Many applications, such as imaging and laser engi-neering, require the dynamic correction of wavefrontaberrations that can be well characterized by Zernikemodes. In the past, unimorph and bimorph piezo-electric deformable mirrors have been frequentlyused to compensate both dynamic and static aber-rations [1–4]. Desired features of the deformablemirrors are a large stroke and a high fidelity of themirror deformation. In many cases [5–7], the correc-tion of aberrations with low spatial frequencies is suf-ficient because they usually have a higher statisticalweight than the higher frequency terms. This makesunimorph and bimorph mirrors excellent candidatesbecause they can generate low-order Zernike modeswithout any actuator print-through. Actuator print-through is typical of mirrors that rely on push–pull-type actuators. It leads to surface errors of highspatial frequencies, which scatter light into largeangles. This is detrimental for many applications,in particular, laser resonators. Typical experimen-tally observed aberrations with low spatial frequen-cies consist of only a few low-order Zernike modes.
Zernike modes represent a complete, orthonormalset of functions. Deformable mirrors can be treatedas linear systems to a good approximation. We there-fore analyzed and compared different unimorph mir-rors with respect to the amplitude and the fidelitywith which they can create certain low-order Zernikemodes. For unimorph and bimorphmirrors, these am-plitudes and fidelities depend critically on their ac-tuator pattern. A variety of patterns have been usedsince the early days of adaptive optics [8]. However, asystematic study of the merits of the different pat-terns has not yet been published. In this paper, theinfluence of the actuator pattern on the mirror defor-mation is studied by analytical reasoning and by fi-nite element modeling (FEM). The various actuatorpatterns are analyzed by calculating the mirror’s re-sponse to the activation of each actuator. From the in-fluence functions, we can calculate amplitudes andfidelities of arbitrary Zernike modes [9]. We foundvery good agreement of these calculationswith the ex-perimental results of novel unimorphmirrors that wedeveloped.
2. Finite Element Model
FEM of complicated piezoelectric structures has al-ready been used successfully in the past [4,10,11].Analytical models are also described in the literature
[12–15], but all analytical models are based on moreor less severe approximations. FEM provides moreprecise results. For this study, unimorph mirrors con-sisting of a piezoelectric disk sandwiched betweentwo metallic electrodes and bonded to a passive glassdisk are modeled using the commercial FEMsoftware package COMSOL Multiphysics 3.4. Thediameter of the disks was 10, 20, or 25mm,depending on the electrode pattern that was se-lected. For all the mirrors, the deformation across anoptical aperture of 10mm was calculated. This aper-ture is suitable for intracavity laser applicationswhere beam diameters are small. The thickness ofthe piezo disk is 200 μm, and the thickness of theglass disk is 100 μm. The mirrors were assumed tobe supported by a 2:5mm wide ring of 300 μm thickelastomer. We applied fixed boundary conditions forthe outer circumference of this elastomer ring. Theelectrode, which is situated between the glass andthe piezo disk, serves as the common ground elec-trode for all actuators. The other electrode of thepiezo disk is divided into segments that can be acti-vated separately with different voltages. This formsthe actuator pattern of the mirror. Themirror geome-try, along with the boundary conditions, is shown inFig. 1. Because the investigated electrode patternsdo not have rotational symmetry, three-dimensionalFEM models have been used. The hysteresis of thepiezoceramic, nonlinear effects, and the influenceof the bonding layer are ignored in the present study.This slightly affects the accuracy of our calculationsbut has no influence on the relative comparison of theelectrode patterns. The validity of the FEM calcula-tions is, however, highly dependent on the shape andthe density of the numerical mesh. It is, therefore,important to optimize the mesh parameters with re-spect to the accuracy of the result and the computingtime. In the model, a fine mesh is used in criticalareas, i.e., regions where the solution gradient ishigh and the geometry features are small, and acoarse mesh is used in less critical areas. The meshoptimization is based on error estimates in the com-puted solutions and leads to an adaptively refinedmesh that finally converges to approximately250,000 nodes. The large aspect ratio, i.e., the ratioof the disks’ diameters to their thicknesses, has beentaken into account by a 25-times higher mesh densityin the z direction, the direction of the disk’s thick-ness. The mesh of a model built in COMSOL, as well
as the calculated surface deformation under activa-tion of a single electrode, is presented in Fig. 2.The material properties are listed in Table 1.
