Deformable 2-D mirror using multilayered electrostrictors

Takuso Sato, Hiromi Ishikawa, Osamu Ikeda, Shoichiro Nomura, and Kenji Uchino

A compact 2-D deformable mirror is proposed which consists of multiple electrostrictive laminar sheets

bonded to the backside of a mirror glass plate. A set of desired wave fronts is considered; the sheets are as-

signed to generate each wave front separately. The electrode of each sheet is designed so that the exact volt-

age distribution required to deform the mirror surface is approximated optimally by a few discrete voltages.In actual operation, the deformation of the mirror is determined by the superposition of properly weightedcontributions from all the actuators. The usefulness of the construction is demonstrated experimentally.

1. Introduction

Precise wave front control with as small a number ofparameters as possible and compact construction is acommon and basic requirement for adaptive opticalsystems.' For example, continuous surface deformablemirrors may be more desirable than segmented mirrorsfrom the viewpoint of precision. 2 -4 Bimorphous de-formable mirrors,5 -7 which consist of a laminar mirrorglass plate and a PZT or PVDF piezostrictive sheet witha suitable on-off supply voltage distribution, have beendeveloped to reduce the control parameters.

Recently, we proposed a bimorphous deformablemirror using a PMN [Pb(Mg1/3Nb2 /3)03)1 electro-strictor as the actuator. It has neither hysteresis noraging. 8 An extension of the single layered PVDF de-formable mirror to the multilayer system was proposedto make the construction compact.9 The deformationin that system, however, is restricted to 1-D due to sig-nificant anisotropic characteristics of the d constantsof available PVDF film as the actuator.

In this paper, by combining the merits of the abovetwo methods, a compact 2-D deformable mirror isconsidered, that is, a deformable mirror consisting ofmultilayered PMN electrostrictive sheets. The elec-trode of each PMN sheet is designed so that it deformsthe mirror surface corresponding to the wave front de-sired. Deformation of the mirror in actual operationis controlled by choosing the proper combinations of thesupply voltages to the layers.

All authors are with Tokyo Institute of Technology, Japan; K.Uchino and S. Nomura are in the Department of Physical Electronics,Ookayama, Meguro-ku, Tokyo 152, and the other authors are at theGraduate School at Nagatsuta, 4259 Nagatsuta, Midori-ku, Yoko-hama 227.

Received 21 May 1982.0003-6935/82/203669-04$01.00/0.© 1982 Optical Society of America.

A design of the deformable mirror and basic experi-mental results are given in the following sections.

11. Design of the Multilayered Deformable Mirror

Construction of the multilayered deformable mirror(MDM) is shown schematically in Fig. 1. It consists ofmultiple PMN laminar sheets bonded to a laminar glassplate with a mirror surface. Let us consider a set ofdesired wave fronts. The electrode of each PMN sheetis designed so that the exact voltage distribution re-quired to deform the mirror surface to the desired wavefronts is approximated by a few discrete voltages.

The differential equation that governs the static be-havior of the MDM is given from Ref. 9 by

A[a 4fl/ax 4 + 2(d4f;Iax 2ay2) + a4fl/ay 4 ]

+ B[d3l (a2Vi/ax 2) + d32(a2V,/ay2)] = 0, i = 1, 2, . . ., I,

where A and Bi are constants, d31 and d32 are the pi-ezoelectric constants, Vi = Vi (x ,y) and f = f (x ,y) arethe applied voltage distribution and the resulting de-flection of the ith layer, respectively, and I is the num-ber of layers.

Equation (1) is an expression for the case where thepiezostrictors are used, but it may also be applied whereelectrostrictive sheets are used, provided the nonlinearrelation between the electric field and the electric strainis taken into account.

We may put the condition d31 = d32 for the case of thePMN.8 If we impose the conditions that fi (xy) = 0 forVi(x,y) = 0, i = 1, 2, . . , I, then Eq. (1) is reduced to

V2fi(x,y) + CiV(x,y) = 0 i = 1,2,..., I, (2)

where Ci is the constant which is different for each layerand V2 is the Laplacian.

As a typical case let us consider a rectangular de-formable mirror of size 1, X ly, supported along thefour edges. In this case the boundary conditions aregiven by

15 October 1982 / Vol. 21, No. 20 / APPLIED OPTICS 3669

(1)

ControlVoltages 9GGlass Plate

VIL-) rrMirror Surface

Fig. 1. Schematic of the MDM using PMN electrostrictive sheets.Each actuating layer has a discrete version of the exact voltage dis-tribution required to deform the mirror surface to each one of the

desired wave fronts.

