MARS Model of Biom deformable Markets and Autopilot actuators

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1 Deformable mrror model for open-loop adaptve optcs usng multvarate adaptve regresson splnes Dan Guzmán, 1, 2,* Francsco Javer de Cos Juez, 3 Fernando Sánchez Lasheras 4 Rchard Myers 1 and Laura Young 1 1 Physcs Department, Durham Unversty, South Road Laboratores, Durham, DH1 3LE, UK 2 Electrcal Engneerng Department, Catholc Unversty of Chle, Vcuña Mackena 4860, Santago, Chle 3 Mnng Explotaton and Prospectng Department, C/Independenca nº13, Unversty of Ovedo, Ovedo, Span 4 TecnProect S.L.,Marqués de Pdal 7, 3004 Ovedo, Span *dan@astronventons.com Abstract: Open-loop adaptve optcs s a technque n whch the turbulent wavefront s measured before t hts the deformable mrror for correcton. We present a technque to model a deformable mrror workng n open-loop based on multvarate adaptve regresson splnes (MARS), a nonparametrc regresson technque. The model s nput s the wavefront correcton to apply to the mrror and ts output s the set of voltages to shape the mrror. We performed experments wth an electrostrctve deformable mrror, achevng postonng errors of the order of 1.2% RMS of the peakto-peak wavefront excurson. The technque does not depend on the physcal parameters of the devce; therefore t may be ncluded n the control scheme of any type of deformable mrror Optcal Socety of Amerca OCIS codes: ( ) Actve or Adaptve Optcs; ( ) Atmospherc Correcton; ( ) Atmospherc Turbulence. References and lnks 1. F. Hammer, F. Sayede, E. Gendron, T. Fusco, D. Burgarella, V. Cayatte, J. M. Conan, F. Courbn, H. Flores, I. Gunouard, L. Jocou, A. Lancon, G. Monnet, M. Mouhcne, F. Rgaud, D. Rouan, G. Rousset, V. Buat, and F. Zamkotsan, The FALCON Concept: Mult-Obect Spectroscopy Combned wth MCAO n Near-IR, Proc. ESO Workshop (2002). 2. F. Assémat, E. Gendron, and F. Hammer, The FALCON concept: mult-obect adaptve optcs and atmospherc tomography for ntegral feld spectroscopy - prncples and performance on an 8-m telescope, Mon. Not. R. Astron. Soc. 376(1), (2007). 3. D. Guzmán, A. Guesalaga, R. Myers, R. Sharples, T. Morrs, A. Basden, C. Saunter, N. Dpper, L. Young, L. Rodríguez, M. Reyes, and Y. Martn, Deformable mrror controller for open-loop adaptve optcs Proc. SPIE 7015, 70153X 70153X 12 (2008). 4. J. Fredman, Multvarate adaptve regresson splnes, Ann. Stat. 19(1), 1 67 (1991). 5. C. Hom, P. Dean, and S. Wnzer, Smulatng electrostrctve DM: I nonlnear statc analyss, Smart Mater. Struct. 8(5), (1999). 6. D. Andersen, M. Fscher, R. Conan, M. Fletcher, and J. P. Veran, VOLT: the Vctora Open Loop Testbed Proc. SPIE 7015, 7015OH-7015OH-11 (2008). 7. E. Laag, D. Gavel, and M. Ammons, Open-loop woofer-tweeter control on the LAO mult-conugate adaptve optcs testbed n Adaptve optcs for ndustry and medcne, C. Danty. (Imperal College Press, 2008), pp T. Bfano, P. Berden, H. Zhu, S. Cornelssen, and J. Km, Megapxel wavefront correctors, Proc. SPIE 5490, (2004). 9. C. Blan, O. Guyon, R. Conan, and C. Bradley, Smple teratve method for open-loop control of MEMS deformable mrrors, Proc. SPIE 7015, (2008). 10. K. Morznsk, K. Harpsoe, D. Gavel, and M. Ammons, The open-loop control of MEMS: modelng and expermental results, Proc. SPIE 6467, 6467OG-6467OG-10 (2007). 11. J. Stewart, A. Douf, Y. Zhou, and T. Bfano, Open-loop control of a MEMS deformable mrror for largeampltude wavefront control, J. Opt. Soc. Am. A 24(12), (2007). 12. J. Hardy, Wavefront Correctors n Adaptve Optcs for Astronomcal Telescopes (Oxford 1998), pp S. Sekulc, and B. R. Kowalsk, MARS: a tutoral, J. Chemometr. 6(4), (1992). (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6492

2 14. L. Breman, J. H. Fredman, R. A. Olshen, and C. G. Stone, Classfcaton and Regresson Trees., Wadsworth Internatonal Group, Belmont, CA (1984) 15. Q.-S. Xu, M. Daszykowsk, B. Walczak, F. Daeyaert, M. R. de Jonge, J. Heeres, L. M. H. Koymans, P. J. Lew, H. M. Vnkers, P. A. Janssen, and D. L. Massart, Multvarate adaptve regresson splnes - studes of HIV reverse transcrptase nhbtors, Chemom. Intell. Lab. Syst. 72(1), (2004). 16. P. Craven, and G. Wahba, Smoothng nosy data wth splne functons: estmatng the correct degree of smoothng by the method of generalzed cross-valdaton, Numer. Math. 31, (1979). 