We compared the fidelities of the low-order Zernikemodes of each mirror in open-loop control. In ourapproach, the shape of the deformable mirror surfaceis modeled as a weighted sum of the deformationsϕiðx; yÞ contributed by each individual actuator,where ϕiðx; yÞ is commonly known as the actuator in-fluence function [3]. The surface shape sðx; yÞ of thedeformable mirror can thus be described by
sðx; yÞ ¼Xni¼1
vivtest
· ϕiðx; yÞ; ð1Þ
where n is the number of the actuators, ϕi is the influ-ence function of the ith actuator describing thesurface of the mirror and has the unit μm, vtest isthe voltage applied for determining the influencefunctions, and vi is the voltage to be applied to theith actuator. Here we use the simplification thatthe influence is a linear function of applied voltage.In vector form, Eq. (1) can be rewritten as
sðx; yÞ ¼ ~vivtest
·~ϕðx; yÞT ; ð2Þ
where ~ϕðx; yÞ ¼ ½ϕ1ðx; yÞ;…;ϕiðx; yÞ;…;ϕnðx; yÞ� and~v ¼ ½v1;…; vi;…; vn�. The influence function of eachactuator was numerically calculated by applying atest voltage of vtest ¼ 100V to the actuator while set-ting the voltages of all other actuators to zero. Eachinfluence function was then approximated by a 90-dimensional expansion into Zernike polynomials Zjthrough a least-squares fit according to
ϕiðx; yÞvtest
¼X90j¼1
aij
vtestZjðx; yÞ; ð3Þ
Fig. 1. (Color online) Sketch of the unimorphmirror geometry used for modeling. The sketch is not to scale: the thickness (z direction) hasbeen magnified by almost an order of magnitude.
Fig. 2. (Color online) Optimized FEM mesh with 260,000 ele-ments (left) and surface deformation of themirror under activationof a single electrode (right).
whereZjðx; yÞ are the Zernikemodes (Zernike polyno-mials) and aij are the corresponding Zernike coeffi-cients used to fit the influence function. Finding thebest number of Zernike modes to be used for the ex-pansion is difficult. A small number of modes leadsto a less precise surface fit. On the other hand, a largenumber of modes can result in a higher influence ofnumerical errors. For our calculations, the best re-sults have been obtained for 65 or 90 modes, so weused 90 modes in all the calculations. Again, the pre-vious equation can be rewritten in vector form as
~ϕðx; yÞTvtest
¼ IM ·~Zðx; yÞT ; ð4Þ
where ~Zðx; yÞ ¼ ½Z1ðx; yÞ;…;Zjðx; yÞ;…;Z90ðx; yÞ� isthe 1 × 90 Zernike vector and IM ¼ ½aij�=vtest is the n ×90 influencematrix thatdescribes themirror. Thema-trix inversion of IM can be carried out by a singular-value decomposition (SVD) leading to the so-calledcontrol matrix CM ¼ IM�, the pseudoinverse of IM.Small singular values of the SVD-decomposed influ-ence matrix IM need to be set to zero before invertingthis matrix, because the corresponding modes aredominated by numerical errors.
