Multi-spectral piston sensor for co-phasing giant segmented mirrors and multi-aperture interferometric arrays

Transcription

1 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors Multi-spectral pisto sesor for co-phasig giat segmeted mirrors ad multi-aperture iterferometric arrays Fraçois Héault UMR 6525 CRS H. Fizeau US, OCA Aveue icolas Coperic, 0630 Grasse Frace Abstract. This paper presets the optical desig of a multi-spectral pisto sesor suitable to co-phasig giat segmeted mirrors equippig the Future Extremely Large Telescopes (ELTs). The geeral theory of the sesor is described i detail ad umerical simulatios have bee carried out, demostratig that direct pisto ad tiptilt measuremets are feasible withi accuracies respectively close to 20 m ad 0 ao-radias. Those values are compatible with the co-phasig requiremets, although the method seems to be perturbed by ucorrected atmospheric seeig. Keywords: Telescopes, Fourier optics, Phase retrieval, Phase-shiftig iterferometry PACS: , Kq, Rx, Bg Itroductio From the Hooker telescope of 00 diameter build durig the 920s o Mout Wilso to the achievemet of the Very Large Telescope (VLT) array i Chile, the 20th cetury has uquestioably demostrated the superiority of large reflective telescopes i the field of astroomical observatios. It is commoly believed, however, that the classical operatios of maufacturig, polishig ad supportig large glass mirrors will soo be cofroted to their techological limits, ad that i view of 0-m class (or higher) groud-based telescopes, the primary mirrors will eed to be composed of several smaller idividual reflective facets (or segmets), a major choice havig bee validated o the two 0-m Keck telescopes. For space observatories alteratively, the mirror diameters are rather limited by the space available uder the coe of the lauchig rocket, leadig to a curret maximum below 4 meters. Hece the James Webb Space Telescope (JWST, to be operated i 204) will be equipped with a 6.5-m segmeted mirror, beig deployable ad optically adjustable i space. I the case of a giat segmeted mirror, all the reflective facets must be idividually adjusted i pisto (alog a directio parallel to the telescope optical axis) ad tip-tilt (rotatios aroud two axes perpedicular to the telescope optical axis) so that the assembled segmets ideally mimic the theoretical, cotiuous surface of the mirror. This operatio is sometimes called the co-phasig of the telescope ad must be carried out withi a give accuracy, which could be λ/4 Peak-to-Valley (PTV) or λ/3.4 Root Mea Square (RMS) accordig to either the Rayleigh or Maréchal criteria, where λ is the wavelegth of the electro-magetic field. I this paper is chose a target accuracy of λ/0 RMS, which is frequetly quoted i papers relevat to the co-phasig of telescopes ad sparse apertures iterferometers (see for example [-2]), ad costitutes a reasoable magitude order at least whe coroagraphic applicatios are ot evisaged. For a telescope of diameter 5 m, this requiremet correspods to approximate pisto ad tip-tilt toleraces of respectively 25 m ad 0 ao-radias i

2 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 2 the visible rage (λ = 0.5 µm). Those requiremets are extremely demadig ad ecessitated to develop specific aligmet techiques, such as those summarized below: - The first pioeerig works were udeiably udertake at the Keck telescope: Chaa et al [3-4] upgraded some already well-kow Wavefrot Sesors (WFS) cocepts, such as the Shack-Hartma or curvature WFS, i order to give to them the capacity to discrimiate pisto errors: here it has to be oticed that, sice their basic priciple cosists i measurig the local slopes or curvatures of the wavefrots (WFE) before recostructig them digitally, those WFS are ot aturally well-suited for pisto sesig. It was the show that ehaced hardware ad algorithms searchig for local slope or curvature discotiuities ca be used to determie the pisto errors. However these methods are hardly applicable to diluted apertures iterferometers, a coditio that we seek to satisfy i this study. - Aother optio is to employ phase retrieval or phase diversity digital procedures, sice the latter have already bee applied successfully to the determiatio of phase errors o both moolithic telescopes ad multi-aperture optical systems [5-7]. However the techique usually requires sigificat post-processig times, which prevets them from beig operated o-groud i a adaptive optics (AO) regime: thus a ideal WFS should ideed combie the ability to perform direct WFE measuremets i quasi real-time, which adds aother striget requiremet to the system. - Some alterative WFS cocepts based o a Mach-Zehder iterferometer (or o a equivalet priciple) istalled at the focal plae of the optical system to be co-phased have also bee proposed by differet authors [8-0], but oe of them seem to have bee validated o-sky. I this paper we fially choose to re-examie the cocept of a multi-wavelegth, phase-shiftig Telescope-Iterferometer (TI), aother focal plae WFS that has bee described ad studied recetly i its moochromatic versio [-3]. Its priciple, makig use simple umerical algorithms could allow to quickly ad directly evaluate the WFEs (icludig pisto errors) created o either giat segmeted mirrors or multi-aperture iterferometers. We first recall briefly the moochromatic theory of the TI i sectio 2., before extedig it to the case of multiple wavelegths i sectio 2.2. A tetative optical desig based o the combiatio of a phase-shiftig stage ad a multi-spectral stage is the described i sectio 3. A ed-to-ed umerical model iteded to evaluate the WFS performace is briefly described i sectio 4 ad its prelimiary results i terms of pisto ad tip-tilt measuremet accuracy are preseted, before a short summary is provided i sectio 5. 2 Theory I this sectio is first recalled the priciple of WFE measuremets performed usig a moochromatic phase-shiftig TI, before geeralizig it to the polychromatic case. Basically, the proposed techique cosists i addig a secod, referece optical beam ito the mai pupil i order to geerate modulated ad phase-shifted poit spread fuctios. The searched phase errors ca the be extracted from demodulatig the Fourier trasforms of the PSFs i the Fourier plae. More details about the TIs ad their theory ca be foud i Refs. [-3]. 2. The moochromatic Phase-Shiftig Telescope Iterferometer (PSTI) Let us cosider a telescope of 5-m diameter whose primary mirror is costituted of idividual segmets disposed followig a hexagoal arragemet as depicted i Figure. It is assumed that the cetral segmet does ot exist, ad is replaced by a smaller, circular referece mirror of diameter d = 2r cetred o poit O. amed referece pupil i the remaider of the text, it is supposed this mirror ca be displaced logitudially alog the Z optical axis, thereby itroducig a phase-shift φ m of the cetral referece pupil with respect to the other mirror segmets (φ m will further take M differet values, with m M, see below). The expressio of the complex amplitude A m (P) i the exit pupil plae OXY therefore writes (see Figure ): [ m ] + Am (P) = Br (P) exp iφ BD(P - P )exp[ ik (P - P )], () = where B r (P) is the amplitude trasmitted by the referece pupil (here equal to the pillbox or tophat fuctio of radius r), B D (P) is the two-dimesioal amplitude trasmissio fuctio of the hexagoal segmets, P is the cetre of the th segmet, (P) is the WFE to be measured ad k = 2π/λ. I this sectio will oly be cosidered the pisto ad tip-tilt errors ξ ad t of the segmets, hece:

3 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 3 ( P - P ) = ξ + t P P, (2) where t stads for the uitary vector perpedicular to the th facet. I the frame of scalar diffractio theory, the complex amplitude distributio  m (M') i the telescope image plae O X Y is equal to the Fourier trasform of A m (P):  m (M') = Bˆ r (M')exp[ iφm ] + Bˆ D(M' ) exp[ik(ξ + t P P) ]exp[ ikop O' M' /F] =, (3) where Bˆ r (M') ad Bˆ D (M') respectively are the Fourier trasforms of B r (P) ad B D (P), ad F is the focal legth of the segmeted telescope. The Poit-Spread Fuctio (PSF) of the system is by defiitio equal to the square modulus of  m (M'), i.e. after multiplyig with its complex cojugate: PSF (M') = PSF (M') PSF (M') + exp m r + T [ iφm ][ Bˆ r Bˆ D ](M') exp[ik(ξ + t P P) ]exp[ ikop O' M' /F] = + exp m exp[ ik(ξ + t P P) ]exp[ikop O' M' /F] = * [ iφ ][ Bˆ r Bˆ D ] (M'), (4a) where PSF r (M ) ad PSF T (M ) respectively stad for the PSFs of the referece pupil ad of the whole segmeted telescope: 2 PSF (M') = Bˆ (M' (4b) r r ) 2 PSFT (M') = Bˆ D (M') exp[ik(ξ + t P P) ]exp[ ikop O' M' /F] = exp[ ik(ξ + t P P) ]exp[ikop O' M' /F]. (4c) = Oe of the two basic priciples of the PSTI ow cosists i physically measurig the poit spread fuctios PSF m (M ) o a CCD camera (or aother type of detector array), ad i computig digitally their associated Optical Trasfer Fuctios (OTF) by meas of a iverse Fourier trasform. From Eq. (4), oe fids a expressio of OTF m (M ) that is composed of four differet terms: OTF (P) = OTF (P) OTF (P) m r + T m (P - P ) = [ iφ ] exp[ik(ξ + t P ] [ Br BD ] + exp P) m (P + P ) = [ iφ ] exp[ ik(ξ + t P ] [ Br BD ] + exp P), (5) with symbol * deotig a covolutio product. The secod essetial poit of the PSTI procedure is the to isolate the third term of Eq. (5) by meas of a appropriate set of phase-shifts φ m allowig to liearly combie all the computed OTFs: M M C(P) = OTF (P) exp m M m= M m= + = [ i φ ] = exp[ iφ ] [ OTF (P) OTF (P)] m m r + [ B B ] exp[ ik(ξ + t P P) ] r D (P - P ) M + exp[ 2iφm ] exp[ ik(ξ + t P P) ] [ Br BD ](P + P ), (6) M m= = ad it is foud that the secod term ca be easily separated from the others if the selected phase-shifts are satisfyig the both coditios: M m= M [ φm ] = exp[ 2iφm ] = 0 exp i. (7) m= T

