distributed actuators deformable mirror for adaptive optics

Optics Communications 284 (2011) 3467–3473

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Optics Communications

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Discussion

Distributed actuators deformable mirror for adaptive optics

S. Bonora ⁎CNR-IFN, Laboratory for Ultraviolet and X-ray Optical Research, via Trasea 7, 35131 Padova, ItalyAdaptica srl, via Tommaseo 77, 35131 Padova, Italy

⁎ Tel.: +39 0 497897308.E-mail address: [email protected].

0030-4018/$ – see front matter © 2011 Elsevier B.V. Aldoi:10.1016/j.optcom.2011.03.010

a b s t r a c t
a r t i c l e i n f o
Article history:Received 26 November 2010Received in revised form 22 February 2011Accepted 4 March 2011Available online 2 April 2011

OCIS codes:(010.1080) Adaptive optics(230.3990) Microstructures devices(330.4460) Ophthalmic optics

Keywords:Adaptive opticsDeformable mirrorImage sharpening

In this paper we present a Modal Deformable Mirror (MDM) based on the continuous voltage distributionover a resistive layer. This Deformable Mirror (DM) can correct spherical aberration and coma in addition tothe low order aberrations (defocus and astigmatism) using just nine contacts thus limiting the requirementsof high voltage lines with respect to state of the art devices. We demonstrate the advantages of thistechnology presenting a mathematical description of the mirror, an experimental characterization and acomparison with discrete actuator DMs. In order to demonstrate the effectiveness of this design in sensorlessapplications we present the experimental results for image sharpening optimization. The easiness of usesuggests that it can be useful for many applications from imaging systems to laser process optimization.

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© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Adaptive Optics (AO) technology is nowadays present in a lot ofremarkable experiments [1]. It is a fundamental tool for large telescopes[2,3] and it is routinely used in extremely high power laser facilities[4,5]. Nonetheless AO is not wide spread diffused technology and its useis limited to touch and go experiments. There are several types ofdeformable mirrors on the market [1,6,7]. The most popular aremembrane mirrors, which have obtained their diffusion thanks totheir price [8] but their use is limited by the low damage threshold. Forhigh power laser applications bimorph mirrors are the most commonchoice because they can be easily associated to heat sinks [9]. Strongimprovements in AO devices have been carried out by the introductionofmagnetic deformablemirrorswhich can achieve very high stroke, andMEMS (Micro Electro-Mechanical Systems) deformable mirrors whichhave very high resolution (up to about thousands of actuators). All thosetechnologies use discrete actuators which are connected to a series ofhigh voltage or current amplifiers in order to deform themirror surfaceresulting in the need for complex and large electronic devices andcables. Inmost of the applications they are actuated by the calculation ofthe desired shape using the influence functions matrix [1,6–10].Moreover recently, it has been demonstrated that it is possible todrive the deformable mirror shape using a photo control addressing[11]. This powerful technique limits drastically the requirements of highvoltage lines since the DM response is related to the intensity

distribution of an auxiliary light source modulated by a Liquid CrystalPanel. The quadratic dependence of the DM shape upon the auxiliarylight intensity uniformity suggests that a preliminary calibration of theDM is necessary.

Hence, in order to deform the mirror, a preliminary knowledge ofthe deformation given by each actuator must be acquired with awavefront sensor (WFs). In some cases this is not possible, thus,sensorless applications have been developed based on an imageoptimization [12–16] and optimization algorithm [17].

Several studies have been carried out in order to reduce thecomplexity of adaptive optics devices trying to limit the number ofactuators [18] because important application fields, such as laserphysics [17], ophthalmology [19–22] and astronomy [2,7,9], showthat, in most of the cases, 90% of the aberration weight is in the firstthree aberration orders. In many of those cases the benefit and theconvenience of the use of AO are overcome by the skills andinstrumentation required or by the convergence time. The devicepresented addresses three problems: 1) the use of least actuators aspossible, 2) the introduction of the DMmodal control in a continuousactuators arrangement, and 3) reduce the algorithmic complexity forthe correction of optical systems or enhancement of laser processes.The main feature of the MDM is that the actuator response is directlyrelated to the optical aberrations allowing for a more versatile andstraightforward use than conventional discrete actuator deformablemirrors. We will show that the voltages necessary to drive the mirrorcan be directly computed from the Zernike decomposition terms ofthe target aberrations leading to an ideal device for modal control insensorless applications.

http://dx.doi.org/10.1016/j.optcom.2011.03.010

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http://www.sciencedirect.com/science/journal/00304018

http://dx.doi.org/10.1016/j.optcom.2011.03.010

Fig. 1. Diagram of the Distributed Actuator Deformable Mirror which shows theelectrode contacts and the resistive layers' layout.