We want to evaluate how well a mirror can gener-ate a certain Zernike mode Zj, which we call the “tar-get Zernike mode.” Therefore, the control matrix CMis used to calculate the vector of control voltages ~vrequired to produce a mirror shape that is the best
fit of the desired jth Zernike mode. ~v is expressedin terms of the “Zernike coefficient target vector”~atarget;Zj
:
~v ¼ CM ·~aTtargetZj
: ð5ÞThe Zernike coefficient target vector has a singlenonzero element if the mirror is supposed to createa pure Zernike mode. The amplitude of each Zernikecoefficient target vector ~atargetZj
was increased untilthe first actuator reached the voltage limit of −60Vor þ100V. These voltage limits of −60V and þ100Vwere calculated from the piezo disk thickness and themaximum allowed electric field strengths of thepiezoelectric material, limited by reverse polingand electric breakthrough, respectively. A second, in-dependent limit for the maximum amplitude of theZernike mode was set by the fidelity of the mirrorsurface. We define fidelity as the inverse of the rmsdeviation σΔs of the actual mirror surface from thetarget surface
σ2Δs ¼1A
ZZA
½sactualZj− stargetZj
�2dxdy
¼ 1A
ZZA
�Σ90
j¼1ðaactualZj
− atargetZjÞ · Zjðx; yÞ
�2dxdy;
ð6Þ
where A is the area of the beam footprint. We call thearea of the beam footprint on the mirror the “activearea” of the mirror. It is a circle of 10mm diameter inall mirrors we analyzed. σΔs was allowed to increaseup to the Maréchal criterion λ=14 [16] as the ampli-tude of the target Zernike mode was graduallycranked up. Therefore, the maximum Zernike ampli-tude that a mirror can produce can be limited byeither the maximum allowed actuator voltage(“actuator saturation”) or by the surface fidelity.
OurFEManalysis revealed that, for some electrodepatterns, the amplitudes of several Zernikemodes arelimitedbysaturationofoneor severalactuators,whilethe surface fidelity is still much better than theMaré-chal criterion. Theperformance of these electrode pat-terns might be underestimated, because leaving thesaturated actuator’s voltage constant while increas-ing all other voltages could result in a larger ampli-tude of the Zernike mode while still maintaining theMaréchal criterion. In order to determine the maxi-mum amplitude within the limit of the Maréchal cri-terion, we used an iterative control scheme proposedby Bonora and Poletto [17]. At first, the requiredvoltage vector~v is calculated usingEq. (5). This vectoris subdivided into a vector ~v0 ¼ fvi ∈~vjvi < vmaxg,consisting of all unsaturated actuators and a vectorv~″ ¼ fvi ∈~vjvi ≥ vmaxg consisting of the saturatedactuators. The influence matrix is divided into thesubmatrices IM0 ¼ fIMijvi < vmaxg and IM″ ¼fIMijvi ≥ vmaxg. Now we can calculate better voltagevalues~v0 of the unsaturated actuators, according to
Table 1. Material Properties and Dimensions Used in NumericalModels—Also Relevant Data for Unimorph Mirrors We Manufactured
Component Property Value
Piezo disk Material PIC255Disk thickness (μm) 200Disk diameter (mm) 10–25Maximum voltage (V) −60=þ 100Poisson ratio vp 0.36Young’s modulus Ep (GPa) 62.9Piezoelectric strain constantd31ðmV−1Þ
whereCM0 is the pseudoinverse of the submatrix IM0.The new voltage vector is given by~v ¼~v0∪~v″. This pro-cedure is repeated iteratively until all actuators aresaturated or the rms deviation σΔs of the actual sur-face from the target surface reaches the Maréchallimit.
If a Zernike mode is identical or nearly identical toone of the mechanical eigenmodes of the mirror, themirror is easily deformed into this Zernike mode andseveral actuators are saturated when the maximumamplitude of this Zernike mode is reached. In thiscase, the voltage range is the limiting factor and theprocedure proposed by Bonora yields considerably in-creased amplitudes. However, a Zernike mode thathas little or no similarity to one of the eigenmodesof the mirror quickly exceeds the Maréchal limit.For such a Zernike mode, which in some sense is“orthogonal” to the eigenmodes of the mirror, the op-timization routine remains without effect. How closea Zernike mode is to an eigenmode or a superpositionof eigenmodes of a mirror can be quantified by the so-called purity, as reported by Bonora and Poletto [17].The purity is defined as the projection of the normal-ized, actual Zernike vector ~̂aactual;Zj
on the targetZernike unit vector ~̂atarget;Zj
. The normalization ofthese vectors is denoted by the ^ symbol:
pj ¼ ~̂atarget; Zj• ~̂aT
actual;Zj: ð8Þ
If the mirror is able to perfectly reproduce a requiredZernike mode, the purity value for this mode is 1.