17. D. L. Massart, B. Vandegnste, L. Buydens, S. De Jong, P. Lew, and J. Smeyers-Verbeke, In: Handbook of Chemometrcs and Qualmetrcs vol. 20 A., Elsever, Amsterdam (1997) 18. J. W. Evans, B. Macntosh, L. Poyneer, K. Morznsk, S. Severson, D. Dllon, D. Gavel, and L. Reza, Demonstratng sub-nm closed loop MEMS flattenng, Opt. Express 14(12), (2006). 19. Y.F. L, S.H. Ng, M. Xe, T.N. Goh. A systematc comparson of metamodelng technques for smulaton optmzaton n Decson Support Systems. Appled Soft Computng, In Press, Corrected Proof, Avalable onlne 24 December do: /.asoc M. Carln, T. Kavl, and B. Lllekendle, A comparson of four methods for non-lnear data modellng, Chemom. Intell. Lab. Syst. 23(1), (1994). 21. E. Deconnck, M. H. Zhang, F. Pettet, E. Dubus, I. Iaal, D. Coomans, and Y. Vander Heyden, Boosted regresson trees, multvarate adaptve regresson splnes and ther two-step combnatons wth multple lnear regresson or partal least squares to predct blood bran barrer passage: A case study, Anal. Chm. Acta 609(1), (2008). 22. B. R. Oppenhemer, D. Palmer, R. Dekany, A. Svaramakrshnan, M. Ealey, and T. Prce, Investgatng a Xnetcs Inc. deformable mrror, Proc. SPIE 3126, (1997). 1. Introducton Open-loop adaptve optcs (AO) s a technque devsed for mult-obect adaptve optcs (MOAO), whch s one of the types of AO proposed to wden the lmted feld-of-vew of classcal closed-loop astronomcal AO systems [1,2]. In MOAO, wavefront sensors are locked on natural and/or laser gude stars n the feld, measurng uncorrected, turbulent wavefronts. One or more pck-off mrrors are placed at scence target locatons n the mage plane, drectng the lght to deformable mrrors (DMs) n the correspondng pupl planes. The turbulence s thus corrected for the specfc lne-of-sght to the scence obect, usng estmatons based on the measured gude star wavefronts. In open-loop AO, the DM shape cannot be controlled by the wavefront sensor as n closed-loop AO. In Guzmán et al [3], the problem has been tackled usng a dedcated wavefront sensor measurng the DM shape and closng a local control loop on t. In ths paper a mathematcal model s presented to control an electrostrctve DM n open-loop, usng multvarate adaptve regresson splnes (MARS). The use of MARS to buld a DM model presents some advantages compared to more tradtonal technques such as multple lnear regresson and neural networks. Possbly the man advantage s the fact that MARS s able to descrbe a gven response (e.g. actuator voltage) startng from a large number of predctors (e.g. DM facesheet postons), from whch the best predctors are automatcally selected, therefore the varables space s kept under control. Compared to neural networks and partal least squares (PLS) [4], MARS models are easer to nterpret, snce the orgnal varables can be drectly found n the resultng model and even nteractons between the varables are ndcated. Thus, MARS s able to buld flexble models wthout the dsadvantages of the more black-box methods, as PLS and neural networks are sometmes called. Applyng voltages to a grd of actuators underneath the mrror deforms the contnuous facesheet of the DM. The fnal shape of the mrror depends on factors such as the rgdty of the facesheet and the relatonshp between voltage appled and actuator dsplacement. The poston of the facesheet at any gven pont depends upon the dsplacement of all actuators n ts neghborhood as well as other parameters such as hysteress of the actuators and operatonal temperature. As t wll be seen n ths paper, the fnal shape s not a lnear combnaton of actuator dsplacements; therefore t s not straghtforward to mplement a smple model of a DM. Prevous works modelng deformable mrrors have explored the physcs behnd the actuators and the mrror facesheet. Hom et al [5] presents a non-lnear model of an electrostrctve DM, ncurrng a 13% underestmaton error wth respect to the real mrror poston n a statc test and ~40 nm rms errors when cancelng a gven wavefront. Andersen et al [6] and Laag et al [7] report results wth a voce-col actuator DM made by (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6493