3468 S. Bonora / Optics Communications 284 (2011) 3467–3473

2. Device

TheMDM is a membrane electrostatic deformablemirror composedof a silvered nitrocellulose membrane (5 μm thick) suspended 70 μmover the actuators by polyamide spacers (Fig. 1). The membrane toelectrodes distance is calibrated in order to generate a stroke of about10 μm with the maximum voltage. This operational regime allows to

Fig. 2. Simulation results. Left column: voltage distribution, electrostatic pressure and megeneration of the astigmatism aberration. The application of both positive and negativemembrane which corresponds to a zero electrostatic pressure (dotted line 0 P).

work in the safe area, preserving the membrane from snap down to theactuators. The prototype mounts a membrane of 19 mm in diameter(initial astigmatism of 0.1 μm peak to valley in the active region)designed for an optimal active region of 10 mm. The deformable mirrorwas driven by a multichannel electronic driver for deformable mirrors(Adaptica IO32) which can supply 260 V over 32 channels while themembrane was connected to a voltage reference of 130 V in order togenerate on the membrane both positive and negative voltages.

The DM is actuated by the electrostatic pressure p(x,y) betweenthe actuators and the metalized membrane which deforms the mirrorM(x,y) according to the Poisson equation [8]:

ΔM x; yð Þ = 1Tp x; yð Þ = ε0

TV x; yð Þ2

d2ð1Þ

where T is the mechanical tension of the membrane.The device is composed of 3 actuators placed on three concentric

rings: the first actuator has one contact, the second and the third oneshave 4 contacts each (Fig. 1). The actuators are composed of a graphitelayer 35 μm thick which presents a resistivity of 350 Ωm whichcontinuously distributes the voltage. The current estimated for eachchannel applying the maximum voltage is about 60 μA with a powerconsumption for each actuator of about 10 mW.

The voltage distribution over the resistive layer with resistivity ρcan be computed solving the Laplace equation for the scalar electricpotential [23]:

1ρΔV x; yð Þ = 0: ð2Þ

By a proper design of the deformable mirror the third actuator canbe used to generate piston, tilt and astigmatism, the second actuator

mbrane shape obtained applying 130 V to the 9th contact. Right column: example ofvoltages generates a zero voltage distribution (dotted line 0 V) in the middle of the

image of Fig.�2

Fig. 3. Measurements and simulation result comparison for astigmatism and comaaberration (top and middle panel). The figures show the peak to valley aberrationmagnitude in function of αa and αc respectively. Bottom panel: measurements of thegeneration of spherical aberration and defocus in function of the parameter Asfe withAdef=0.13.

3469S. Bonora / Optics Communications 284 (2011) 3467–3473

can be used for the generation of the coma and defocus, and the firstactuator is used for the generation of spherical aberration. In order tosolve numerically both formulas (1) and (2) we applied the recursivefinite difference method. Fig. 2 reports some examples of voltagedistribution, electrostatic pressure and mirror deformation.

The simulations show how the voltage of a single actuator isrelated to the mirror shape. Vdovin et al. [18] demonstrated thatoutside the active area are necessary 2N+1 actuators for thegeneration of N aberration orders. Here we show that, exploitingthe continuous distribution of the electric field and that theelectrostatic pressure is proportional to the square of the voltage,driving the mirror with both positive and negative voltages it ispossible to use less than 2N+1 actuators. Fig. 2 shows that theapplication of opposite sign voltages to adjacent contacts generates asort of virtual electrode, i.e. an area of zero voltage and pressure (seeFig. 2, OV and OP dotted lines) leading to the generation of theastigmatism with just 2N contacts outside the active area.

