Telescope intefeometes: an altenative to classical wavefont sensos F. Hénault UMR 6525 CNRS H. FIZEAU UNS, OCA, Avenue Nicolas Copenic, 06130 Gasse Fance ABSTRACT Seveal types of Wavefont Sensos (WFS) ae nowadays available in the field of Adaptive Optics (AO). Geneally speaking, thei basic pinciple consists in measuing slopes o cuvatues of Wavefont Eos (WFE) tansmitted by a telescope, subsequently econstucting WFEs digitally. Such pocess, howeve, does not seem to be well suited fo evaluating co-phasing o piston eos of futue lage segmented telescopes in quasi eal-time. This communication pesents an oiginal, ecently poposed technique fo diect WFE sensing. The pinciple of the device, which is named Telescope-Intefeomete (TI), is based on the addition of a efeence optical am into the telescope pupil plane. Then incident WFEs ae deduced fom Point Spead Function (PSF) measuements at the telescope focal plane. Heein ae descibed two diffeent types of TIs, and thei pefomance ae discussed in tems of intinsic measuement accuacy and spatial esolution. Vaious eo souces ae studied by means of numeical simulations, among which photon noise sounds the most citical. Those computations finally help to define the application ange of the TI method in an AO egime, including main and auxiliay telescope diametes and magnitude of the guide sta. Some pactical examples of optical configuations ae also descibed and commented. Keywods: Wave-font sensing, Fouie optics, Telescope-Intefeometes, Phase measuement, Astonomical optics 1. INTRODUCTION The pinciple of Adaptive Optics (AO) was fist poposed by Babcock [1] in 1953, and encounteed continually gowing success afte a few decades. Today the lagest obsevatoies on Eath ae all equipped with this technology that demonstates outstanding capacities to pass beyond the seeing limit and eveal unsuspected details about numeous types of sky objects, paticulaly in the infaed egion of the electomagnetic spectum. But adaptive optics is continually faced to new challenges, such as coveing low-wavelength spectal domain, o attaining exteme Stehl atios fo the detection of exta-sola planets with new geneation, planet-finding instuments [2], whee moe than one thousand mio actuatos ae needed. In addition, AO has to cope with the inceasing size of futue Extemely Lage Telescopes (ELTs), with diametes anging fom 30 to 50 metes. Fo such facilities, it is expected that the pimay mio will be made of an aay of smalle eflecting segments, like the Keck, GanTeCan o JWST (James Webb Space Telescope) aleady ae. In that case, one of the most citical poblems becomes to adjust (o co-phase) the individual pistons of the segments in ode to appoximate the continuous theoetical suface of the pimay mio within accuacies typically bette than one tenth of wavelength. Looking deepe into the futue, the imaging hype-telescope poposed by Labeyie [3] will also impose to develop obust co-phasing capacities. The measuement of the Wavefont Eo (WFE) emeging fom a telescope can be caied out in seveal diffeent ways. In the field of AO, the most common method is to sense WFE by means of a pupil plane Wavefont Senso (WFS), such as Shack-Hatmann [4], cuvatue [5], pyamidal [6] o optical diffeentiation sensos [7]. In the most geneal case howeve, these devices ae not suitable fo co-phasing mio segments, because thei basic pinciple consists in measuing phase slopes and then etieving WFE using digital pocedues. Hence they do not ecognize piston eos. To ovecome this difficulty, anothe way is to employ image plane estoation techniques such as phase etieval [8] o phase divesity [9]. Howeve, such pocesses ae not well matched to AO opeation because they usually equie significant post-pocessing times. The ideal wavefont senso should indeed combine advantages of both methods, i.e. the ability to pefom diect WFE measuements in quasi eal-time. Such a new geneation of WFS is aleady ising howeve. Let us mention as examples the woks fom Angel [10] and Labeyie [11], whose ideas ae to move the WFS fom the pupil down to the image plane, whee its design would be based on a Mach-Zehnde intefeomete, eventually using hologaphic techniques.