3 ALPAO (France), achevng errors of the order of nm rms. Laag et al used a smple lnear superposton of actuators to determne the fnal shape of the mrror. As stated by Laag et al, the ALPAO mrror desgn consderatons ncluded reducton of nonlneartes and crosscouplng; thereby resultng n a mrror that s easer to control n open-loop. MEMS DMs have become popular n AO because of ther hgh densty of actuators and neglgble hysteress [8, 9]. Electrostrctve actuators, as well as MEMS actuators, have a quadratc response to voltage [8, 12]. Morznsk et al [10] and Stewart et al [11] have developed openloop models for MEMS DMs based on a physcal model of the electrostatc force for the actuator and a thn plate equaton for the facesheet; both groups report errors on the order of 15 nm rms. Our MARS model was traned wth DM surface measurements taken wth a Fsba Twyman-Green nterferometer, whch works wth monochromatc lght at 633 nm, dsplayng the surface of the mrror n a 512 x 512 pxels phase map n Angstroms. The mrror we modeled s a Xnetcs wth 97 electrostrctve actuators. Once traned, the model has a set of non-recursve equatons that can be easly mplemented n real-tme. Usng ths technque, we are proposng a general technque to operate a DM n open loop and n real-tme. We stress that hysteress, whch s nherent to the behavor of electrostrctve actuators (and of clear mportance n open loop control), was present durng both the tranng and the testng of our model. It s true that the level of hysteress demonstrated by ths type of actuator ncreases strongly f the temperature s reduced towards the Cure pont, whereas our DM was operated around 20 Celsus where the maxmum hysteress-nduced error s lmted to around 2-4% of the total stroke. However, we argue that an electrostrctve DM employed at temperatures where hysteress could greatly exceed ths range would be a very odd choce for open-loop control. There are several alternatve actuator technologes whch can provde smlar or better hysteress performance at lower temperatures; these nclude harder pezoelectrc ceramcs, electrostatc MEMS DMs and magnetc-based actuators. We therefore beleve that our model has been derved and tested wth hysteress present at the maxmum levels lkely to be present n any open loop control desgn. 2. Multvarate adaptve regresson splnes Multvarate Adaptve Regresson Splnes, MARS, s a multvarate nonparametrc regresson technque ntroduced by Fredman [4, 13] n The space of the predctors (e.g. DM voltages) s splt nto several (overlappng) regons n whch splne functons are ft. The man purpose of the MARS model s to predct the values of a contnuous dependent varable, y n 1, from a set of ndependent explanatory varables, X. The MARS model can be represented as n Eq. (1): n p y= f ( X ) + e (1) where e s an error vector of dmenson (n x 1). MARS can be consdered as a generalzaton of Classfcaton and Regresson Trees (CART), and s able to overcome some lmtatons of CART [14] Due to ts advantages, ts use has become popular n the last years for dfferent purposes [15]. MARS does not requre any a pror assumptons about the underlyng functonal relatonshp between dependent and ndependent varables. Instead, ths relaton s uncovered from a set of coeffcents and pecewse polynomals of degree q (bass functons) that are entrely drven from the regresson data (y, X). The MARS regresson model s constructed by fttng bass functons to dstnct ntervals of the ndependent varables. Generally, pecewse polynomals, also called splnes, have peces smoothly connected together. In MARS termnology, the onng ponts of the polynomals are called knots, nodes or breakdown ponts. These wll be denoted by the small letter t. For a splne of degree q each segment s a polynomal functon. MARS uses two-sded truncated power functons as splne bass functons, descrbed by Eqs. (2) and 3: (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6494

4 ( ) q < q t x f x t ( x t) = + 0 otherwse ( ) q > q t x f x t + ( x t) = + 0 otherwse (2) (3) Where q 0 s the power to whch the splnes are rased and whch determnes the degree of smoothness of the resultant functon estmate. When q= 1, whch s the case n ths study, only smple lnear splnes are consdered. A par of splnes for q= 1 at the knot t= 0.5 s presented n Fg. 1. Fg. 1. A graphcal representaton of a splne bass functon. The left splne (x<t, (x t)) s shown as a dashed lne; the rght splne (x>t, +(x t)) as a sold lne. The sold lne represents the rght-sded splne, x> t, + ( x t), whch s postve for all obects located at the rght sde of the knot t. The dashed lne represents the left-sded splne, x< t, ( x t), whch s postve for all obects located at the left sde of the knot t. The twosded truncated functons of the dependent varable are bass functons, lnear or nonlnear, that descrbe the underlyng phenomena. The MARS model of a dependent varable y wth M bass functons (terms) can be wrtten as Eq. (4): M yˆ = fˆ ( x) = c + c B ( x) (4) M 0 m m m= 1 where ŷ s the dependent varable predcted by the MARS model, c 0 s a constant, Bm ( x ) s the m-th bass functon, whch may be a sngle splne functon or a product (nteracton) of two or more splne bass functons, and c m s the coeffcent of the m-th bass functon. Both the varables to be ntroduced nto the model and the knot postons for each ndvdual varable have to be optmzed. For a data set X contanng n obects and p (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6495