3. Device characterization

In the following section we compare the simulation andmeasurement results for the generation of the tilt, defocus, comaand spherical aberrations. The measurements were carried out byinterferometric technique of the deformable mirror surface (Zygo).Introducing the Zernike polynomial decomposition of a wavefrontW(x,y) according to OSA (Optical Society of America) standards [24]:

W x; yð Þ = ∑i; j

cij Z x; yð Þij

where cji are the aberration coefficients and Z

ij x; yð Þ are the Zernike

polynomials, the performance is evaluated computing the rmsresidual error with respect to the polynomial target Zm

n x; yð Þ, themagnitude of the Zernike terms and the spectral purity according tothe formula [25]:

Pnm =

cnmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑i; j

cij� �2r ð3Þ

where m and n are the indexes of the target Zernike aberration. Thenpurity is one (Pmn =1) if the mirror shape is an ideal Zernike m, npolynomial, while it is zero if the decomposition of the mirror shapeover the Zernike base generates components on other base vectorsbut the Zm

n x; yð Þ. Hence the spectral purity represents a method formeasuring how well the target shape Zm

n x; yð Þ has been generated.The tilt aberration described by the coefficients c1

−1, c11 can begenerated by applying an electrostatic pressure distributed over atilted plane to the first electrode. This goal can be achieved byapplying the following voltages to the contacts 6–9, (see Fig. 1):

V6 =ffiffiffiffiffiffiffiffiAtilt

p Vmax

21 + sin αt + π = 2ð Þ½ �


p Vmax

21 + sin αt + πð Þ½ �


p Vmax

21 + sin αt + 3π = 2ð Þ½ �


p Vmax

21 + sin αtð Þ½ �

8>>>>>>>>>>><>>>>>>>>>>>:

ð4Þ

where Atilt = ktffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic−11

� �2 + c11� �2q

and kt is a constant which nor-

malizes Atilt between (0 and 1) and controls the magnitude of the tilt

while the angle is: αt = tan−1 c−11

c11

!. Vmax is the maximum voltage

supplied to the membrane (Vmax=130 V). The simulations andmeasurement results show that the magnitude of the tilt is constantwith the angle α with an average purity of 0.9963.

In order to generate astigmatism (c2−2, c22) we followed the samerule which was valid for the tilt with the exception that the useof both positive and negative voltages yields to the generation ofa symmetric electrostatic pressure (see for example Fig. 2 secondcolumn).

image of Fig.�3


Thus, the astigmatism aberration can be generated again on thefirst actuator by the following voltages:

f v6 =ffiffiffiffiffiffiffiffiAast

pVmax sin αa + π = 2ð Þ½ �

v7 =ffiffiffiffiffiffiffiffiAast

pVmax sin αa + πð Þ½ �


pVmax sin αa + 3π = 2ð Þ½ �


pVmax sin αað Þ½ �

V6−9 = kl αað Þv6−9

ð5Þ

where the parameter Aast = kaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic−22

� �2 + c22� �2q

controls the magni-

tude (ka is a parameter that normalizes Aast between 0 and 1) and

αa = tan−1 c−22

c22

!, kl αað Þ = 1

max v6−9ð Þ= is a linearization factor. The

simulations and measurement results are reported in Fig. 3 whichshows the magnitude of the astigmatism aberration as a function ofαa. The error in the generation of the astigmatism was measured bythe spectral purity over the first five Zernike orders and has anaverage value of 0.9840 with an average rms error on the target of57 nm. Using the second actuator we have been able to generate comaaberration(c3−3, c33) over a biased membrane [9] as reported in Fig. 3both theoretically and experimentally.

In this case the formulas are:

V2 =ffiffiffiffiffiffiffiffiffiffiffiffiAcoma

pVmax sin αc + π = 2ð Þ½ �


pVmax sin αc + πð Þ½ �


pVmax sin αc + 3π = 2ð Þ½ �


pVmax sin αcð Þ½ �

8>>><>>>:

ð6Þ

where Acoma = kcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic−33

� �2 + c33� �2q

, kc is a constant which normalizes

Acoma between 0 and 1. The value of αc is tan−1 c−33

c33

!. The voltages

computed by formula (6) generate a combination of coma and tiltwhich can be compensated using Eq. (4) in dynamic applications. Theaverage spectral purity of the coma aberration is 0.92 with a meandeviation from the target of about 54 nm. As shown in Fig. 3 the Peakto Valley (PtV) value of astigmatism and coma presents somedeviation in function of αa and αc from the simulations because ofsome misalignment in the parallelism between the membrane andthe actuators. In astigmatism this deviation from the ideal case hastwo symmetric lobes because of the double periodicity of thisaberration.

Finally, both defocus and spherical aberrations can be obtainedusing contacts 1–5. In order to generate those aberrations indepen-

Fig. 4. Electrode configuration and active area of the nine electrodes (DM9) and eleven elecmirror (MDM).

dently we used the following procedure. We computed the projectionof the membrane shape S2, generated by the actuator 2 puttingV2=V3=V4=V5 and the actuator 1 (S1) over the Zernike termdefocus Z0

2 and spherical aberration Z04.

