded to the back of the mirror. The proper-ties of these wires are such that any forces ormoments applied to the mirror laminate are suffi-ciently small to leave the mirror figure unaffected.Resistive heating due to the narrow wire cross sec-tion is not a concern because the wires carry onlyvery short duration currents of microampere to milli-ampere levels. Figure 10 shows a schematic diagramof this connection scheme.

4. Shape Control

The RMS surface error (deviation from nominalshape) is a simple scalar measure of the shape-related performance of a mirror that is particularlyconvenient for comparison purposes; it will be uti-lized as a criterion for mirror design [39]. It shouldbe noted that the high-spatial-frequency componentsof the RMS error will be governed by the mirror sur-face roughness, which is related to manufacturingtechniques and processes that cannot be addressedwith shape correction. On the other hand, minimiza-tion of the low- to mid-frequency components of theRMS error may be achieved through the use of a suf-ficient number of actuators that bend the mirror intothe desired shape. These effects are addressed in thissection, together with the numerical techniques formirror surface control.

A. Influence Functions

Consider m sampling points (nodes) distributed onthe surface of a general mirror surface, and an asso-ciated control system with n actuators. Associatedwith the ith actuator is a column vector, ai ∈ Rm,i � 1…n, obtained from the nodal deflections of themirror due to a unit input (e.g., 1 V) to the ith actua-tor, while all other actuators are turned off. This col-umn vector is known as the influence vector ofactuator i, since it determines the influence thatthe actuator has on the mirror surface. It is linearly

independent from the other n − 1 vectors, corre-sponding to the other actuators. Example influencefunctions are shown in Fig. 5. The influence vectorsare assembled into the influence matrix, A:

A � � a1 a2 … an � ∈ Rm×n: (5)

It is assumed that all deviations from the initialsurface shape are small with respect to the diameterof the mirror. This assumption allows linear combi-nations of the influence vectors to be used to predictthe mirror deflections. Hence, the influence matrixcan be used to transform a control vector, u ∈ Rn,consisting of the actuator input values, into a shapedeflection vector, δ ∈ Rm, which contains the deflec-tion of all nodal points of themirror. Thus, the controlvector and shape deflection vector are related via theinfluence matrix by

Au � δ: (6)

The correction of the mirror from its current shape,s1 ∈ Rm, to a desired shape, s2 ∈ Rm, requires a de-flection δ � s2 − s1. This deflection vector will, in gen-eral, not belong to the range space of A. Therefore,the appropriate control vector is obtained from theleast squares solution of Eq. (6).

For generality, the nodal deflections are weightedby appropriate surface areas, Si, to make the shapecontrol formulation independent of meshing or sam-pling nonuniformities. In the present study, the val-ues to Si were found by calculating the Voronoi area[40] surrounding each node. These area weights arearranged along the diagonal of a matrix, W ∈ Rm×m,and Eq. (6) is then modified to

WAu � Wδ: (7)

The weighted, least-squares solution of Eq. (7) canbe calculated by using the QR factorization or othermethods, and software packages such as MATLABhave built in functionality to compute these solutionsefficiently. If the available actuator inputs are con-strained to a certain range, then a constrained,weighted, linear least squares solution would be re-quired in order to find the optimal u.

Once the solution u has been determined, the dif-ference between the approximation and the originalis the residual vector or residual shape error,r � Au − δ ∈ Rm, or, accounting for the weights inthe residual,

r̂ � WAu −Wδ ∈ Rm: (8)

For convenience, the weights inW can be redefinedas the square roots of Si nondimensionalized by thetotal mirror surface area. Thus, the 2-norm of r̂(usually with piston, tip, and tilt removed) is thenequivalent to the RMS surface error:

Thin mirrorSteel spheres

PCB

MagnetThin wire(Au or Al)

Electrode

PCB access hole

PCB padWirebond

Fig. 10. Cross section of magnetic mounting scheme. Pairs of pol-ished, magnetized spheres pinch the mirror at three places aroundthe rim to hold it on a PCB. Tiny wire-bonded electrical connec-tions jump from the board to the mirror.