4 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 4 Figure 2 provides a graphic illustratio of possible solutios for M = 3 ad M = 4 i the complex plae it has to be oticed that oly the quadruplet {φ ;φ 2 ;φ 3 ;φ 4 } = {0;π/2;π;3π/2} was origially cosidered i the previous publicatios [-3]. Whe the coditios (7) are respected the fial expressio C(P) of the liearly combied OTFs becomes: = + t P P) ] [ Br BD ](P - P ) exp[ik(ξ + t P ] BD(P - P ) = C(P) = exp[ik(ξ P), (8) ad, if the spatial dimesios of the referece pupil are sigificatly smaller tha those of the hexagoal facet, the fuctio B r (P) ca be replaced by the Dirac distributio δ(p), ad the phase of C(P) ca be cosidered as costat over the whole segmet area, beig simply proportioal to the searched pisto ξ. The justificatio ad implicatios of this approximatio have bee extesively discussed i Ref. [4]. The fial step of the PSTI procedure cosists i a phase extractio that is carried out by isolatig the phase errors of the th segmet, multiplyig both sides of Eq. (8) with the fuctio B D (P-P ): C(P) BD(P - P ) = exp[ik(ξ + t P P) ] BD(P - P ), (9) ad the estimated phase simply writes: ϕ = ( ξ + t P P) λ = arcta{ Im[ C(P) BD(P - P )] Real[ C(P) BD(P - P )]} 2π.(0) All the previous steps are schematically summarized by the flow-chart of Figure 3. The overall performace of the method was studied i more detail i Refs.[-2], leadig amog others to the followig coclusios. The itrisic measuremet accuracy of the PSTI is better tha λ/5 PTV ad λ/00 RMS, which makes it competitive with respect to the best existig WFS. However i moochromatic light the measuremet rage stays limited by the so-called 2π ambiguity i.e. the phase ca oly be retrieved modulo 2π from Eq. (0). The observed light source does ot eed to be purely moochromatic: maximal spectral badwidths δλ/λ of 20% ca be tolerated without sigificat loss of performace. Although the method is particularly efficiet i space, it may also be operated o groud i a adaptive optics regime: depedig o the seeig coditios ad o the size of the referece pupil, it ca accept guide stars up to the magitude i the V bad. ext sectio is ow devoted to the extrapolatio of this moochromatic WFE measuremet method to several differet wavelegths ad spectral bads i order to exted its capture rage beyod the 2π ambiguity. 2.2 Multi-wavelegth operatio Extedig the measuremet rage of the Optical Path Differeces (OPD) betwee differet separated telescopes is a commo problem i multi-aperture iterferometry that ca be solved by combiig phase measuremets simultaeously performed i differet spectral bads. Such methods ca ideed be extrapolated to the measuremet of pisto errors ad to the co-phasig of large segmeted mirrors. For example, we describe i the Appedix how to adapt the dispersed speckles method recetly proposed by Borkowski et al [2][5-6] to the case of a polychromatic PSTI. It turs out, however, that the required umber of spectral chaels is probably too importat, ad would be detrimetal to the phase measuremet accuracy (that was show to be iversely proportioal to the spectral width of a idividual measuremet chael [2]). Hece the followig study will be restricted to methods that make use of a miimal umber of wavelegths, ispired from the techiques of multi-colour phase shift iterferometry [7-8] ad o-cotact legth ad distace metrology [9-20]. Give a optical distace to be determied ξ, the basic priciple cosists i combiig its fractioal phases ϕ l measured for L differet wavelegths λ l, with l L. Limitig the total wavelegth umber to L = 3, the problem reduces to solvig the followig equatios system: ξ = + ϕ λ = + ϕ λ = + ϕ λ () ( ) ( 2 2 ) 2 ( 3 3 ) 3 The latter is actually a 3 system of 4 ukow variables, which are ξ ad the positive or egative iteger umbers, 2 ad 3. Due to the iteger ature of l, this uder-costraied system ca evertheless be solved over a limited domai of pisto values [-λ S /2, λ S /2], where λ S is defied as the

5 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 5 sythetic wavelegth. Herei we select a heuristic expressio of λ S, which still belogs to Tilford s origial family of solutios [9]: λ = λ 2λ + λ, (2) S ( ) assumig that λ < λ 2 < λ 3. This eables to defie a simple multi-spectral OPD uwrappig procedure whose pseudo-code is provided i Figure 4 ad major steps are described below (here the idices have bee omitted for the sake of clarity): ) Startig from the fractioal phases ϕ, ϕ 2 ad ϕ 3 acquired for the three differet wavelegths, a first guess of the pisto error is estimated as δ 0 = λ S (ϕ - 2ϕ 2 + ϕ 3 ), ad δ 0 is further brought back ito the [-λ S /2, λ S /2] rage if ecessary. 2) First guess δ 0 ad the fractioal phases together allow to estimate the iteger umbers, 2 ad 3. 3) Three improved estimatios of the pisto error δ, δ 2 ad δ 3 ca ow be derived from those iteger umbers ad the fractioal phases. 4) The fial pisto error ξ is defied as the arithmetical mea of δ, δ 2 ad δ 3. 5) A pisto measuremet accuracy idex oted σ δ ad equal to the variace of δ, δ 2 ad δ 3 is also computed, servig as a quality estimator for the obtaied result. The fifth ad last step of the umerical procedure is perhaps the most importat, sice it may be compreheded as a saity check of the whole measuremet sequece: errors i step 4 ca origiate either from the itrisic ucertaity δϕ affectig the measured fractioal phases, or from false estimatios of, 2 or 3. I the latter case, the fial measuremet ucertaity σ δ should be at least equal to λ /3. If o the cotrary the three iteger values are ubiased, σ δ ca be estimated as: /2 [ λ + λ + λ ] λ δϕ 3 σδ = δϕ , (3) ad it is therefore possible to defie a simple criterio (e.g. σ δ < λ /0) allowig to accept or reject the curret estimatio of the pisto error. To coclude, it must be emphasized that the major advatages of the here above preseted multispectral OPD uwrappig procedure are the fact that it oly ivolves very simple mathematical relatioships suitable to real-time computig, o the oe had, ad the possibility of a self saity check allowig to reject umerical results corrupted by real measuremet errors, o the other had. The followig sectio ow aims at defiig a tetative optical layout for the evisaged multiwavelegth, phase-shiftig pisto sesor. 3 Descriptio of the pisto sesor The theoretical cosideratios that have bee exposed i the previous sectio ca serve as a startig poit for desigig the prelimiary optical architecture of a WFS based o the priciple of the multispectral PSTI. The major requiremets may be summarized as follows: ) This is basically a focal plae wavefrot sesor, whose volume ad dimesios ear the telescope focus (or oe of its images) should be kept as small as possible i order to esure good mechaical ad thermal stabilities. 2) The WFS shall iclude a phase-shiftig device located at a image plae of the segmeted mirror to be characterized (assumed to be the etrace pupil of the telescope). 3) The WFS shall provide the capacity of acquirig simultaeously or withi a very short lap of time the PSFs of the telescope i several arrow spectral chaels. Two very prelimiary desigs aswerig to the two first requiremets have already bee described i Ref. [3]: i the first oe the beams are divided by a arragemet of M- beamsplitters ad the PSFs are acquired simultaeously o M differet, sychroized detector arrays. I the secod cofiguratio all the PSFs are measured sequetially o oe sigle camera withi a total acquisitio time ot exceedig 0 msec. I additio to the advatage of requirig oly oe detector array, the latter solutio was foud the most favourable i terms of radiometric performace, especially whe photo oise domiates detector oise (i.e. for bright sky objects). I the herei preseted desig, the phaseshifts φ m are itroduced sequetially while the measuremets i differet spectral bads are simultaeous. It is assumed that the primary mirror of the telescope icludes a referece segmet of superior image quality (i.e. diffractio-limited), correspodig to the referece pupil area where the phase-shifts are itroduced (two examples of practical implemetatio of the referece pupil are 2 3