S2 = aZ02 + bZ0

4

S1 = cZ02 + dZ0

4

8<: ð7Þ

Since the Zernike modes are orthogonal the parameters a, b, c, andd can be computed by the scalar products:

a = S2⋅ Z02

b = S2⋅ Z04

c = S1⋅ Z02

d = S1⋅ Z04

8>>>>>><>>>>>>:

ð8Þ

Now, in order to generate independently combinations ofspherical aberration and defocus, the following parameters must beused:

Z04 =

S2a

ba

+dc

� �−S3c

ba

+dc

� � = a1 S2 + b1 S1

Z02 =

S3b

ab

+cd

� �−S3d

ab

+cd

� � = c1 S2 + d1 S1

→V2−5 = Vmax

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAsfea1 + Adef c1

qV1 = Vmax

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAsfeb1 + Adef d1

q :

8><>:

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð9Þ

The control coefficients Asfe = ks Z04 and Adef = kd Z

02 determine the

magnitude of the spherical and defocus aberrations while theconstants ks ans kd fix the range of Asfe and Adef between [0,1] and[−1,1] respectively. In thosemeasurements we kept fixed the defocuswhile the spherical aberration was linearly changed. Those valueshave been used for the measurements carried out in the bottom panelof Fig. 3. We achieved a spectral purity of 0.9948 with an average rmserror of 14 nm.

We then investigated the superposition of the aberrations that wecan generate using Eqs. (3)–(9). It has been demonstrated that themembrane shape obtained acting on different electrodes can becomputed by algebraic sum of the influence functions [6,8,10]. Thenastigmatism (first ring), defocus and spherical aberration (second andthird ring) add linearly. This is not true if we generate two aberrationsacting on the same ring because the electrostatic pressure is

trodes DM11 of deformable mirrors and for the modal distributed actuator deformable

image of Fig.�4

Fig. 5. Experimental layout used for the image sharpening optimization. The sample isimaged on the CCD camera through the lenses f1, f2 and the X3 telescope. Theaberrations are introduced by a low optical quality window plate.


quadratically related to the voltage distribution and the doubleproduct of the two factors adds a spurious contribution. This happensin case of coma and defocus, and, tilt and astigmatism. In the formercase, since defocus means the addition of a constant value over thesecond ring the effect of the double product is just to amplify thecoma. Hence in image sharpening applications this is not a problem ifthe defocus is corrected before the coma as illustrated into details inthe last paragraph. In the latter case tilt and astigmatism are notindependent because the non-linear relationship which arisesbecause of the virtual electrode formation. From simulations weestimated the influence of the tilt on the astigmatism value in about20% of its peak to valley amplitude.

Following from the previous section, the MDM response can bedirectly derived by the Zernike decomposition of the incomingwavefront realizing a modal control of the DM rather than from theinfluence functions (zonal control).

Thus, the voltages which generate the aberrations described by theZernike coefficients are given by:

V1−9 = Vdefocus + Vastigmatism + Vcoma + Vspherical

= V6−9 c−11 ; c11

� �+ V1−5 c02

� �+ V6−9 c−2

2 ; c22� �

+ V2−5 c−13 ; c13

� �+ V1−5 c04

� �ð10Þ

where the complete description of the voltages can be found in Eqs.(5–6), (9). Thus, using Eq. (10) it is possible to generate an aberrationstarting from the knowledge of its Zernike spectrum.

4. Simulations

In order to understand the advantages of the MDMwith respect tothe state of the art deformablemirrors [26] we carried out simulationsof aberration generation for electrostatic DMwith 9 electrodes (DM9)and with 11 electrodes (DM11) distributed over one ring outside andtwo inside the active area as illustrated in Fig. 4 following theprocedure described in Ref. [6]. Since bipolar voltages play afundamental role in the use of MDM, for a better comparison weuse it for both DM9 and DM11 as well. Results are summarized inTable 1. DM9 uses the same number of high voltage lines as MDM butcan generate only tip-tilt, defocus, coma and spherical aberration.Bipolar supply does not increase performances because the virtualactuators are blocked between adjacent actuators while in the case ofMDM it floats depending on actuator voltages. Increasing the deviceof one actuator per ring, increases performances allowing for thegeneration of polynomials of radial order 2 such as astigmatism

Z22; Z

−22

� �and Z2

4; Z−24

� �even if with poor spectral purity because of

the asymmetries which arise using 5 electrodes for generatingrespectively 2 and 4 lobes. Table 1 shows in the 6th column theresults of the MDM which allows to generate the same aberrations ofDM11 using 9 high voltage lines and with better spectral purity. In

Table 1Results of the generation of the 4th order aberrations for a 9 and 11 electrodes DM andfor the DMD. Gray background cells indicate the polynomials generated with a spectralpurity larger than 0.9.