5 explanatory varables, there are N = n p pars of splne bass functons, gven by (2) and (3), wth knot locatons x (=1, 2,, n; =1, 2,, p). A two-step procedure s followed to construct the fnal model. Frst, n order to select the consecutve pars of bass functons of the model, a two-at-a-tme forward stepwse procedure s mplemented [4]. Ths forward stepwse selecton of bass functons leads to a very complex and overftted model. Such a model, although t fts the data well, has poor predctve abltes for new obects. To mprove the predcton, the redundant bass functons are removed one at a tme usng a backward stepwse procedure. To determne whch bass functons should be ncluded n the model, MARS utlzes the generalzed cross-valdaton [16] (GCV). The GCV s the mean squared resdual error dvded by a penalty dependent on the model complexty. The GCV crteron s defned as n Eq. (5): ( ) 1 n ( y fm ( x )) m= 1 GCV M = n 2 C M n ( 1 ( ) / ) where C(M) s a complexty penalty that ncreases wth the number of bass functons n the model and whch s defned as n Eq. (6): 2 (5) C( M ) = M + dm (6) where M s the number of bass functons n Eq. (4), and the parameter d s a penalty for each bass functon ncluded nto the model. It can be also regarded as a smoothng parameter. Large values of d lead to fewer bass functons and therefore smoother functon estmates. For more detals about the selecton of the d parameter, see [4]. In our studes, the parameter d equals 2, and the maxmum nteracton level of the splne bass functons s restrcted to 3. The man steps of the MARS algorthm as appled here can be summarsed as follows: 0. Select the maxmal allowed complexty of the model and defne the d parameter. Forward stepwse selecton: 1. Start wth the smplest model,.e. wth the constant coeffcent only. 2. Explore the space of the bass functons for each explanatory varable. 3. Determne the par of bass functons that mnmses the predcton error, and nclude them n the model. 4. Go to step 2 untl a model wth predetermned complexty s derved. Backward stepwse deleton: 5. Search the entre set of bass functons (excludng the constant) and delete from the model the one that contrbutes least to the overall goodness of ft usng the GCV crteron. 6. Repeat 5 untl GCV reaches ts mnmum. The predetermned complexty of MARS model n step 3 should be consderably larger than the optmal (mnmal GCV) model sze M*. Choosng the predetermned complexty of the model as more than 2M* s enough n general [4]. In our case, t was equal to 3025 and ths value s called Mmax ANOVA decomposton of the MARS model It s possble to analyse a MARS model usng surface plots that vsualse the nteractons and effects between the bass functons. To llustrate ths, some defntons wll be ntroduced. (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6496

6 Let f ( x ) be the set of all sngle varable bass functons,.e., bass functons that contan only x. Smlarly, let f ( x, x ) be the set of all two-varable bass functons that contan the pars of varables x and x, and fk ( x, x, x k ) the set of all three-varable bass functons that contan the trplets of varables x, x and x k. The MARS model can be rewrtten as n Eq. (7): f ˆ( x ) = c + f ( x ) + f ( x, x ) + f ( x, x, x ) 0 k k (7) where the frst sum s over all sngle-varable bass functons, the second sum s over all strctly two-varable bass functons, and the thrd sum represents all three-varable bass functons. Equaton (7) s called the ANOVA decomposton due to ts smlarty to the decomposton by ANOVA of expermental desgn [17]. The two-varables nteracton of a MARS model, If ( x, x ), s gven by Eq. (8): If ( x, x ) = f ( x ) + f ( x ) + f ( x, x ) (8) Hgher level nteractons can be defned n a smlar way. The graphcal presentaton of the ANOVA decomposton