6 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 6 provided i Ref. []). The optical layout of the pisto sesor is schematically preseted i Figure 5: it is essetially composed of three mai sub-systems, amely the phase-shiftig stage, the multi-spectral stage ad the CCD camera that are described below. umerical simulatios of the performace of the complete optical system are further preseted i sectio Phase-shiftig stage This sub-system is essetially composed of oe collimatig les imagig the telescope etrace pupil (i.e. the giat segmeted mirror) o a referece flat mirror that is pierced at the locatio of the telescope referece pupil. A madrel carryig a small, flat ad high-precisio optical surface is piezoelectrically shifted alog the optical axis, thus geeratig the required successive phase-shifts φ m, for all idices m comprised betwee ad M. However, may other desigs could certaily be evisaged owig to the recet progress of Micro-opto-electromechaical Systems (MOEMS) techology. 3.2 Multi-spectral stage The core fuctio of the multi-spectral stage is to perform simultaeous measuremets of PSF m (M ) as defied i Eq. (4a) i differet chaels of moderated spectral width. The proposed desig makes use of a image/pupil iversio that was origially suggested by Courtès ad Vito for the observatio of gaseous ebulae ad distat galaxies [2]. It cosists i settig a trasmissive diffractio gratig ear a itermediate image plae X Y, spreadig the beams i differet directios depedig o their wavelegths, ad to re-arrage them i the spectrally dispersed pupil plae XY (see the middle part of Figure 5, which is ot to scale). This ca be achieved by meas of a few optical desiger tricks: - The employed diffractio gratig is egraved o a coverget optical elemet (here represeted as a les) i order to re-imagig the dispersed etrace pupil o its slicer with high demagificatio ratio, therefore allowig to defie spectral chaels δλ l with sharp edges (it must be oticed that although the diffractio gratig preseted i Figure 5 is a trasmissive oe, ay reflective gratig may be used as coveietly). - The wavelegth separatio is realized by a pupil slicer located at the dispersed output pupil plae, also servig to redirect the beams towards differet areas of the detector array. This pupil slicer ca be made of a micro-les array associated to a covergig les as represeted i Figure 5, a desig that was already maufactured ad test for the istrumet MUSE of the VLT [22]. The pupil slicer could also be fully reflective: the realizatio of such mii arrays of optical elemets is owadays well mastered (see for example Ref. [23]). - Fially, the focal legths of the idividual elemets of the pupil slicer are ot idetical, but iversely proportioal to the wavelegths λ l i order to compesate for the liear magificatios of the measured fuctios PSF m (M ) with respect to λ l. The real ecessity of this wavelegth rescalig has bee cofirmed by the umerical simulatios preseted i the ext sectio. The rescalig might also be performed digitally (e.g. usig iterpolatio algorithms) rather tha optically, however it has bee chose here to simplify the data processig software as much as possible, i order to stay compatible with real time applicatios. 3.3 CCD camera This fial stage of the pisto sesor cosists i a high performace detector array, either of the CCD or CMOS type, typically characterized by high quatum efficiecy ad low read-out oise over the cosidered spectral domai, ad equipped with a 2-bit aalogue-to-digital coverter. 4 umerical simulatios I this sectio are preseted the umerical results of a ed-to-ed optical model (described i sectio 4.) iteded to assess the expectable performace of the described WFS i terms of phase ad pistos measuremet errors. For a give set of telescope ad istrumetal parameters (sectio 4.2), the simulated cases cover static pisto ad tip-tilts errors (sectio 4.3), the ifluece of the selected spectral badwidths δλ l (sectio 4.4), ad the effects of atmospheric disturbace for a groud-based sesor (sectio 4.5).