Electrode Plane

(a) -_tittH _

_ LIT _ L

Optically EffectiveArea'

(b) ADeflection by UnitVoltage Application

(d)

Shape of Electrode

(e) X

Voltage Distribution

(C) _=Zye I 1 1 1 1

Voltage Distribution

Fig. 2. Procedure of producing patterns of the electrodes.

fi(x,y) = O2fi(x,y) Ix 2 = 0 for x = 0 and 1x,

fi(Xy) = a2f (x,y)lay 2 = 0 fory = 0 and 1y, (3)

i = 1,2,. I.

Then the explicit solutions are obtained as

fi(Xy) = (Ci/7r2 ) apq sin(prx/lx) sin(q7ry/l),p q ()2 q~

i=1,2,. I, (4)

where

aipq = (4/11xy) ' f Vi(x,y) sin(prx/1x) sin(q7ry/1y)dxdy.

(5)

The procedure for producing the patterns of theelectrodes is as follows (see Fig. 2):

(a) Only a central part of the mirror surface is usedas the optically effective area, since the outer part isdifficult to deform because of the boundary conditionsadopted. The electrode of each layer that covers thewhole area is divided into J identical elements.

(b) The deflection for the application of a unit volt-age to each electrode element is calculated using Eqs.(4) and (5) as

at1uij(x,y) = (16Ci/7r4 ) E

pql pq [(p)2+ (q)2]

X sin(prtj/l1,) sin(q7rqj/ly)

X sin(pru/21.,) sin(qrv/21y)

X sin(prx/lx) sin(q7ry/ly), j =1, 2,.J, (6)

where (Qj,,j) is the center position of the jth electrodeelement with the same area u X v, and the suffix i sig-nifies the ith layer.

(c) The voltages Vij, j = 1, 2,. . . , J for generating thedeflection fi (x,y) are determined so that the mean-square error between the generated and the requireddeflections over the optically effective area is mini-mized:

f1V [J 121x0 ~)j/ aVij = 0,,9 ( E L;2 Vijuij (XnY) -fi (XnYn)5 j=O

j=1,2,...,J, (7)

where N > J and (xn,yn) are the positions of equispacedpoints in the optically effective area. The spacing ofthe equispaced points, A = x - xn_- or = yn -Yn-1,is taken to be small so that the functions fi (x,y) anduij (x,y) are almost piecewise linear over A. Then fromEq. (7) we obtain

3670 APPLIED OPTICS / Vol. 21, No. 20 / 15 October 1982

Vi = (U 1U)-lUiFi, (8)

where(C)

(a) +f()

Eleactrode ()

Plane (d)L (e)

Vl V2 :V3 :V4 :V5: V6

= 3.9: -3.5*1-l :35 -39

Fig. 3. Example of an electrode pattern for generating coma aber-ration: (a) electrode plane and optically effective area; (b) electrodepattern obtained; (c) and (d), interferograms of a coma aberrationgenerated by the pattern over the whole region and the optically ef-fective area, respectively; (e) desired ideal interferogram of the coma

aberration in the optically effective area.

Wave FrontRequired to Gen-rate

I _ _ D . _ _ _ _Oecompositionof Wav Font Si.

' --- P........s ' Ol S

,', r

,f-m.bl. Mr

He-Ne Laser0.6328 a

> Shutter

/ BS

Fig. 4. Schematic of the experimental system: BThe shutter is opened only when the holograr

V2i _ \ \\\\VtB --l X~~~~VL z~~~~~~~v!

Wa ,

(a) Top View

tg ft t2 t3

wiYt!

Vi = i .. ** vij )

[uji(ri) ... uil(rN)

Ui = I,uiJ(r) . . uiJ(rN)

(9)

F = Li(ri) ... fi(rN)],

rn = (n,yn), and r denotes the transverse operation.(d) Elements with the same quantized discrete

voltages are united so that a pattern which consists ofseveral divisions is obtained.

(e) The voltages to be applied to the divided regions,[Viji], i = 1, 2, . . . , I (see Fig. 1), are determined as in(c).