facltates the nterpretaton of the MARS model. The effect of a one-varable bass functon can be vewed by plottng f ( x ) aganst x. Two-varable nteracton can be vewed by plottng If ( x, x ) aganst x and x n a surface plot Predcton ablty of the MARS model The predcton ablty of the MARS model can be evaluated n terms of the Root Mean Squared Error of Cross-Valdaton (RMSECV) and the squared leave-one-out correlaton 2 coeffcent ( q ). To compute RMSECV, one obect s left out from the data set and the model s constructed for the remanng n-1 obects. Then the model s used to predct the value for the obect left out. When all obects have been left out once, RMSECV s gven by Eq. (9): RMSECV = n = 1 ( y yˆ ) where y s the value of dependent varable of the -th obect, yˆ s the predcted value of the dependent varable of the -th obect wth the model bult wthout the -th obect. 2 The value of q s gven as n Eq. (10): n 2 (9) q = 1 n 2 = 1 n = 1 ( y yˆ ) ( y y) 2 2 (10) where y s the mean value of the dependent varable for all n obects. 3. Nonlnear behavor of the DM The behavor of the facesheet of our DM s non-lnear, as t s n MEMS DMs [18]. The smplest approach of a lnear combnaton of ndvdual actuator deformatons produces large errors when tryng to represent the fnal shape of the facesheet. An example of ths nonlnearty s presented n Fg. 2: we experment wth a sector of 9 x 5 actuators, whch are centrally located on the DM. We appled a half-range offset to all actuators to place them at a (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6497

7 nomnal condton for an AO system and then we exercse each actuator to +12 volts wth respect to ths offset. The data was taken wth our nterferometer (please see secton 5 for detals on our expermental setup), whlst subtractng the half-range offset reference. We obtaned the sum of all ndvdual actuators (top-left panel) and then we appled +12 volts to all actuators at once ( ont poke, top-rght panel). We evaluated the dfference between both results, as an mage (bottom-left panel) and as a slce of that mage (bottomrght panel). Ths smple exercse shows that the sum of actuators produces an excessve result wth respect to the ont poke of about 200 nm. Therefore, modelng the fnal shape of a DM s not as smple as combnng the ndvdual contrbutons of the actuators. Fg. 2. non-lnear behavor of DM actuators, n a 9 x 5 actuators example. Top-left panel: sum of ndvdual actuators, poked to +12v (for an explanaton for the tlted surroundng, please see secton 5). Top-rght panel: combned effect when pokng all actuators together to +12v. The Z coordnate n the top panels s n nanometers. Bottom-left panel: dfference between top panels, wth resdual RMS for the area beng poked. Bottom-rght panel: slce of the bottomrght panel along the central column, showng ndvdual actuators, the sum of them and the ont poke. 4. Actuator s area of nfluence Tradtonally, the nfluence functon s the concept used to descrbe the shape of the DM facesheet around an actuator beng poked. However, the fnal shape of the mrror surface depends on the postons of the neghbor actuators. Dependng on the physcal characterstcs of the facesheet, ths effect may or may not extend beyond adacent neghbor actuators. Knowng the extenson of ths area of nfluence s mportant for our purposes, snce t determnes the complexty of the model we are attemptng to produce, because surroundng actuator postons (defned by ther appled voltage) can ultmately affect the fnal poston of a central actuator. For a complete characterzaton of the area of nfluence of an actuator, we chose to exercse all neghbor actuators and not to lmt the study to the tradtonal nfluence functon. For ths experment, we poke a 5x5 actuator sector wth random values, usng smlar exercsng parameters to the ones used n the tranng data to feed our MARS model (see next secton for detals). (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6498