7 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 7 4. The umerical model The umerical model of the multi-wavelegth pisto sesor is actually based o the moochromatic model that was described i Ref. []. It is ideed split ito two major modules, respectively simulatig the PSF m (M ) fuctios recorded by the camera, ad subsequetly computig the OTFs via iverse Fourier trasforms ad applyig the whole data processig defied by Eqs. (8-2). The multispectral OPD uwrappig procedure of Figure 4 is etirely icluded ito the secod module. The first module is basically a ray-tracig software havig the ability of modellig differet apertures of various shape ad sizes, takig ito accout aligmet errors i both pisto ad tip-tilt ad atmospheric seeig. This istrumet simulator the computes the moochromatic PSFs ad sums them icoheretly over the cosidered spectral chaels. The program ca also add various types of detectio oises ad digitizatio errors to the PSFs, eve if the latter possibility was ot utilized i the frame of this study. All these operatios are repeated for the M phase-shifts φ m ad the L differet spectral chaels cetred o the wavelegths λ l. 4.2 Defiitio of the system ad iput parameters We cosider a telescope of diameter 5 m ad focal legth 50 m, hece beig ope at F/0. It is composed of = 8 hexagoal facets arraged as show i Figure (this cofiguratio was origially iteded to mimic the JWST. Those parameters would of course eed to be updated i the case of the future groud-based ELTs, although it is likely that the major coclusios would ot chage sigificatly). For all simulatios, it is assumed that the phase-shifted, referece pupil is cetred o the optical axis of the telescope, which should ot hamper the coclusios. Radom pisto errors ragig from 0 µm to +0 µm are added to each telescope segmet, as well as tip-tilt errors comprised betwee ad + micro-radia aroud the X ad Y-axes. Those pistos ad tip-tilts values are supposed to represet realistic aligmet errors of the segmets resultig from the employed prealigmet techique (for compariso purpose, they are twice more optimistic tha the iitial errors measured o the Keck 2 telescope [3]). All the umerical values are provided i Table ad the global WFE map of the segmeted mirror is depicted by the grey-scale map ad the perspective view of Figure 6. Table : Iitial, measured ad differece values of the pisto ad tip-tilt errors for a spectral badwidth δλ/λ = 0% ad a referece pupil radius r=500 mm. Iitial errors Measured errors Differeces Segmet Pisto Tilt X Tilt Y Pisto Tilt X Tilt Y Pisto Tilt X Tilt Y umber (µm) (µrad) (µrad) (µm) (µrad) (µrad) (µm) (µrad) (µrad) Average

8 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 8 PTV RMS I order to miimize the total time requested for the acquisitio of the M phase-shifts φ m, we choose M = 3 ad cosequetly {φ ;φ 2 ;φ 3 } = {0;2π/3;4π/3}. The selectio of the umber of wavelegths L, of the cetral wavelegths λ l ad of the spectral widths δλ l of each spectral chael is ot a straightforward task, ad was here carried out i a iterative maer, tryig to establish a compromise betwee the followig adverse tedecies: The pisto capture rage icreases as more wavelegths are added ad their values are closed oe to the other. However, this criterio teds to reduce the spectral widths δλ l. For radiometric reasos, the measuremet accuracy icreases with δλ l, but this is detrimetal to the capture rage ad augmets the risk of failure of the OPD retrieval algorithm (the poit is further discussed i sectio 4.4). We fially adopted L = 3 ad the triplet {λ ;λ 2 ;λ 3 } = {0.456;0.502;0.552} µm. The other umerical parameters are provided i Table 2. For such values of the three cetral wavelegths, the sythetic wavelegth λ S is equal to µm accordig to the relatio (2), which yields a capture rage of the pisto errors beig theoretically equal to [ , ] µm, largely compliat with their actual figures. Moreover, settig M = L = 3 allows to limit the total umber of axial displacemets of the referece mirror i the phase-shiftig stage to seve, beig respectively equal to 0 (commo to all spectral chaels), ad the triplets {λ /3;λ 2 /3;λ 3 /3} ad {2λ /3;2λ 2 /3;2λ 3 /3} i terms of OPD. The total computatioal load is essetially domiated by the seve required Fast Fourier Trasform (FFT) operatios: assumig WFE map samplig of 52 x 52 as i the preset study, it is foud that quasireal time operatio requires approximately 20 GFlops, a performace that is easily fulfilled by moder laptop computers. Table 2: umerical parameters of the three selected spectral chaels. Cetral wavelegth Maximal spectral badwidth δλ/λ Effective focal legth (µm) (µm) (%) (m) λ λ λ Pisto ad tip-tilt sesig capacity Let us firstly cosider the case of static pisto ad tip-tilt errors resultig from a prelimiary aligmet of the segmeted primary mirror. We set the maximal spectral badwidth δλ/λ of the three measuremet chaels to 0 % ad the referece pupil radius to its maximal possible value, i.e. r = 500 mm (top left pael of Figure 7). The umerical results of the simulatio are provided i Table 2 ad illustrated i Figure 7, showig the ucorrected PSF (middle row) ad the retrieved WFE at the surface of the exit pupil (bottom left pael), as well as its differece with respect to the origial errors (bottom right pael). It must be oticed that the umerical values provided i Table 2 result from liear regressios of the WFE map at the surface of each idividual segmet, limited to a 700 mm useful diameter circle i order to maximize the pisto measuremet quality estimator σ δ defied i sectio 2.2. It is foud that the pisto ad tip-tilt estimatio errors are fially equal to 7 m ad /2 ao-radias (with respect to the X/Y axes) i RMS sese, a satisfactory result that is early compliat with the origial goals defied i sectio. 4.4 Ifluece of spectral badwidth As was already metioed i sectio 4.2, The WFE measuremet accuracy is expected to improve with the spectral width of the three measuremet chaels, o the oe had, but this beefit is couterbalaced by a smaller capture rage ad the possibility of false estimatios of the iteger umbers, 2 ad 3 (see 2.2), o the other had. The latter ca be evaluated quatitatively by meas of a OPD retrievig cofidece ratio defied as follows: ρ = δ / T (4)

