Computing the Influence Functions of an AdaptiveOptics Large Deformable Mirror: the NumericalMethod and the Experimental Data

C. Del Vecchio˚,1, R. Briguglio1, M. Xompero1, A. Riccardi1, D. Gallieni2 and R. Biasi31INAF-OAA, 2ADS International, 3Microgate˚L. Enrico Fermi 5 I-50125 Firenze, Italy, cdelvecchio@arcetri.astro.it

Abstract:Adaptive Optics units are mounted

on many large telescopes, thus enablingdiffraction-limited astronomical imagesthrough compensation of the effects of at-mospheric turbulence. The design of thecontrol system of the Deformable Mirrorincludes a feed-forward block, aimed for in-creasing the efficiency of the control loop.Such a block is based on the stiffness ma-trix of the Deformable Mirror, defined asthe force required to move a single actua-tor while all the others are constrained atzero positions. The mirror shape caused bythis force pattern is called Influence Func-tion. This paper discusses the computationmethods adopted to calculate the influencefunctions of the LBT and VLT DeformableMirrors, grounded in the capabilities of theComsol/Matlab interactions. The compar-ison of the Comsol results with the dataavailable from the optical tests shows thatthe methods analyzed in this paper are suit-able for accurately predicting the delicateoptomechanics of the Deformable Mirrors.

Keywords: Adaptive Optics, Optomechan-ics, Influence Function, FEA

1 IntroductionThe Adaptive Optics (AO) units imple-mented on many large telescopes allow toobtain diffraction limited astronomical im-ages by compensating the effects of theatmospheric turbulence. The control sys-tem of a typical AO unit, sketched infigure 1, provides the correction of the dis-torted wavefront deforming the DeformableMirror (DM) by means of several actuators.The closed-loop response is dramatically in-creased when a feed-forward, open-loop cor-rection is added to the servo system, as dis-

cussed in [2]. This correction is based on thestiffness matrix of the DM, which is oper-atively defined by arbitrarily displacing oneof the DM actuators along the normal to thethin mirror surface, while all the other actu-ators are constrained at their nominal posi-tions, and calculating all the reaction forces.In the AO terminology, the shape of the DMwhen poking a single actuator is known asthe influence function (IF), whose numeri-cal computation is discussed in this paper.The method adopted to calculate the IF ofthe concave DM of the Large Binocular Tele-scope (LBT), described in [6], and of the con-vex DM of the Very Large Telescope (VLT),described in [1], is based on some Matlabscripts and functions, found on the capabil-ities of LiveLink for Matlab.

Figure 1: Scheme of an AO control system.

Starting from the optical and physi-cal parameters of the thin mirrors andthe geometrical layouts of the actuators, afirst script builds the geometry, applies thephysics required to solve the problem andsets up the meshes. In fact, the complexityof the actuator geometry (the LBT and the

VLT DMs have 672 and 1170 actuators, re-spectively) and the accuracy required by theoptics when modeling the geometry of theactual glass shell makes the Comsol GUI in-effective. Moreover, as many auxiliary coor-dinate systems as the number of actuatorsshould be defined in order to properly ap-ply the required constraints. Unfortunately,this large number increases dramatically thedata to be stored by Comsol — which, as amatter of fact, makes this choice unworkable.Thereby, an alternative method is carriedout: the constraints are implemented viaa user-defined interpolation function, whoseinput is defined by the Matlab algorithm dis-cussed in section 2.2. Finally, a further func-tion runs the computational loop and per-forms the post-processing of the results, asexamined in section 3.

The comparison of the displacements ac-quired by interferometric analyses and of thethe actuator forces derived from electricalmeasures — such data have been collectedduring the test runs, as explained in [6] —with the Comsol results shows a good agree-ment, analyzed in section 4.

2 Static assumptions andsimplifications

2.1 The model

LBT VLTRo 455.5 mm 558 mmRi 28 mm 48 mmtm 1.6 mm 2.0 mmRb 1994.9 mm 4575.30 mmKb 0 0Rf 1974.24 mm 4575.3 mmKf ´0.7330 ´1.66926N 672 1170

Table 1: The optical and physical mainparameters of the LBT and VLT Zerodur

DMs. Ro and Ri designate the physical outerand inner radii, respectively; Kf and Kb the

front and back surface conic constants,respectively; Rf and Rb the front and back

surface optical radii, respectively; tm the meanthickness; N the total number of actuators.