As an example let us consider the case of coma aber-ration. In this case f(x,y) = CcX (X 2 + y2 ), where Cc =

constant. Both the whole electrode and the opticallyeffective area are divided as shown in Fig. 3(a) into 8 X8 elements to calculate V by Eq. (8). The pattern ob-tained is shown in Fig. 3(b), which consists of only sixdivisions and has the fixed ratio of supply voltages. Thedeflection profile obtained by this electrode pattern isshown in the form of an interferogram over the wholeplane in Fig. 3(c). The same pattern in the opticallyeffective area is shown in Fig. 3(d). The result in Fig.3(d) shows fairly close agreement with the desired idealshown in Fig. 3(e).

Ill. Basic ExperimentThe experimental system is shown schematically in

Output Fig. 4. A He-Ne X = 0.6328-,m laser was used as theWave light source. First a hologram is obtained using the

HoFoilrteC slant collimated reference wave and the wave reflected

S, beam splitter. by the MDM. This is used as the holographic spatiala is formed. filter to compensate for the static residual minute de-

formations on the mirror surface.7 Then the spatialfilter gives the collimated plane wave for the samesupply voltages as those used in the hologram formationand the resulting spatially modulated wave for differentvoltages.

When wave fronts to be generated are given as com-binations of the functions assigned to the layers, the

h Electrode voltages proportional to the weights for the functionsI3 for Refocusing

. , k-PMN with Electrodefor Coma Aberration

/ P PMN without ElectrodeGlass Plate

tb) Cross Section

Fig. 5. MDM capable of generating wave fronts of refocusing andcoma aberrations. Six voltages whose ratio is fixed are used for comaaberration, while a single voltage is used for refocusing. The MDMis mounted in a holder by supporting the four edges of the glass

plate.

Table 1. Materials Used for the MDM

Young's Thickness Sizemodulus (N/M2 ) (Am) (mm)

Glass plate Eg = 6.6 X 1010 tg = 130 Wg = 30PMN E, = 1.1 X 10" ti = 300 W = 18

i = 1,2,3

15 October 1982 / Vol. 21, No. 20 / APPLIED OPTICS 3671

3

V }~~~~~~~EtactrostricbveVI s Z 'Materiat PMN

a Gass Plate

Fig. 6. Interferograms showing deformations by the MDM.

are supplied. On the other hand, for arbitrary spatialdistribution of wave fronts the optimum weights forthese functions are obtained, for example, by least-mean-square error fitting.

An MDM with electrode patterns for refocusing andcoma aberrations was used in the experiments, that is,the two functions are assigned to two layers; see Fig. 5and Table I. The electrode pattern for coma aberrationis that shown in Fig. 3(b). A PMN sheet without elec-trode was inserted as shown in Fig. 5(b) to put theneutral plane outside of the two PMN sheet actua-tors.

Typical experimental results are summarized in Fig.6, where the three experimentally obtained interfero-grams of refocusing aberration, coma aberration, anda combination of the two are compared with the corre-sponding ideal ones. Good agreement is seen in eachcase. The slight disagreement may be mainly due to theanisotropic elasticities of the mirror glass plate used.These results demonstrate the validity of the superpo-sition of deformations and the appropriateness of themethod of producing the electrode patterns.

It was also observed in other experiments that theMDM responds linearly up to 500 Hz for sinusoidalinput voltage.

IV. Conclusion

A compact multilayered deformable mirror using theelectrostrictive sheets was proposed, and its effective-ness was demonstrated experimentally. The mirrormay be used for the control of wave fronts, especially incases where they can be expanded using a small numberof desired wave functions.

References1. J. W. Hardy, Proc. IEEE 66, 651 (1978).2. T. R. O'Meara, J. Opt. Soc. Am. 67, 306 (1977).3. T. R. O'Meara, J. Opt. Soc. Am. 67, 318 (1977).4. J. E. Pearson and S. Hansen, J. Opt. Soc. Am. 67, 325 (1977).5. E. Steinhaus and S. G. Lipson, J. Opt. Soc. Am. 69,478 (1979).6. N. T. Adelman, Appl. Opt. 16, 3075 (1977).7. T. Sato, H. Ishida, and 0. Ikeda, Appl. Opt. 19, 1430 (1980).8. K. Uchino, Y. Tsuchiya, S. Nomura, T. Sato, H. Ishikawa, and 0.

Ikeda, Appl. Opt. 20, 3077 (1981).9. T. Sato, H. Ishikawa, and 0. Ikeda, Appl. Opt. 21, 0000 (1982).

The authors would like to express their sincere thanksto the reviewers for their kind and careful advice andsuggestions.

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-Layer I Shapes of Electrode-

I -St

2-nd