9 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 9 where T is the total umber of measuremet poits at the surface of a idividual mirror segmet, ad δ is the umber of poits for which the criterio σ δ < λ /0 is fulfilled (σ δ beig defied by Eq. (3) i 2.2). For each segmet of the previous simulated case, the evolutio of ρ as a fuctio of the relative spectral badwidth δλ/λ is tabulated i Table 3 ad graphically illustrated i Figure 8, where the cofidece ratio is ecoded ito a grey scale: white areas are those where the saity check σ δ < λ /0 was foud successful, while other toes progressively darke as the discrepacies are icreased (black toe represetig error values superior or equal to µm). It ca be see that the safe areas where pisto errors ca be uambiguously determied progressively shrik as δλ/λ teds to its maximal value of 0%, possibly leadig to low cofidece ratios such as 8% ad 9% for segmets 9 ad 0 respectively. It is quite remarkable however that correct pisto estimatios ca still be achieved, probably makig advatage of the high umber T of available measuremet poits ad of the self saity check iheret to the method. Aother coclusio derived from the preseted umerical results is that the worst cofidece ratios are ot oly associated to large pisto errors, but also to the iitial tip-tilt aligmet errors of the segmets, ad this effect stays oticeable eve for small pisto errors (see for example segmets 7, ad 5). Table 3: Ifluece of the spectral badwidth δλ/λ o the OPD retrievig cofidece ratio for each mirror segmet. Spectral badwidth δλ/λ Segmet umber 3% 5% 7.5% 0% Ifluece of atmospheric perturbatios Keepig the same pisto ad tip-tilt static errors as above, we fially add moderate seeig perturbatios beig characterized by a Fried s radius r 0 = 500 mm [24], ad try to recover the iitial wavefrot ad geometrical aligmet errors. It ca the be readily observed that the method breaks dow whe sigificat referece pupil dimesios or spectral badwidths are simulated by the umerical model. As a example, the Figure 9 illustrates the results obtaied for a referece pupil radius r = 200 mm ad δλ/λ = 7.5 %, showig the perturbed WFE (top right pael), the resultig PSF (middle left pael, logarithmic scale), the recostructed WFE (middle right pael) ad a raw differece map betwee both WFEs (bottom left pael). It has bee oticed that most of the residual error origiates from oe particular segmet (the 0, where the cofidece ratios ρ were foud lower, see Table 3). Removig this segmet from the mai pupil allows to greatly improve the global measuremet accuracy (see the bottom right pael of Figure 9), however this stays isufficiet for

10 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 0 eterig withi the requiremets. It ca be cocluded that atmospheric perturbatios severely limit the capacities of the method, which caot be employed for disetaglig the pisto errors from the seeig. Hece the method should probably be restricted to spatial applicatios, or to groud telescopes already equipped with a adaptive optics system, or with a moo-mode WFE filterig subsystem such as sigle-mode fibers or itegrated optics [25]. 5 Coclusio I this paper was preseted the optical desig of a multi-spectral pisto sesor suitable for co-phasig giat segmeted mirrors equippig the future Extremely Large Telescopes (ELTs). The geeral theory of the sesor has bee preseted i detail ad umerical simulatios demostrated that direct pisto ad tip-tilt measuremets are feasible withi accuracies respectively close to 20 m ad 0 aoradias i RMS sese. Those values are compatible with the usual co-phasig requiremets, although the method is severely perturbed by ucorrected atmospheric seeig. It must be emphasized that, although it was preseted herei i the sole framework of the ELT, this method is fully applicable to sparse aperture iterferometers. Appedix. Coectio with the dispersed speckles method The dispersed speckles method was origially proposed by Borkowski et al for the co-phasig of groud or space multi-aperture iterferometers [2][5]. It basically cosists i reorgaizig multispectrally measured PSFs o a three-dimesioal data-cube, whose third dimesio is scaled i terms of the waveumber σ (the iverse of the wavelegth λ). A three-dimesioal iverse Fourier trasform of the data-cube the allows to recover iformatio about the pisto errors alog the third vertical axis. However the floor map of this data-cube beig described by relatio (4b) ofte presets a complex structure, makig it difficult to retrieve the origial errors. This problem ca be solved usig phaseretrieval algorithms hardly applicable to real time computig [6]. The previous dispersed speckles procedure ca easily be adapted to the case of a multi-spectral phaseshiftig TI as illustrated i Figure 0. For that purpose the relatio (8) should be rewritte as follows: C(P, σ) exp[2iπσξ ] B (P - P. (A) = D ) The Fourier trasform of Eq. (A) with respect to the variable σ is defied as: + Ĉ (P,ξ) = C(P,σ)exp[ 2iπσξ] dσ. (A2) Combiig the two previous relatioships readily leads to: Ĉ(P,ξ) = + = BD (P - P ) exp[ 2iπσ(ξ -ξ )] dσ, (A3) which, usig a elemetary property of Fourier trasformatio, reduces to: Ĉ(P,ξ) = B (P - P )δ(ξ -ξ ), (A4). = D δ(ξ) beig the Dirac distributio. It turs out that the th sub-pupil will be shifted of a quatity ξ alog the vertical axis of the trasformed data-cube, allowig i priciple a fast ad accurate determiatio of its pisto error. From a practical poit of view however, the Fourier trasform alog the pisto axis must obey to the classical relatioship: = C 2ξMax δξ, (A5) where C is the total umber of spectral chaels, ad ξ Max ad δξ respectively are the half-rage ad measuremet accuracy of the pisto errors. Assumig ξ Max = 0 µm ad δξ = 50 m as i the preset study we fid C = 400, a excessive umber that would certaily degrade the sigal-to-oise ratio of the sesor, ad therefore its measuremet accuracy. It is worth metioig, however, that a related method is beig curretly developed for the PRIMA istrumet at the focus of the VLTI, makig use of a dramatically reduced umber of spectral chaels [26]. REFERECES