8 To determne the area of nfluence of an actuator n the presence of neghbor actuators beng exercsed, we computed the correlaton coeffcents matrx R (or cross-correlaton ), whch s calculated as follows. If C s the covarance matrx of the set of data, the cross correlaton R s defned as n Eq. (11): R, = C C, C,, In practcal terms, each row and column of R represents one actuator, and the values at postons (,) n R correspond to the correlaton between actuators and Usng ths defnton, t follows naturally that the dagonal elements of R wll be untary. (11) Fg. 3. Left panel: correlaton coeffcents matrx R; central panel: correlaton coeffcents for the central actuator, taken from matrx R.; rght panel: comparson between a tradtonal nfluence functon and the correlaton coeffcents Fgure 3 shows the results of ths experment, where the most mportant concluson s that only the closest neghbors are sgnfcant when defnng the fnal poston of an actuator. The area of nfluence of an actuator spans ts mmedate neghbors, n partcular the closest ones (the ones whch form a cross together wth the one under study). Counter-ntutvely, the corner actuators do not contrbute to the fnal shape of the central actuator, whch mght be related to the rgdty of the facesheet. The rght panel n Fg. 3 compares the correlaton coeffcents wth a normalzed, tradtonal nfluence functon for the actuator. The heght of the contnuous DM facesheet was measured only at actuator coordnates to make the comparson vald: ths s why the plots appears not to be contnuous. Fgure 3 shows that correlaton coeffcents gves slghtly dfferent results wth respect to the nfluence functon, at one and two actuators away from the one under study. We beleve ths method s better suted to descrbe the trul behavor of the DM membrane. Ths result may be dfferent for other types of deformable mrror (for nstance, for MEMS DMs, whch have a much thnner membrane), but the result of ths secton s useful n order to have a pror knowledge of the complexty of a DM model. 5. Data takng methodology Our expermental setup conssts of a commercal Twymann-Green nterferometer (a Fsba µphse 2 HR ) coupled to a 100 mm telecentrc lens, nstalled n front of the DM. The lght source used by the nterferometer s a temperature-stablzed 632 nm laser. The complete 75 mm DM aperture can be sampled by the nterferometer, wth each pxel correspondng to 0.2 x 0.2 mm of the DM pupl. The Z coordnate (DM membrane deformaton) s measured by the nterferometer and measured n Angstrom (Å). The typcal deformaton we appled to the DM was around ± Å = ± 2700 nm = ± 2.7 µm. The maxmum excurson we can measure s lmted by the ablty of the nterferometer to unwrap the phase nformaton from the nterference frnges, whch n turn s lmted by the spatal resoluton of the frnge samplng. Ths maxmum excurson s reasonable for typcal AO correctons [12]. We operate the DM at half ts range (V offset = 50 volts), applyng postve and negatve voltages around ths plateau voltage, n order to model the DM n accordance wth a typcal AO system. Our nterferometer was calbrated for each (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6499

9 run wth ths plateau poston, whch was subtracted automatcally for all phase maps. Ths s why our data has a zero mean value, around whch there are postve and negatve excursons. A phase map wth the Fsba takes approxmately 15 seconds to be computed, therefore data takng s a long process, usually spannng weeks. Throughout any gven run, there were varatons n the envronment, whch produced small long-term drfts n the measurements we were gatherng. We settled on modelng the man central area of our deformable mrror (11 x 5 actuators), leavng two statc and symmetrc areas n the DM around the modeled area, whch allows us to measure the drfts n the measured poston of the DM surface. Ths method would stll model a large area of the mrror (55 actuators), whle overcomng a lmtaton n our measurement equpment. Fgure 4 presents the area of study and the postons we are usng n the surroundng area to measure these drfts. We obtaned the actuator postons (wthn the nterferometer phase map) by dong ndvdual pokes for each actuator and recordng ts phase map. Then we measured a typcal full-wdth at halfmaxmum (FWHM) of the nfluence functon for the actuators and used ths value as a parameter for a two-dmensonal Gaussan nfluence functon. The Gaussan was used wth a least-squares fttng procedure to fnd the coordnates of each ndvdual poke. The coordnates of the Gaussan are the measured poston of each actuator n phase map space (where X and Y coordnates are pxel numbers). Edge actuators have largely asymmetrcal nfluence functons, unlke the rest of the actuators; therefore the fttng procedure wth the Gaussan dd not produce satsfactory results for the edge actuators. We decded to dentfy the actuator postons manually for