∥r̂∥2 ��r̂T r̂

p�

��Au − δ�TW2�Au − δ�

q(9)

≡

��Pir

2i SiP

i Si

s. �10�

B. Control Implementation

To be consistent with the overall approach of mini-mizing the mass of the mirror, the mass and volumeof the control electronics should also be as low aspossible. Rather than attaching a high-voltage am-plifier to each channel, a single amplifier is usedto control all channels, following the scheme shownin Fig. 11(a). The mirror shape is measured by awavefront sensor, which then passes the informationto a controller.

Themirror controller is based on a design proposedby Song et al. [41]. It consists of a microcontrollerwhose analog output is amplified to the range−500 to �500 V by a single high-voltage amplifier,and then multiplexed into the individual actuatorchannels that electrically act as capacitors. The con-troller cycles through each actuator and sets thechannel voltage. The voltage level is then held withsome minor leakage until the next refresh cycle. Thisallows control over a large number of actuators byusing a single controller board and amplifier boardat the expense of control frequency bandwidth,which is ideal for low-frequency disturbances(thermal, creep, etc.). The multiplexing concept isillustrated in Fig. 12, and a photograph of a proto-type 42-channel multiplexer is shown in Fig. 11(b).

This controller operates around 1 Hz. The mirrorcould potentially be operated at higher frequencies,with different electronics; however the implicationsin terms of heating of the mirror should then beevaluated.

5. Experiments

A. Mirror Prototypes

Although several prototype mirrors were con-structed, the results presented in this section wereall obtained from the same prototype mirror. Thismirror was based on an active laminate consistingof a 200 μm thick silicon wafer and a single 20 μmthick layer of P(VDF-TrFE) coated with segmentedannular electrodes, Fig. 13(a). An alternative, finerelectrode pattern is also shown in Fig. 13(b).Figure 13(c) shows the whole mirror assemblymounted on nanopositioners that provide piston,tip, and tilt adjustment, and Fig. 13(d) shows thefront of the mirror surface with a 100 nm reflectivegold coating. Note that this mirror prototype was notdesigned to be thermally balanced, and so even smallchanges in the laboratory thermal environment canbe expected to affect any measurements.

WavefrontSensor

data

signal out

D/Aconverter

High voltageamplifier

ground

Pas

sive

sub

stra

te

Act

ive

laye

r

solid

sta

te s

witc

hes

Opt

ical

ly is

olat

ed

Mul

tiple

xer

Micro-controller

Mirror

wavefront

chan

nel

addr

ess

(a) (b)

Fig. 11. (a) Schematic diagram of control system showing wavefront feedback from the mirror passed to a controller; the controller thenuses a high-voltage amplifier and multiplexer to apply a new set of voltages to the mirror electrodes. (b) Photograph of a 10 cm × 10 cmprototype high-voltage multiplexer board populated with solid-state switches is shown on the right; it is capable of running 42 channels to�500 V.

Time

Vol

tage

InputCh. 1Ch. 2Ch. 3Ch. 4

Ch.1 switch openCh.1 switch closed

Ch.3 switch openCh.3 switch closed

Fig. 12. Traces showing time-varying high-voltage input (heavyblack) and multiple actuator channels (light colored) that shows asteady quasi-DC level. Channel decay between refresh points isexaggerated for illustration purposes.


B. Optical Measurement Setup

In order to measure the shape of the mirror, an opti-cal testbed based around a ThorLabs WFS150–7ARShack–Hartmann wavefront sensor was used. Thesensor utilizes an array of lenslets to form an arrayof spots on an image sensor; the deviation of the spotsfrom a perfect grid is proportional to the local slopeerror in the wavefront. Figure 14 shows a diagram ofthe experimental setup. It consists of a 633 nm laserbeam filtered with a pinhole, collimated, reflected offthe deformable mirror under test, and then passed tothe wavefront sensor by means of a beam splitter andlens. This arrangement was chosen so as to reimagethe mirror pupil to a smaller size that will fit insidethe sensor aperture. With a good alignment of allcomponents, the wavefront sensor provides a meas-urement of the surface figure of the mirror. TheShack–Hartmann sensor had been calibrated by themanufacturer. A thick, flat mirror was used to alignthe setup; the deformable mirror was substituted in,and the Shack–Hartmann sensor was moved inpiston in order to zero the defocus and capture the

image of the pupil (mirror). In other words, the shapemeasurement was relative to the closest sphere. Themeasurement area of the mirror was constrainedby the smaller clear aperture of the 75 mm diameterobjective lens.