Starting from the optical parameterslisted in table 1, the aspherical and sphericalaxially symmetric surfaces of both the DMscan be generated in Matlab as zf “ zf prq

and zb “ zbprq, with r “a

x2 ` y2. Becauseof the values of table 1, the mean surfacez “ p1{2qpzf ` zbq as well as the thicknesst “ |zf ´ zb| are very accurately fitted by thepolynomials vprq and qprq, respectively, ofdegree M “ 9. Moreover, also the angleϕ “ ´arctanpdz{drq, defined in section 2.2,can be accurately approximated by the poly-nomial pprq of degreeM . Naming V, P, andQ the M ` 1 coefficients of vprq, pprq, andqprq, respectively, z, t, and ϕ can be analyt-ically defined in the Comsol Finite ElementModel (FEM) according to (1), (2), and (3),respectively. As a consequence, the geome-try generation is quite easy and all the rele-vant variables are available in Comsol.

z “M`1ÿ

i“1

V piqrM`1´i (1)

ϕ “M`1ÿ

i“1

P piqrM`1´i (2)

t “M`1ÿ

i“1

QpiqrM`1´i (3)

Both the VLT and LBT DMs strokes areactuated by magnetic forces, as describedin [3], which are applied perpendicularlyto the DM by a puck built of a magnetand a glass disk glued to it via three gluespots, angularly equispaced at a distancerg “ 2 mm from the puck axis — the bluearrow in figure 2. Modeling the full puckwith a solid FEM, restrained at the threeglue/DM interfaces and loaded by a bodyforce inside the magnet, allows to calcu-late Kp, the overall stiffness of the puck asKp “ 112.5 Nˆ µm´1. As the spot diame-ters are as small as « 50µm, the magneticforces applied to the Zerodur DM, mod-eled as a shell, can be approximated by thethree trusses∗ of length lp “ 10 mm† per-pendicular to the DM surface (pictured inblack in figure 2) and the three rigid beamsof length rg “ 2 mm (pictured in yellow infigure 2). The actual response of the fullpuck is well approximated by assigning a

∗The truss elements are modeled as beam elements with negligible moments of inertia and torsional constant,because of a known Comsol issue about dealing with a large number of trusses†As no force is applied perpendicularly to the actuator axis, lp an be arbitrarily chosen.

null density to the truss and beam elementsand by the following two inputs: naming Ap

the truss cross sections and Ep its Young’smodulus, Ap must satisfy the relationshipKp “ 3EpAp{lp; the beam elastic and crosssection data are chosen in order to definea beam rigidity several orders of magnitudelarger that the truss and DM stiffnesses. Fi-nally, the three beam/truss interfaces are re-strained along the red and green directionsin figure 2, perpendicular to the blue axis,identifying the stroke direction.

Figure 2: Approximation of the puck. See thetext for a discussion.

The effects of the central flat mem-brane, described in [4], placed perpendic-ularly with respect to z, whose z stiff-ness is Km, are simulated by the edge loadp´Km{p2πRiqq ˆ pw ´ 0q and the prescrib-ing a null radial and tangential (with respectto the xy plane) displacement of the edgesat r “ Ri.

The VLT and LBT obtained models con-sist of 32824 triangular elements and 7020beam elements and 19144 triangular ele-ments and 4032 beam elements, respectively.

2.2 Coordinates functioning

As the actuators axes are perpendicular tothe DM surface, the natural manner to prop-erly define displacements and forces withinComsol is the implementation of as manyauxiliary coordinate systems as the numberof the actuators. Defining ϕ the angle be-tween the optical axis Z and the normalto the DM at the point P ” pX,Y q, withR “

a

X2 ` Y 2 and θ “ arctanpY {Xq, the“global-to-local” and “local-to-global” trans-

formation matrices G and L “ G´1 are de-fined as in (4) and (5), respectively:«

cospϕq cospθq cospϕq sinpθq sinpϕq-sinpθq cospθq 0

-cospθq sinpϕq -sinpϕq sinpθq cospϕq

ff

(4)