11 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors [] A. Labeyrie, Resolved imagig of extra-solar plaets with future 0-00 km optical iterferometric arrays, Astroomy ad Astrophysics Supplemet Series vol. 8, p (996). [2] V. Borkowski, A. Labeyrie, F. Martiache ad D. Peterso Sesitivity of a "dispersed-speckles'' pisto sesor for multi-aperture iterferometers ad hypertelescopes, Astroomy ad Astrophysics vol. 429, p (2005). [3] G. A. Chaa, M. Troy, F. G. Dekes, S. Michaels, J. elso ad T. Mast, D. Kirkma, Phasig the mirror segmets of the Keck telescopes: the broadbad phasig algorithm, Applied Optics vol. 37, p (998). [4] G. A. Chaa, M. Troy ad E. Sirko, Phase discotiuity sesig: a method for phasig segmeted mirrors i the ifrared, Applied Optics vol. 38, p (999). [5] R. G. Paxma ad J. R. Fieup, Optical misaligmet sesig ad image recostructio usig phase diversity, J. Opt. Soc. Am. A vol. 5, p (988). [6] J.R. Fieup, J.C. Marro, T.J. Schulz ad J.H. Seldi, Hubble Space Telescope characterized by usig phase-retrieval algorithms, Applied Optics vol. 32, p (993). [7] M. R. Bolcar ad J. R. Fieup, Sub-aperture pisto phase diversity for segmeted ad multi-aperture systems, Applied Optics vol. 48, p. A5-A2 (2009). [8] R. Agel, Imagig extrasolar plaets from the groud, i Scietific Frotiers i Research o Extrasolar Plaets, D. Demig ad S. Seager eds., ASP Coferece Series vol. 294, p (2003). [9] A. Labeyrie, Removal of coroagraphy residues with a adaptive hologram, for imagig exo-earths, i Astroomy with High Cotrast Imagig II, C. Aime ad R. Soummer eds., EAS Publicatios Series vol. 2, p. 3-0 (2004). [0]. Yaitskova, K. Dohle, P. Dierickx ad L. Motoya, Mach-Zehder iterferometer for pisto ad tip-tilt sesig i segmeted telescopes: theory ad aalytical treatmet, J. Opt. Soc. Am. A vol. 22, pp (2005). [] F. Héault, Coceptual desig of a phase shiftig telescope-iterferometer, Optics Commuicatios vol. 26, p (2006). [2] F. Héault, Sigal-to-oise ratio of phase sesig telescope iterferometers, J. Opt. Soc. Am. A vol. 25, p (2008). [3] F. Héault, Telescope iterferometers: a alterative to classical wavefrot sesors, Proceedigs of the SPIE vol. 705, 705X (2008). [4] F. Héault, Aalysis of stellar iterferometers as wavefrot sesors, Applied Optics vol. 44, p (2005). [5]. Tarmoul, D. Mourard ad F. Héault, Study of a ew cophasig system for hypertelescopes, Proceedigs of the SPIE vol. 703, 7033U (2008). [6] F. Martiache, Global wavefrot sesig for iterferometers ad mosaic telescopes: the dispersed speckles priciple, J. Opt. A: Pure Appl. Opt. vol. 6, p (2004). [7] Y.-Y. Cheg ad J. C. Wyat, Two-wavelegth phase shiftig iterferometry, Appl. Opt. vol. 23, p (984). [8] K. Creath, Step height measuremet usig two-wavelegth phase-shiftig iterferometry, Appl. Opt. vol. 26, p (987). [9] C. R. Tilford, Aalytical procedure for determiig legths from fractioal friges, Appl. Opt. vol. 6, p (977). [20] P. J. de Groot, Extedig the uambiguous rage of two-color iterferometers, Appl. Opt. vol. 33, p (994). [2] G. Courtès ad M. Vito, U filtre à bades passates multiples, réglables et simultaées destié à l aalyse spectrophotométrique des images télescopiques, Aales d Astrophysique vol. 28, p (965). [22] F. Lauret, F. Héault, E. Reault, R. Baco ad J.P. Dubois, Desig of a Itegral Field Uit for MUSE, ad results from prototypig, Publicatios of the Astroomical Society of the Pacific vol. 8, 849, p (2006). [23] F. Lauret, F. Héault, P. Ferruit, E. Prieto, D. Robert, E. Reault, J.P. Dubois, R. Baco, CRAL activities o advaced image slicers: optical desig, maufacturig, assembly, itegratio ad testig, ew Astroomy Reviews vol. 50, 4-5, p (2006). [24] D. L. Fried, Limitig resolutio lookig dow through the atmosphere, J. Opt. Soc. Am. vol. 56, p (966). [25] F. Malbet, P. Ker, I. Schae-Duport, J.-P. Berger, K. Rousselet-Perraut ad P. Beech, Itegrated optics for astroomical iterferometry I. Cocept ad astroomical applicatios, Astro. Astrophys. Suppl. Ser. vol. 38, p (999).