Our analysis consists of five steps: (i) setup of theFEMmodel, (ii) calculation of the influence functionsand the control matrix, (iii) calculation of the controlvoltages required for the generation of each targetZernike mode, (iv) evaluation of the calculated sur-face shapes (calculation of σΔs), and (v) increasingthe control voltages and iteration from steps (iv)to (v).
3. Electrode Geometries
Our aim is to find the electrode pattern that can gen-erate certain low-order Zernike terms with the high-est amplitudes and the highest fidelities (i.e., thelowest rms deviation σΔs). Generally, the numberof electrodes directly corresponds to the available de-grees of freedom. Therefore, a higher number of elec-trodes allows us to generate higher order Zernikemodes with good fidelity, so wavefront distortionswith higher spatial frequency can be corrected inan adaptive optics system. However, the appropriatemaximum number of electrodes is also determinedby physical and technical limitations. If one appliesdifferent voltages to two adjacent electrodes, the re-sultant field strength in the gap between both elec-trodes has to stay below the depolarization fieldstrength. This condition results in a minimum elec-trode spacing of approximately 150 μm for the max-imum voltage of 100V that can be safely applied to
the 200 μm thick piezo disks that we used in our FEMmodels. Increasing the number of electrodes thus re-duces the ratio of the area covered by electrodes tothe total area of the mirror. This decreases the am-plitude of low-order Zernike modes, for which manyneighboring electrodes have similar voltages, due tothe smaller effective electrode area. The dimension ofeach electrode should also be larger than the thick-ness of the piezoelectric material, so that the influ-ence of the stray field on the adjacent electrodesremains negligible.
The mirrors we analyzed were designed for an ac-tive area of 10mm diameter, the beam footprint.Using the previous reasoning, this results in approxi-mately 24 electrodes under the beam footprint.Figure 3 shows the first 12 Zernike modes, with theexception of the piston term. The Zernike modes arenumbered, using the single-index notation of WyantandCreath [16]. Inaddition, the cumulative inflectionlines of all previous Zernike modes up to the Zernikemode that is shown in false shades are shown in eachgraph. The last inflection line pattern can be used as afirst starting point for sectioning the electrodes be-cause inflection lines of the mirror surface can onlybe induced by neighboring electrodes with differentsigns of the applied voltage. The cumulative inflectionlines of all Zernikemodes up toZ12 in Fig. 3would cre-ate already 48 electrodes (16 azimuthal segmentstimes 3 radial segments). However, the middle ringis very narrow; its width is of the order of the piezothickness. It is therefore reasonable to abandon thisring and make only one radial segmentation insteadof two.Wehave selected the radial inflection line ofZ8,see Fig. 3, which has a radius of r1 ¼ 2:7mm, for ouractive area of 10mmdiameter. This radius is found bysolving
∂2Z8
∂r2¼ 0; ð9Þ
for a disk with simplified free boundary conditions,i.e., no circumferential stresses. The optimum radialand angular segmentation of the electrodes can be de-termined more precisely by the FEM simulation,taking into account the real boundary conditions.For the optimization of the radial segmentation, wehave modeled a mirror without any azimuthal seg-mentation. The variable parameters for the optimiza-tion have been the voltages V1 and V2 of the twoelectrodes and the radius r1 of the electrode segmen-tation. The optimum radius r1 is then determined bysearching for the parameter triple (V1, V2, r1) corre-sponding to the surface shape with the least residualrms deviation σΔs from the Zernike surface Z8, whichrepresents spherical aberration. The resultant opti-mum radial electrode segmentation is r1 ¼ 2:85mm,slightly larger than our analytical result of 2:7mm.By making just one radial segmentation, we have re-duced the number of electrodes to 32, still more than24.We further reduced thenumber of electrodes bydi-viding thecircularareawithin r1 intoonly8azimuthalsegments of 45° each instead of the 16 azimuthal
segments shown in Z12 of Fig. 3. We again justify thisby the fact that otherwise exceedingly small electro-des would be created. The cumulative inflection linepattern of Z12 in Fig. 3 would create a segmentationof the second ring into 30° and 15° segments. For com-parison, we also made FEM simulations of a mirrorthat has 16 azimuthal segments of 22:5° each. To de-monstrate the improvement in Zernike mode fidelityand amplitude due to the radius optimization, wehave simulated another mirror with 22:5° segmentsand a nonoptimized radial segmentation at r1 ¼3:25mm for comparison.