«

cospϕq cospθq -sinpθq -cospθq sinpϕqcospϕq sinpθq cospθq -sinpϕq sinpθq

sinpϕ 0 cospϕq

ff

(5)

Nevertheless, the definition of a number ofcoordinate system as large as 672 or 1170causes a Comsol crash — likely because sev-eral hundreds of nodes isn’t manageable bythe program. For this reason, an alternativeway to handle forces and constraints mustbe set up. Naming XIi,j and YIi,j , withi “ 1, 2, . . . , 4N and j “ 1, 2, 3, the X andY coordinates of PI — the three “interfacenodes” identified by the intersection of thetrusses and the beams in figure 2, whose an-gles with respect to the actuation axis are ψj

(ψ takes only the three values 0 and ˘p2{3qπfor j “ 1, 2, 3) — andXFi

and YFitheX and

Y coordinates of PF — the “actuation node”of the ith actuator, identified by the intersec-tion of the three yellow beams in figure 2 —,we can define a 4N ˆ 5 matrix whose rowsfrom i to i`4 of the table are defined in (6).

»

—

—

–

XIi,1 YIi,1 XFiYFi

ψ1

XIi,2 YIi,2 XFi YFi ψ2

XIi,3 YIi,3 XFiYFi

ψ3

XFiYFi

XFiYFi

0

fi

ffi

ffi

fl

(6)

The matrix defined in (6) can be easily gen-erated in Matlab and written as a file, whichcan be used as table data source of the Com-sol nearest neighbor interpolation functionΓpx, yq. Such a function, with the nodalvariables ζi,j “ XFi

, and ηi,j “ YFi, al-

low to associate to each of the three PI co-ordinates the coordinates of the correspon-dent PF ; moreover, Γ defines for each PI

the three angles Θi,j “ ψj . Let’s defineρ “

a

ζ2 ` η2, θ1 “ arctanpη{ζq, and, simi-

larly to (2), ϕ1 “M`1ÿ

i“1

P 1piqρM`1´i, and sub-

stitute θ with θ1 and ϕ with ϕ1 in (4): thelocal displacement ul “ rul; vl;wls “ Gu,where u “ ru; v;ws is the displacementvector in the global coordinate system, isdefined in the coordinate system relativeto each actuation axis. If the kth (k “

1, 2, . . . , N) actuator is displaced by w˚

along its axis, we can define N pointwiseconstraints as in in (7), in order to apply the

correct strokes to PF , and 3N pointwise con-straints as in (8) and (9), in order to radi-ally and tangentially (in the cylindrical co-ordinate system of each actuator, i.e. alongthe red and green directions, respectively, infigure 2) restrain PI . As a result, addinga small set of analytic function allows thecomputation of the N IFs avoiding any ad-ditional auxiliary coordinate system.

wlk “

#

0 if k ‰ i

w˚ if k “ i(7)

ulk cospψjq ` vlk sinpψjq j “ 1, 2, 3 (8)

-ulk sinpψjq ` vlk cospψjq j “ 1, 2, 3 (9)

3 ResultsA Matlab function, after reading the modelfile built by a script discussed in sections2.1 and 2.2, runs the IF, taking the actu-ator number(s) as input vector and comput-ing the forces and the displacements in theproper coordinate system.

3.1 LBTRunning the above described Matlab func-tion gives the LBT DM 672 by 672 stiffnessmatrix, whose main diagonal is plotted infigure 3. An example of a single IF — theone of the actuator # 145 — is plotted infigure 4 in terms of actuator forces, whilefigure 5 shows the DM deformations of theportion of glass centered in the location ofthe actuator # 145.

Figure 3: The main diagonal of the 672 by 672LBT DM stiffness matrix vs. the actuator

geometry.

Figure 4: LBT DM: the actuator forces whenthe actuator #145 is displaced by w˚

“ 1µm.

Figure 5: LBT DM: displacement normal tothe surface when the actuator #145 is

displaced by w˚“ 1µm.

3.2 VLT

The process described in section 3.1 givesthe VLT DM 1170 by 1170 stiffness matrix,whose main diagonal is plotted in figure 6.Again, an example of a single IF — the oneof the actuator # 377 — is plotted in figure 7in terms of actuator forces, while figure 8shows the DM deformations of the portionof glass centered in the location of the actu-ator # 377.