the edge actuators, assumng they are located under the tallest area of the nfluence functon (where the gradent changes sgn). As can be seen n Fg. 4, the poston of these actuators do not comply wth a square grd as the rest of actuators do: ths s a result of the boundary condtons of the membrane; but t s not relevant that t s not a square grd, as long as we sample the heght of the actuators at coordnates that respond to the dfferent postons that the actuators can adopt. In summary, we are samplng the Z coordnate of the phase maps at actuator postons and we use these values to represent the DM surface. We are also samplng sx ponts of the surface n areas of the DM we left statc, whch we use to ft a plane that represents the poston of the DM membrane not deformed by the actuators. The poston of ths plane accounts for any drfts throughout the run and s subtracted from the Z coordnate measured at actuator postons. We found ths method to be smple and gve consstent results. Ths s the reason one sees a tlted surroundng n Fg. 2, gven the fact that the 9 x 5 actuators were poked one by one and t took tme to complete the run. Durng ths perod, there were drfts n the measured value of the DM surface, whch were compensated n the mrror data (ths s why the aperture crcle s not tlted), showng the effect of the compensaton n the outer areas of the phase maps, whch otherwse have a null value. (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6500

10 Fg. 4. the 11 x 5 actuators area beng modeled by our MARS model: The 55 actuator postons found are presented here wth black crosses, and the surroundng area, sampled at 6 postons and presented wth blue Xs, are used to ft a plane to account for tlt drfts throughout the runs. The DM s shown wth a typcal random deformaton spannng a few mcrometers n the Z coordnate Our Xnetcs DM s fed by hgh voltage amplfers able to output 0 to 100 volts, for a 0 to 10 volts nput. A typcal voltage/stroke plot s presented n Fg. 5. Fg. 5. voltage/stroke plot for a central actuator Our MARS model receves 55 heght (Z coordnate) nputs to produce 55 voltage outputs. We exercse the DM to ts lmts, n order to gather realstc data for an AO system: the DM s frst rased to half ts range and from that pont the 55 actuators are gven random voltages, to shape the DM surface to a completely random phase. The range of the random values s the maxmum permtted wthout damagng the surface of the mrror. For our Xnetcs DM ths s acheved by applyng +/ 12 volts to the actuators on top of the rased half range heght. Ths s smlar to an AO system that needs to compensate postve and negatve phases. As an example, Fg. 6 presents the frst 9 random postons of one of our runs. It can be seen that there s an average value of 0 Angstrom (because the nterferometer subtracts the purely rased mrror as ts reference) and postve (redder) and negatve (bluer) bumps. (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6501

11 Fg. 6. Frst 9 random pokes of one of the runs wth our Xnetcs DM. Z coordnate (color bar) s n nanometers 6. Results and analyss We traned our MARS model wth 6000 random postons, such as the ones depcted n Fg. 5. To test the ablty of our model to generate the correct voltages to shape the facesheet, we expermented wth two types of tests: Qualtatve test: we generate some arbtrary combnaton of Zernke polynomals and feed the model wth the values of the combnaton of polynomals evaluated at actuator coordnates. The model outputs the voltages for the 55 actuators, whch we run wth the DM, measurng the surface shape wth our nterferometer Quanttatve test: we produced an addtonal 1000 new random postons wth the same parameters as the ones used for tranng and feed the model wth them. The model produces the predcted voltages to acheve such random shapes, whch we run wth the DM, acqurng phase maps. We call them quanttatve and qualtatve tests because n the latter case, the ablty of the DM to reproduce a certan Zernke polynomal s lmted by ts spatal resoluton, ncurrng n what s usually known as fttng error, therefore we only present qualtatve results. The quanttatve case does not have that problem, allowng us to obtan a postonng or GoTo error, or the dfference n poston between the desred shape and the acheved phase by the mrror. We decded on usng purely random voltages to test our model to ts lmts, snce the spatal frequency response of unformly dstrbuted random data s more strngent at hgh spatal frequences compared to Kolmogorov turbulence for example. Once the MARS model s calculated t can be mplemented n any computer as a set of equatons [19 21]. For an Intel Core 2 Duo, at 2,26GHZ, wth 4GB of RAM (standard desktop computer), t takes 190 mcroseconds to perform a predcton for 55 actuators. Typcal operaton cycle of AO computers s Hz (or 2-3 mllseconds), therefore our model would not add any sgnfcant delay to an AO computer. (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6502