Recently, a new appoach was suggested fo diect WFE sensing [12-13], combining some of the pevious tends since the measuement is diectly achieved at the telescope focal plane, on the one hand, and the telescope is equipped with an additional, specific module, on the othe hand. Indeed, the idea consists in tansfoming the telescope itself into a phase sensing appaatus by ceating sets of intefeence finges into the focal plane. Pactically, this is ealized by adding one efeence am at the pupil plane of the telescope a modification equiing subsequent additional opto-mechanical hadwae. Infomation about WFE is then extacted fom the measued signal by means of easonably simple data pocessing algoithms. The geneal pinciples of such a device, which I called Telescope-Intefeomete (TI) ae summaized in section 2, whee two diffeent TI families ae descibed. Then a geneal eo analysis of both types of TIs is povided in section 3, including andom noise as well as systematic eos (o bias). Typical applications and thei ange of validity ae also discussed in that section, and a peliminay tade-off between both types of TIs is conducted. Finally, two examples of pactical implementation of phase-shifting TIs ae descibed in section 5. 2. THEORY Two diffeent types of Telescope-Intefeometes wee descibed so fa, namely the off-axis and phase-shifting TIs. Thei basic theoies ae detailed in Refs. [12] and [13] espectively, and the eade is invited to look at those papes. In ode to pesent a self-content communication howeve, I povide heein two shot and altenative appoaches, based on the diect evaluation of Optical Tansfe Functions (OTFs) by means of coss-coelations. In any case, the fundamental pinciple of the TI method consists to: acquie one o seveal Point-Spead Functions (PSFs) at the telescope focal plane, then compute numeically thei associated OTFs by means of an invese Fouie tansfom, and finally etieve the Wavefont Eo of the telescope fom the phase(s) of the OTF(s). That pocedue is suitable to Adaptive Optics applications, since the used algoithms ae athe simple and compatible with quasi eal-time opeation. The majo diffeence between the off-axis and phase-shifting TI esides in types and shapes of the TI output pupils, such as depicted in Fig. 1. In both cases we conside a main telescope apetue of diamete D = 2R and full geometical aea S R, and an additional apetue, named efeence pupil, of diamete d = 2 and geometical aea S. The key hypothesis consists in assuming that is significantly smalle than R, i.e. the atio C = S /S R is negligible with espect to unity (C << 1). C is one basic chaacteistic of any telescope-intefeomete, called its contast atio. In the next sections ae used the following scientific notations: k = 2π/λ whee λ is the wavelength of the incoming light (supposed to be monochomatic), and and espectively stand fo the coss coelation and convolution poducts. Y Clea apetue adius R Refeence pupil adius Refeence pupil: - Radius - Phase-shift φ Y Clea apetue adius R X X Baseline B Off-axis TI Phase shift TI Fig. 1. Useful pupil aeas fo the off-axis (left) and phase-shifting TIs (ight).