Zernikemode

9 electrodes DM 11 electrodes DM Distributed actuator 9 contacts

Unipolar Bipolar Unipolar Bipolar Bipolar

111, 11−1 0.93 0.99 0.95 0.99 0.9920 1 1 1 1 1222, 22−2 – – 0.83 0.93 0.98311, 31−1 0.93 0.92 0.95 0.96 0.98333, 33−3 – – – – –

40 0.99 0.99 0.99 0.99 0.99422, 42−2 – – 0.66 0.81 0.92444, 44−4 – – – – –

order to reach the same performance of theMDM at least 13 actuatorsare necessary.

5. Application in image sharpening

We finally demonstrate the effectiveness of this mirror using it in aImage Sharpening (IS) setup (Fig. 5).

A sample (Honeybee wing) is imaged through the spherical lensesf1 and f2 to a CCD camera and magnified 4 times. An aberrator plate(plexiglass disk 4 mm thick) is positioned between f1 and the MDMand generates a strong image degradation as illustrated in Fig. 7a. Inorder to correct for the aberrations introduced by the aberrator platewe applied a merit function based on the optimization of low spatialfrequencies [16]. The merit function is the integral of the Fouriertransform of the acquired image (I(x,y)) over an annular region R:

Me = ∬RFT I x; yð Þf gdudv ð11Þ

where FT is theFourier transformoperator and (u,v)are the coordinates inthe Fourier plane. R corresponds to the regionm1b |u|bm2,m1b |v|bm2.In this case Booth et al. demonstrated that the merit functionMe can be

Fig. 6. Merit function evolution during the aberrations correction. Scatter datarepresent the acquisitions which are fitted by the continuous lines. The best pointsare indicated by the arrows.

image of Fig.�5

image of Fig.�6

Fig. 7. a) Initial image after the insertion of the aberrator plate. Images acquired after the correction of: b) defocus, c) astigmatism, d) spherical aberration, and e) coma. (Online only)Movie 1: optimization sequence of a Honeybee Leg. The movie shows the identification of the optimum image trying the correction of defocus (D), astigmatism (A, αAst) and coma(C, αComa).


approximated as the sum of the square of the aberration coefficients.This means that exploiting the independent generation of aberrations ofthe MDM we can correct the image acting independently over eachaberration in sequence, thus reducing algorithmic search complexity. Itis important to note that this is not valid for zonal deformable mirrors.

The correction was performed computing the merit function toimages acquired applying aberrations in the following order: defocus,astigmatism 0° and 45°, spherical aberration, and coma 0° and 90°. Toreduce the number of acquisitionswe sampled themerit functionover 5or 7 values of the aberration coefficients and then used a polynomialfit to find the optimum value for each aberration. Fig. 6 shows theimprovement of themerit functionwhich increases its value during thecorrection. Fig. 7 shows the optimum images after the correction of eachaberration. This method demonstrated to be effective over a largenumber of trialswithmanydifferent samples andalways using less than36 measurements. (Online only) Movie 1 shows an example ofoptimization sequence on a honeybee leg sample.

6. Conclusions

In conclusion we have shown that the technique of the distributedactuator can be used for the realization of a modal deformable mirrorwith a minimum number of electrodes introducing a modal control oflow order aberrations with an excellent quality and linearity.Moreover, the MDM technique allows to directly connect the effectof the actuators to the optical quality through the direct generation ofZernike polynomials. The convenience of use of the electrostatic MDMhas been demonstrated in an image sharpening setup. The concept ofthis design, which exploits resistive layers, can be implemented inother technologies such as bimorph mirrors.

Supplementarymaterials related to this article can be found onlineat doi:10.1016/j.optcom.2011.03.010.

Acknowledgments

The realization of this device was funded by Adaptica srl (Patentpending M3101733/IT). The author is grateful to Cosmo Trestino forthe device design and to Fabio Frassetto and Tommaso Occhipinti forthe useful discussions.

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image of Fig.�7


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distributed actuators deformable mirror for adaptive optics

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