Figure 6: The main diagonal of the 1170 by1170 VLT DM stiffness matrix vs. the actuator

geometry.

Figure 7: VLT DM: the actuator forces whenthe actuator #377 is displaced by w˚

“ 1µm.

Figure 8: VLT DM: displacement normal tothe surface when the actuator #377 is

displaced by w˚“ 1µm.

4 Experimental validation

The optical measurement of the mirror IF ispart of the standard calibration procedurefor deformable mirrors. As we want to com-pare the stiffness obtained from the FEAmodel with that from experimental data, weneed to measure, for each IF, the actuatorforce applied and the actuator displacementproduced.

4.1 Set-up and procedure

IFs are measured by mean of an interfer-ometer, that provides a deformation map ofthe mirror using a reference Helio-Neon laserwhose wavelength is 632.8 nm. The typicalaccuracy of an interferometer is 1 nm, overa capturing range of « 40µm. During theoptical test, the DM is mounted on the tele-scope or in a test tower and illuminated frombelow by the laser. Figures 9 and 10 showtwo IF realizations of the LBT and the VLT,respectively. In both cases, the interferomet-ric map of a random selected actuator IF isshown over the full mirror and over a smallsub-portion of it. In typical conditions, themost relevant noises affecting the measure-ments are convection and vibration. The vi-bration noise is compensated by subtractingtip and tilt from each image. The convec-tion is reduced via a differential samplingof n frames; as a result, the typical resid-uals range from 1 to 5 nm RMS, to be com-pared with a typical actuator displacementof „ 100 nm. Such a differential samplingallows to reject constant and slowly varyingterms. In order to further reduce the mea-surement noise, the images are collected atthe fastest interferometer frame rate (25 Hz).Operatively, the procedure may be summa-rized as follows. Each actuator is sequen-tially commanded n times to the same pos-itive and negative position, which are im-aged (s` and s´, respectively) by the inter-ferometer. The resulting deformation map,

S “1

n

nÿ

i“1

ps`i´ s´i

q

2, provides the optical

shape of the actuator IF. The actuator forceis measured by a current gauge installed oneach actuator, whose reading is then con-verted into a force value via a calibrationcoefficient.

Figure 9: Example of a LBT interferometricdisplacement (see the text for a discussion).

Figure 10: Example of a VLT interferometricdisplacement (see the text for a discussion).

4.2 Data analysis

So far, we collected the IF images and thecorresponding actuator forces. The lattercan be absolutely calibrated by comparingthe vertical component of the DM weightwith the mean force requested to keep theshell floating. However, this procedure isa global calibration, i.e. it is not intendedto compensate actuator by actuator differ-ences — such imprecision affects the stiffnesscomputation. The IF images are analyzed inorder to measure the actuator displacement,according to the following procedure: the lo-cation of the ith actuator is identified in theimage; a mask M0i is drawn around the ac-tuator position: the diameter of the mask

is «1

5of the inter-actuator distance, to en-

close the peak only of the IF; the values ofthe pixels within M0i are averaged together,obtaining the value ai representing the ithactuator position; a large maskM1i is drawnaround the actuator position: the size is «4 times the inter-actuator distance, i.e. largeenough to fully exclude the IF; the values ofthe pixels outsideM1i are averaged together,obtaining the value ri, that is the mirror ref-erence position; finally, the displacement ofthe actuator ith is computed as di “ ai´ ri.The actuator stiffness, defined as in the FEA

model, is computed as ki “fidi.

4.3 Discussion

Figure 11: LBT DM IF: comparison ofcomputed and measured data. In both cases,the IFs are normalized to their maximum

value.

Figure 12: VLT DM IF: comparison ofcomputed and measured data. In both case,the IFs are normalized to their maximum

value.

The LBT and VLT stiffness values are plot-ted in figures 11 and 12, respectively, versusthe actuator radial distance: as the actu-ators are arranged in rings, the points areaccumulated in correspondence of the ringradii. The spread of the experimental stiff-ness values is larger than the FEA modelone, because of some limitations of the sam-pling procedure. The most important lim-itations come from: actuator by actuatorforce calibration, relatively poor image res-olution, tolerances of the actuator locationsin the images, poor IF visibility for the ac-tuators located at the edges and close to thespider arms (the hidden ones are not consid-ered), effects of malfunctioning actuators onthe surrounding ones.