Various other electrode patterns have been usedand published in the past. For example, an additionalring of electrodes could be used that is outside theactive area. In the following text, we call this ringan “out-of-aperture ring.” This would allow bettercontrol of the slope at the circumference of the activearea. Such a design is, for example, used for bimorphmirrors manufactured by AOptix Technologies [3]. Tostudy the influence of the width of the additionalout-of-aperture ring, we simulated such electrodepatterns with an outer diameter of the additionalout-of-aperture electrode ring of 20 and 25mm, whilemaintaining the diameter of the active area of10mm. In some cases, the out-of-aperture electrode
ring is separated from the electrodes of the activearea by an additional inactive ring called the guardring. The purpose of the guard ring is to smooth outthe transition from the out-of-aperture ring to the ac-tive area. In Ref. [18] the authors conclude that mir-rors that can be described by a biharmonic equation,e.g., deformable mirrors with push–pull actuators,two out-of-aperture electrode rings are needed togenerate low-order Zernike modes with high ampli-tude and high fidelity. Furthermore, the authorsstate that, for unimorph and bimorph mirrors thatcan be described by a Poisson equation, one out-of-aperture ring is sufficient to achieve the best ampli-tudes and fidelities. To verify this result, we havesimulated a design with two out-of-aperture rings.
Another prominent example of electrode patternsis a honeycomb pattern of equal-area hexagonal elec-trodes that is frequently used for membrane mirrors.We therefore investigated honeycomb layouts with19 and 37 electrodes.
Electrode patterns derived from Voronoi diagramsare not considered in this work. Such electrode pat-terns are better suited for larger mirrors with ahigher number of actuators.
Figure 4 shows the 11 different electrode patternsfor which we will report the results of our FEM
Fig. 3. (Color online) Low-order Zernike modes. The cumulated inflection lines of all previous Zernike modes up to the Zernike modeshown in false shades are indicated.
simulation in the next paragraph of this paper. Thepatterns can be described as follows:
a. thirty-five-actuator keystone layout with oneout-of-aperture ring and a guard ring similar tothe design used by AOptix,
b. forty-actuator keystone layout with one out-of-aperture ring, a guard ring, and azimuthallysymmetrical electrodes,
c. fifty-six-actuator keystone layout with two out-of-aperture rings and a guard ring,
d. forty-actuator keystone layout with one out-of-aperture ring,
e. forty-actuator keystone layout with one widerout-of-aperture ring,
f. seventeen-actuator keystone layout of our un-imorph prototype mirror, the contact from the groundelectrode on the back side is wrapped around andvisible at the six o’clock position,
g. twenty-four-actuator keystone layout with a22:5° angular segmentation of the second ring witha first radial segmentation at r1 ¼ 2:85mm,
h. twenty-four-actuator keystone layout with a22:5° angular segmentation of the second ring witha first radial segmentation at r1 ¼ 3:25mm (not op-timized),
i. twenty-four-actuator keystone layout with analternating 30° and 15° angular segmentation of thesecond ring,
j. nineteen hexagonal actuators arranged in ahoneycomb pattern, and
k. thirty-seven hexagonal actuators arranged ina honeycomb pattern.
4. Results
In order to validate the numerical model, a prototypemirror has been constructed consisting of a 10mmdiameter, 0:2mm thick PIC255 piezoelectric ceramicdisk bonded to a 10mm diameter, 0:1mm thick BK10glass substrate. The electrode is segmented into one
central pad surrounded by two concentric rings with45° subdivisions [see pattern (f) in Fig. 4]. The elec-trode patterning was done by laser ablation with aQ-switched Nd:YAG marking laser. To measure theinfluence functions of the prototype mirror, a phase-shifting interferometer has been used. The measuredinfluence functions of the mirror were found to be ingood agreement with the numerical predictions, ascan be seen in Figs. 5 and 6. This is proof that ournumerical FEM analysis is sufficiently accurate.As expected for a unimorph deformable mirror, themaximum amplitude of the Zernike mode is observedto be approximately proportional to the inversesquare of the mode’s radial order [19].