C. Influence Functions

Figure 15(a) shows the measured, individual influ-ence functions from the various channels. Thechannel numbers are defined in Fig. 16. These mea-surements were obtained by taking the difference inshape between a reference measurement with allchannels off and a new measurement with a singlechannel turned on and set to 400 V. Figure 15(b)shows the corresponding predictions obtained fromthe finite-element model described in Section 3.C.Visually, there is a reasonable match between mea-surements and predictions, but quantitatively thediscrepancies are such that experimentally based in-fluence functions are required for accurate shapecontrol. The discrepancies are likely due to beammisalignments as well as thermally induced disturb-ances of the unbalanced laminate. For example,channel 9 shows evidence of astigmatism in the ex-perimental measurement, due to misalignments.

D. Focus Control

This test consisted in using all 16 channels in themirror to control a single mode. The mode chosenwas defocus, i.e., the axisymmetric base curvaturecomponent of the mirror surface. A simple, pro-portional derivative feedback controller wasimplemented with nonoptimized gains and, for sim-plicity, the same voltage value was assigned to allchannels. An experiment was carried out in whicha step defocus change of two waves with a long holdwas requested; the step response of the controlledmirror is shown in Fig. 17, together with the appliedvoltage of the controller output.

Fig. 13. Photographs of deformable mirror prototypes: (a) a mirror with a segmented annular electrode pattern, (b) another mirror with atriangular lattice of closely spaced electrodes, (c) rear view of mounted mirror showing three-axis positioning gimbal and mounting board,and (d) front view of mirror showing reflective surface and three low-stress mounting points.

Laser

Test Mirror

Pinhole filter

“Eyepiece”lens

Objective lensShack-

HartmannSensor

Focusing lens

Beamsplitter

Turningmirror

Fig. 14. Schematic diagram of experimental measurement setup.


In the figure, note that after a settling period, de-pendent on the controller gains, the mirror defocus iscontrolled well within a small fraction of a wave-length. The long-term drift in the control voltage isin response to the uncontrolled laboratory thermalenvironment; the oscillatory behavior has not beenstudied.

E. Multiple Mode Control

A more complex scenario is to control many responsemodes by using all of the available actuation chan-nels in order to reduce the mirror aberrations, for ex-ample, to make the mirror as flat as possible. In thistest, the lowest 66 Zernike modes were minimized ina least-squares sense with 16 independent voltages.The control algorithm was implemented by decom-posing each of the 16 measured influence functions

for the mirror into its Zernike components and thenimplementing a proportional derivative feedbackcontroller that reduces the magnitudes of the mea-sured Zernike components of the actual mirrorshape. At each step, the control solution was ob-tained by computing a constrained (to meet the volt-age constraints) least-squares solution of Eq. (6) andmultiplying it by a factor less than unity to ensure adamped response without overshoot and to preventpossible material hysteresis effects. The influencefunctions of the mirror were assumed to be constantand independent of voltage throughout the test.

Figure 18 shows the evolution of the measuredRMS error during this test. The initial RMS errorwas 5.2 waves (at 633 nm), which was reduced to

0.5 1.5-1.0 -0.5 0-1.5 1.0

Deflection (waves)

(1) (2) (3) (4) (5) (6) (7) (8)

(9) (10) (11) (12) (13) (14) (15) (16)

(1) (2) (3) (4) (5) (6) (7) (8)

(9) (10) (11) (12) (13) (14) (15) (16)

(a)

(b)

Fig. 15. Influence functions of preliminary 100 mm mirror prototype with 16 channels with the pattern in Fig. 13(a). (a) Experimentalmeasurements and (b) finite element predictions due to 400 V inputs. Deflection color scale is in units of waves at 633 nm. The diameter ofthe images is only 75 mm owing to objective lens size constraints.