12 Fg. 7. qualtatve results, usng varous Zernke polynomals. The left panels are theoretcal Zernke polynomals; the central panels are the output of the DM for the 11 x 5 actuators beng modeled. The rght panels present a slce n Y from the central panels, to apprecate the dfferences between theoretcal and expermental polynomals (red plot: Zernke polynomal; blue plot: DM surface). The color map for the left and central panels s common, but t has not been ncluded n order to smplfy the fgure The Zernke polynomals selected for Fg. 7 were chosen somewhat arbtrarly, but the selecton crteron was to use Zernke that have more nformaton along the Y-axs (where the maorty of the actuators modeled are) and ncorporate spatal frequences that would be too hgh for ths mrror to shape. The results of ths experment are clearly seen along the rght panels of Fg. 7. For the frst polynomal (at the top of the fgure), the dfference between theory and experment s farly small, but when ncreasng the order of the polynomal and thus the spatal bandwdth, the fttng error becomes sgnfcant, although the areas wth lmted spatal frequences are stll well modeled (see for nstance the central part of the plot at the bottom-rght panel). It s nterestng to note that these results were obtaned wth our MARS model, whch was never traned wth Zernke polynomals. Ths allows us to establsh that our tranng method s general and would be approprate for a mrror n an AO system. (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6503

13 Fg. 8. maxma and mnma of each run. The resdual error s presented for completeness. For more detals on the latter, see Fg. 9. The postonng error or GoTo error was obtaned wth a new 1000 trals random run. The MARS model was fed wth the facesheet postons from ths run and the voltages produced were used for a new run. The phase maps of both runs were subtracted (at actuator coordnates), to produce the results n Fgs. 8 and 9. Fgure 8 shows the maxma and mnma of each tral, to confrm the large span n the data. Fgure 9 s the man result of ths paper, presented as a GoTo error n Angstrom (RMS value for the 55 actuators) and as a percentage of the full-range of actuators excurson (from Fg. 8) (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6504

14 Fg. 9. GoTo error: the top-panel presents the resdual error n Nanometer. The bottom-panel presents the resdual error as a fracton of the full-range of actuators excurson The mean value of the plot at the bottom panel of Fg. 9 s 1.2%, whch we beleve s a remarkable result for an electrostrctve (e.g. wth non-neglgble hysteress) mrror such as our Xnetcs [22]. 7. Conclusons We have presented a model to predct the voltages to apply to a DM n order to acheve a desred poston on ts facesheet, usng a multvarate adaptve regresson splnes model. The model s purely mathematcal and s traned usng a data set of nterferometrc phase maps. Its performance was verfed usng a DM n real condtons, reachng a GoTo error of 1.2% of the full-range of actuator postons. Ths model has the beneft of not developng a physcal model of the DM; therefore the DM modelng strategy can be the same, regardless of the type of mrror n operaton. MARS produced a seres of smple equatons, nvolvng only sums and multplcatons they are fast to compute, unlke an teratve soluton, so t should not add any sgnfcant latency to a real-tme AO computer. Acknowledgments We apprecate frutful dscussons wth Prof. Ray Sharples and Dr. Tm Morrs, from Durham Unversty. We thank Prof. Robert Tyson for useful suggestons to an early draft of ths paper. D. Guzman apprecates support from the Scence and Technology Facltes Councl (STFC), through the Dorothy Hodgkn postgraduate studentshps program. Ths work was funded by STFC, grant PP/E007651/1. (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6505