Figures 7 and 8 show the comparison of the ampli-tudes and purities of all 11 electrode patterns andrepresent the main result of this paper. The figuresindicate that keystone designs with circular symme-try, such as those depicted in Figs. 4(a)–4(i), are espe-cially suited for the generation of low-order Zernikemodes. Moreover, the calculations show that elec-trode patterns with one out-of-aperture electrodering can generate much larger amplitudes for someZernike modes, as shown in Fig. 7. Only the ampli-tude of the defocus mode is reduced by approximately25%. This can be explained by the fact that the bestapproximation of the defocus mode is achieved by ap-plying the maximum allowed voltage to all electrodesin the active area and by applying voltages with anopposite sign to the out-of-aperture ring electrodescalculated by Eq. (5). In this way, the out-of-apertureelectrodes compensate for the influence of the bound-ary conditions imposed by the elastomer. In case ofthe defocus mode, the optimization routine Eq. (7)proposed by Bonora cannot lead to an improvementbecause the voltages of all electrodes inside the ac-tive are already saturated and increasing the elec-trode voltages of the outer ring further reduces theamplitude. To further improve the achievableamplitude of the defocus mode, an evolutionary algo-
Fig. 4. (Color online) Analyzed electrode patterns. The active optical aperture is shaded red/gray.
rithm could be used to determine the best voltagepattern.
The out-of-aperture electrodes also result in signif-icantly higher purity values for all Zernike modes ex-cept for the defocus mode. An example is shown inFig. 9. The out-of-aperture electrodes are increasingthe amplitude of the astigmatism Zernike mode Z4from 1:1 μm to 3 μm. At the same time, fidelity is in-creased as well, as can be seen from the calculatedinterferograms in Fig. 9. This is also reflected inthe purity of 1.0 for the pattern with out-of-apertureelectrodes, compared to the purity of 0.94 for thesame pattern without the out-of-aperture electrodes.Especially the Zernike terms Zm
n with n ¼ jmj, whereZmn is the double-index Zernike polynomial with the
radial order n and azimuthal order �m, i.e., tip/tilt,astigmatism, and trefoil, benefit from the additionalout-of-aperture electrodes. This is in good agreementwith the analytical calculations presented in [18].The comparison of the results obtained for the elec-trode patterns (b), (d), and (e) shows that the achiev-able peak-to-valley amplitudes of these Zernikemodes are proportional to the out-of-aperture ring’swidth, whereas the influence of the out-of-aperture
ring’s width on the other Zernike terms remains neg-ligible. The radial subdivision of the out-of-aperturering into two rings does not lead to any further im-provement, as can be seen by comparing the resultsfor pattern (b) and (c). The amplitudes of the designwith two out-of-aperture rings are generally slightlysmaller than the amplitudes of the design with onlyone out-of-aperture ring. The biggest difference oc-curs for the spherical aberration term where theamplitude is decreased by 7%. As a result, one seg-mented out-of-aperture electrode ring outside is suf-ficient to generate low-order Zernike modes withhigh fidelity and high amplitude.
The optimization of the first radial segmentationimproves the fidelity and amplitude of the sphericalaberration and the secondary astigmatism Zerniketerms Z8, Z11, and Z12. This can be seen in Fig. 7,where the nonoptimized radial segmentation of theelectrode pattern (h) and the prototype mirror elec-trode pattern (f) result in lower amplitudes comparedto the optimized keystone designs (g) and (i). As sta-ted in Section 3, the cumulative inflection line pat-tern suggests a segmentation of the second ring in30° and 15° segments. However, the comparison of
Fig. 5. (Color online) (a) Prototype mirror, (b) experimentally measured influence functions of the mirror, and (c) FEM simulation of theinfluence functions. Shown is the deformation generated by a single electrode activated with a voltage of 100V. The false-shading elevationplots that represent the deformation of the whole mirror are plotted at a position that corresponds to the electrode that is being activated.