Fig. 16. Definition of actuator numbers in 16-channel prototype.

0 20 40 600

200

400

Time (min)

Vol

tage

(V

)

0

2

Foc

us A

mpl

itude

(w

aves

)

Control voltageFocus amplitude

Fig. 17. Demonstration of defocus control using a single inputvoltage.


about 2.3 waves (an improvement of about 55%) inabout four steps. The controller was left runningfor about 10 min to verify its ability to maintain thislow error. Notice that most of the channels hit thecontroller limits of �400 V, which indicates thatthe actuation stroke would first need to be improvedin order to further reduce the error. This could bedone by switching to a mirror design with a morecompliant substrate, by increasing the number ofchannels in the mirror, by increasing the allowablevoltage range, or by using optimized electrode pat-terns, and potentially by updating the influence func-tions. The latter approach would require a longertime to achieve convergence because of the delay in-troduced by remeasuring the influence functions.The controller response time could be decreased tocompensate for this slow down.

6. Discussion and Conclusion

The concept, design, and low-cost fabrication meth-ods of ultralightweight, deformable mirrors havebeen presented in this paper. After an initial inves-tigation of candidate substrate and piezoelectric ma-terials, silicon and PVDF copolymer were selected forfurther study, and a design formula for the maximumcurvature change as a function of the blocked stressof the piezoelectric and the thickness and modulus ofthe substrate was obtained. A minimum thicknessbound on the substrate was established by consider-ing buckling of the laminate mirror during poling,and a general design chart for 20 μm thick PVDFcopolymer active layers on different substrates andthicknesses was derived. Two different actuationpatterns were studied with a finite-element model,and three different densities of the actuators wereconsidered. Correctability and stroke of the mirrorwere evaluated for each design, providing the basisfor future optimization.

From these studies, simple prototype mirrors weredesigned and built using silicon substrates with adiameter of 100 mm and areal density of 0.6 kg∕m2

and were measured to be capable of 20 μm of dis-placement with current geometries and materials.The present solutions may be scaled to larger sizesby using larger silicon wafers or different substratessuch as glass. A kinematic mounting scheme forthese thin, flexible mirrors has been introducedand demonstrated in the laboratory. This schememay be suitable for a variety of operational environ-ments, but launch survivability or vibrations duringoperation may require refinements such as a releas-able restraint system and/or shunted piezoelectricdamping provided by the actuators.

Initial optical testing has been performed on a mir-ror with 16 active channels, and the actuator influ-ence functions have been measured and found tomatch reasonably well to the model predictions.On this preliminary prototype, the residual wave-front error was reduced by about 50% to two waves(RMS wavefront error at 633 nm). Hence it is ex-pected that diffraction-limited performance can beachieved with design optimization and refinementof the proposed concept.

We thank Dr. Harish Manohara (JPL) for provid-ing access to the Microdevices Lab (MDL) cleanroomfacilities for sample fabrication. We thank Dr. RisakuToda (JPL) andMr. Victor White (JPL) for processingequipment training and usage advice at the MDL.We also thank Dr. Namiko Yamamoto (Caltech) forassistance and advice on mirror fabrication. We aregrateful to Dr. Jim Breckinridge (Caltech) for helpfuldiscussions on optics and telescope design, Prof.Chiara Daraio (Caltech) and John Steeves (Caltech)for advice regarding the fabrication of prototype mir-rors, and Dr. Andrew Shapiro (JPL) for advice onmaterial selection.We also appreciate the fabricationfacilities provided by the Kavli Nanoscience Instituteat Caltech. Financial support from the Keck Instituteof Space Studies (KISS) at Caltech is gratefully ac-knowledged. A part of this research was carriedout at the Jet Propulsion Laboratory, CaliforniaInstitute of Technology, under a contract with theNational Aeronautics and Space Administration(NASA).

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0 2 4 6 8 100

2

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rror

(w

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)

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Ultralightweight deformable mirrors · 2017. 2. 27. · relevant mirror technologies. Section 3 describes various aspects of the proposed mirror concept. Section 4 describes the mirror