12 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 2 [26] J. Sahlma, R. Abuter,. Di Lieto, S. Méardi, F. Delplacke, H. Bartko, F. Eisehauer, S. Lévêque, O. Pfuhl,. Schuhler, G. va Belle ad G. Vasisht, Results from the VLTI-PRIMA frige trackig testbed, Proceedigs of the SPIE vol. 703, 703A (2008). FIGURES CAPTIOS Figure : Basic priciple of the phase-shiftig TI. Figure 2: Possible phase shift values φ m for cases M = 3 ad M = 4. Figure 3: Flow-chart of the moochromatic OPD retrieval procedure. Figure 4: Pseudo-code of the OPD uwrappig procedure, with IT stadig for the earest Iteger fuctio. Figure 5: Possible optical implemetatio of the pisto sesor (ot to scale). Plai lies idicate object/image cojugatios, while dashed/dotted lies refer to pupil imagig. Figure 6: Gray-scale map (top) ad three-dimesioal view (bottom) of the WFE of the segmeted mirror affected with radom pisto ad tilt errors (PTV = µm ad RMS = µm). Figure 7: Top row, pupil trasmissio map (left) ad iitial pisto errors (right, PTV = µm ad RMS = µm). Middle row: PSF i the image plae (liear ad logarithmic scales). Bottom row: recostructed WFE (left, PTV = µm ad RMS = µm) ad compariso with iitial errors (right, PTV = 0.07 µm ad RMS = 0.04 µm). Figure 8: OPD estimatio error maps for various spectral badwidths. Figure 9: Same represetatios tha i Figure 7 with additioal seeig perturbatios. Top right pael: perturbed WFE for a Fried s radius r 0 = 500 mm (PTV = µm ad RMS = 4.59 µm). Middle left pael: PSF i the image plae (logarithmic scale). Middle right pael: recostructed WFE (PTV = µm ad RMS = µm). Bottom row: compariso with iitial errors (left, PTV = µm ad RMS = µm for the whole 8 segmets; right, PTV =.403 µm ad RMS = 0.5 µm with segmet 0 excluded). Figure 0: Possible applicatio of the dispersed speckles method to the multi-spectral PSTI. Referece pupil - Radius r - Phase shift φ m Y Exit pupil plae of the telescope O D x 7 0 y P 5 P 8 X F O Y x M y Image plae Z X Figure : Basic priciple of the phase-shiftig TI.

13 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 3 V V C 2 C C 2 2π/3 C 2π/3 O φ C 3 O φ U C 3 C 4 Figure 2: Possible phase shift values φ m for cases M = 3 ad M = 4. Begi For φ m =, M Do Begi Measuremet of PSF m (M ) Computatio of OTF m (P) via iverse Fourier trasform Ed For Combie the M resultig OTFs usig Eq. (8) Phase ad pistos extractio usig Eq. (9) Ed

14 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 4 Figure 3: Flow-chart of the moochromatic OPD retrieval procedure. For each poit P i the output pupil plae: Iputs Sythethic wavelegth: λ S = (/λ - 2/λ 2 + /λ 3 ) - Fractioal phase ϕ (P) at λ Fractioal phase ϕ 2 (P) at λ 2 Fractioal phase ϕ 3 (P) at λ 3 Computatios δ 0 (P) = λ S (ϕ (P) - 2ϕ 2 (P) + ϕ 3 (P)) δ 0 (P) = δ 0 (P) - λ S IT(δ 0 (P)/λ S ) = IT(δ 0 (P)/λ - ϕ (P)) 2 = IT(δ 0 (P)/λ 2 - ϕ 2 (P)) 3 = IT(δ 0 (P)/λ 3 - ϕ 3 (P)) δ = λ ( + ϕ (P)) δ 2 = λ 2 ( 2 + ϕ 2 (P)) δ 3 = λ 3 ( 3 + ϕ 3 (P)) Outputs Measured OPD: δ(p) = (δ + δ 2 + δ 3 )/3 OPD estimatio error: σ δ (P) = [((δ - δ(p)) 2 + (δ 2 - δ(p)) 2 + (δ 3 - δ(p)) 2 )/3] /2 Figure 4: Pseudo-code of the OPD uwrappig procedure, with IT stadig for the earest Iteger fuctio.

15 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 5 Mai Telescope X Phase-shiftig stage Referece segmet Collimatig les X Flat Mirror Secodary mirror M2 Segmeted primary mirror M Focal plae Movig referece facet Multi-spectral stage X X X λ 3 λ 2 Z Flat mirror Focusig les Focusig diffractio gratig Dispersed pupil slicer λ Y CCD camera X λ λ 2 λ 3 Figure 5: Possible optical implemetatio of the pisto sesor (ot to scale). Plai lies idicate object/image cojugatios, while dashed/dotted lies refer to pupil imagig.

16 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 6 OPD (µm) Figure 6: Gray-scale map (top) ad three-dimesioal view (bottom) of the WFE of the segmeted mirror affected with radom pisto ad tilt errors (PTV = µm ad RMS = µm).

17 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 7 OPD (µm) 200 µm 0 m OPD (µm) OPD (µm) Figure 7: Top row, pupil trasmissio map (left) ad iitial pisto errors (right, PTV = µm ad RMS = µm). Middle row: PSF i the image plae (liear ad logarithmic scales). Bottom row: recostructed WFE (left, PTV = µm ad RMS = µm) ad compariso with iitial errors (right, PTV = 0.07 µm ad RMS = 0.04 µm).

18 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 8 δλ/λ = 3 % δλ/λ = 5 % δλ/λ = 7.5 % δλ/λ = 0 % Figure 8: OPD estimatio error maps for various spectral badwidths.

19 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 9 OPD (µm) 600 µm 0 m OPD (µm) OPD (µm) OPD (µm) Figure 9: Same represetatios tha i Figure 7 with additioal seeig perturbatios. Top right pael: perturbed WFE for a Fried s radius r 0 = 500 mm (PTV = µm ad RMS = 4.59 µm). Middle left pael: PSF i the image plae (logarithmic scale). Middle right pael: recostructed WFE (PTV = µm ad RMS = µm). Bottom row: compariso with iitial errors (left, PTV = µm ad RMS = µm for the whole 8 segmets; right, PTV =.403 µm ad RMS = 0.5 µm with segmet 0 excluded).

20 Multi-spectral pisto sesor for co-phasig giat segmeted mirrors 20 Pupil plae Pistos ξ Y ξ 2 ξ ξ 2 ξ X Y X σ = /λ Rescaled ad stacked moochromatic PSFs σ Moo-dimesioal Fourier trasform wrt σ-axis Phase-shiftig ad iverse bi-dimesioal Fourier trasforms Y X Y X Figure 0: Possible applicatio of the dispersed speckles method to the multi-spectral PSTI.