Fig. 6. (Color online) Comparison of the calculated and experimentallymeasured amplitudes of the prototypemirror for different Zernikemodes.
patterns (i) and (g) in Fig. 7 delivers nearly identicalresults for this pattern and the symmetrical segmen-tation in 22:5° sectors. The only difference is aslightly different voltage pattern. The calculations
also reveal that the purity of the Zernike surfacesis almost not noticeably affected by the implementa-tion of a guard ring, as can be seen by comparing pat-terns (b) and (d) in Fig. 8. Figure 7 shows that the
Fig. 7. (Color online) Maximum peak-to-valley Zernike amplitudes for different actuator patterns. The Zernike amplitudes are limited bytheMaréchal criterion for the RMS deviation σΔs or the voltage limits of the piezoceramic. The 11bars for each Zernike mode correspond tothe 11 electrode patterns shown at the top of the figure.
Fig. 8. (Color online) Calculated purity values for the investigated electrode patterns. The purity indicates howwell themirror can createa Zernike mode.
inactive area of the guard ring leads to significantlyreduced amplitudes for some Zernike modes, e.g., Z4,Z5, Z9, and Z10. Besides the results given in thisstudy, the simulations have shown that relaxing therms deviation σΔs of the target surface beyond theMaréchal limit of λ=14 results in smaller differencesbetween the electrode patterns with and without anadditional out-of-aperture ring. This is explained bythe fact that mirror designs with an out-of-aperturering are mainly limited by actuator saturation,whereas designs without an out-of-aperture ring aremainly restricted by the limit of the rms-deviationσΔs. Accordingly, the use of piezoelectric materialswith higher depolarization field strengths (corre-sponding to higher voltage limits) would result inhigher amplitudes for the most Zernike modes if anout-of-aperture ring of electrodes is employed. Howimportant it is to optimize the electrode patterncan be seen, for example, by comparing the ampli-tudes of the astigmatism Zernike modes Z4 and Z5for the hexagonal pattern (k) and the keystone pat-tern (e). The amplitude of the hexagonal pattern of0:69 μm is only 13% of the amplitude achieved bythe keystone layout with one outer ring, which is4:99 μm.
5. Discussion and Conclusion
We have presented a procedure that leads to opti-mum actuator patterns for unimorph or bimorphmirrors. Analytical reasoning regarding the inflec-tion lines of Zernike modes and the maximum num-
ber of electrodes was used to establish a startingpattern. This pattern was then optimized using ex-tensive, experimentally validated, FEM computa-tions with high spatial resolution. We computed andcompared 11 different actuator patterns. About 500hours of CPU computing time on a fast PC was re-quired for the FEM computations of this study.
Obviously, deformable mirrors for different taskswill have different optimum actuator patterns. Wesearchedspecifically fortheoptimumactuatorpatternfor a unimorph mirror of 10mm active area, 0:3mmthickness, and Zernike modes up to Z14 (secondarycoma).Wefoundthatourpattern (e)achieves thehigh-est amplitudes for all Zernikemodes except for the de-focus mode. For this mode, patterns (g) to (i) achieveabout 25%more stroke, but for all other modes, thesepatterns are much worse than pattern (e).
Even though ournumericalFEMcomputations hadto be performed for specific mirror dimensions and fora specific set of Zernikemodes,we can draw some gen-eral conclusions. One ring of actuators that is outsidethe beam footprint on themirror increases the ampli-tude of all Zernike modes with an azimuthal orderidentical to that of the radial order. This outer ringof electrodes has no drawbacks for the other Zernikemodes, only the defocus mode suffers slightly. Honey-comb patterns of hexagonal actuators, which havebeen widely used for micromachined membrane mir-rors, are not very suitable for unimorph and bimorphmirrors.
Fig. 9. (Color online) Astigmatism Zernike amplitudes and purities of the 22:5° electrode pattern with and without an additional outerring outside of the active optical area. The surface deformation, the corresponding interferogram, and the applied voltages are plotted fromtop to bottom.
The general procedure we have outlined for deter-mining the optimum actuator pattern can readily beapplied to any deformable mirror. Input data for theprocedure are the mirror geometry and the requiredamplitudes of the Zernike modes, which have to bederived from the application.
The authors gratefully acknowledge support forthe work presented by the German Ministry for Edu-cation and Research under contract 1726X09 and byThorlabs GmbH.
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