5 Conclusions

Although originated by a flaw of Comsol,a de facto unworkability of a large numberof coordinate systems, the functioning of aFEA definition as delicate as the nodal con-straint reveals the powerful of the flexibilityof Comsol. In fact, the availability of theMatlab computational tools allows to imple-ment a function of the spatial coordinatesable to apply the required displacementseven in a complex geometry, with a com-pact definition of the pointwise constraintsthat allows a solution both fast and accurate.The precision of the results is demonstratedby the interferometric and electric measure-ments of the displacements and forces, re-spectively. In conclusion, the influence func-tions of an Adaptive Optics Deformable Mir-ror can be truthfully evaluated by the nu-merical method discussed in this paper — apowerful and reliable computational tool forthe opto-mechanical design.

References

[1] Roberto Biasi, Mario Andrighettoni,Gerald Angerer, Christian Mair, DietrichPescoller, Paolo Lazzarini, Enzo Anacle-rio, Marco Mantegazza, Daniele Gallieni,Elise Vernet, Robin Arsenault, Pierre-Yves Madec, Philippe Duhoux, Ar-mando Riccardi, Marco Xompero, RunaBriguglio, Mauro Manetti, and MarcoMorandini, VLT deformable secondarymirror: integration and electromechan-ical tests results, Adaptive Optics Sys-tems III (Brent L. Ellerbroek, EnricoMarchetti, and Jean-Pierre Véran, eds.),Proc. SPIE, vol. 8447, SPIE, 7 2012.

[2] Guido Brusa, Armando Riccardi, Valde-maro Biliotti, Ciro Del Vecchio, PieroSalinari, Paolo Stefanini, Paolo Man-tegazza, Roberto Biasi, Claudio Fran-chini, and Daniele Gallieni, Adaptive sec-ondary mirror for the 6.5 conversionof the multiple mirror telescope: firstlaboratory testing results, Adaptive Op-

tics Systems and Technology (DomenicoBonaccini and Robert K. Tyson, eds.),Proc. SPIE, vol. 3762, 1999, pp. 38–49.

[3] Ciro Del Vecchio, Supporting a magnet-ically levitated, very thin meniscus foran adaptive secondary mirror: the opti-mization of the magnetic circuit, Adap-tive Optics and Applications (Robert K.Tyson and Robert Q. Fugate, eds.),Proc. SPIE, vol. 3126, 7 1997, pp. 605–613.

[4] Ciro Del Vecchio, Daniele Gallieni, Hu-bert M. Martin, Armando Riccardi,Guido Brusa, and Roberto Biasi, Designimprovements of the LBT adaptive sec-ondary, Beyond Conventional AdaptiveOptics (Elise Vernet, Roberto Ragaz-zoni, Simone Esposito, and Norbert Hu-bin, eds.), Proc. ESO, vol. 58, ESO, 52001, pp. 435–441.

[5] Armando Riccardi, Guido Brusa, CiroDel Vecchio, Roberto Biasi, Mario An-drighettoni, Daniele Gallieni, Fabio Zoc-chi, Michael Lloyd-Hart, Francois Wildi,and Hubert M. Martin, The adaptivesecondary mirror for the 6.5 m con-version of the multiple mirror telescope,Beyond Conventional Adaptive Optics(Elise Vernet, Roberto Ragazzoni, Si-mone Esposito, and Norbert Hubin,eds.), Proc. ESO, vol. 58, ESO, 5 2001,pp. 55–64.

[6] Armando Riccardi, Marco Xompero,Runa Briguglio, Fernando Quirós-Pacheco, Lorenzo Busoni, Luca Fini,Alfio Puglisi, Simone Esposito, CarmeloArcidiacono, Enrico Pinna, PieroRanfagni, Piero Salinari, Guido Brusa,Richard Demers, Roberto Biasi, andDaniele Gallieni, The adaptive secondarymirror for the large binocular telescope:optical acceptance test and preliminaryon-sky commissioning results, AdaptiveOptics Systems (Brent L. Ellerbroek,Michael Hart, Norbert Hubin, andPeter L. Wizinowich, eds.), Proc. SPIE,vol. 7736, SPIE, 6 2010.