2.1 Off-axis Telescope-Intefeomete In that configuation, the efeence pupil is de-cented of a distance B with espect to the optical axis of the main telescope (see Fig. 1, left side). Let us denote A( P ) the complex amplitude in the pupil plane, as a function of vecto P of coodinates (x,y). The pupil function of the off-axis TI can be witten: A( P) = BR (P) exp[ i k (P)] + B (P P0 ) (1) with ( P ) being the WFE to be measued, and P the baseline vecto of coodinate (B,0). B 0 R ( P ) is the twodimensional tansmission function of a cicula pupil of adius R, unifomly equal to one inside the im, and to zeo anywhee else it must be noticed that this function may not be eally cicula (as in Fig. 1), theefoe R stands fo the clea apetue of the main telescope. Convesely B ( P ) is the tansmission map of the efeence pupil, equal to the tophat function of adius. We seach fo a mathematical expession of C P ( P ), which is the invese Fouie tansfom of the measued PSF at the TI focal plane. But C P ( P ) can also be consideed as the OTF of the off-axis TI, such that [14]: (P) = A(P) A(P) (2) C P Hee will be invoked some classical popeties of coss coelation and convolution poducts, and in paticula: * U(P) V(P) = U (-P) V(P) (3) whateve the functions U( P ) and V( P ) and supescipt * stands fo complex conjugates. Owing to the fact that both B R ( P ) and B ( P ) ae cento-symmetic, C P ( P ) can be developed as follows: CP(P) = BR (P) exp[ i k (P)] BR (P) exp[ i k (P)] + B (P) B (P) + ( P) exp[ i k ( P)] B (P P ) + B (P) exp[ i k (P)] B (P + P ) (4) BR 0 R 0 Fig. 2 shows a typical illustation of one measued PSF at the focus of an off-axis TI unde favoable atmospheic conditions (Fied s adius 0 is equal to 50 mm), while Fig. 3 epesents the modulus of the deived OTF, which is composed of fou diffeent tems as pedicted by Eq. (4). Fig. 2. Example of telescope PSF geneated by an off-axis TI in the pesence of atmospheic petubations with Fied s adius 0 = 50 mm (left: linea scale; ight: logaithmic scale). A weak finges modulation is clealy visible. The fist two tems of Eq. (4) ae popotional to the OTFs of the main and efeence pupils. Denoting them OTF R ( P ) and OTF ( P ) espectively and dividing Eq. (4) by S R, we get: CP (P) = OTFR (P) + C OTF (P) + C BR ( P) exp[ i k ( P)] B(P - P0 )/S + C B (P) exp[ i k (P)] B (P + P )/S (5) The fouth tem is easily isolated fom the othe and e-cented on the oigin: C (P P ) B + (P) = C B (P) exp[ i k (P)] B (P)/S (6) P 0 R R R 0
Fig. 3. Example of Modulation Tansfe Function (MTF) poduced by an off-axis TI in the pesence of seeing with Fied s adius 0 = 50 mm (logaithmic scale). Two symmetic satellites images of the main pupil appea, whose phase is popotional to the telescope WFE. Obviously the method emains valid as long as thee is no ovelaps with the cental tem, one condition that is always fulfilled if B > 3R + [12]. Hee is intoduced the so-called Delta appoximation, which constitutes togethe the main stength and weakness of the method, as will be discussed futhe. It consists in assuming that function B ( P )/S tends towads the Diac distibution δ( P ) as S gets significantly lowe than S R. It implies that: B (P) exp[ i k (P)] C (P P ) B (P)/C (7) R P Eq. (7) is the final phase etieval fomula applicable to an off-axis TI, demonstating that the seached Wavefont Eo ( P ) is popotional to the phase of the complex numbe C P ( P - P 0 ). As a majo consequence, thee emains a 2πambiguity on the phase value, thus the etieved WFE will be enclosed in a ±λ/2 ange. That statement is also valid fo phase-shifting Telescope-Intefeometes. Most geneally 2π-ambiguities can be emoved by means of classical phaseunwapping algoithms, howeve in some specific cases (such as detemination of piston eos of a lage segmented mio), some additional measuements pefomed at diffeent wavelengths might be combined. Fig. 4 and Fig. 5 pesent two examples of numeical simulations whee phase unwapping is efficient. In both cases λ is equal to 0.633 µm, and the main telescope and efeence pupil diametes ae D = 2R = 5 m and d = 2 = 0.5 m espectively. The apetue numbe of the main telescope is 10. Hee the two oiginal WFEs ae showing modeate atmospheic petubations with 0 = 25 mm (Fig. 4), on the one hand, and high-spatial fequency polishing eos of a lage telescope mio that was actually manufactued (Fig. 5), on the othe hand. Attained pefomance is indicated in both Figues, which clealy illustate the phase etieval pocedue showing moduli of cossed OTF tem, ough (o wapped) cossed-tem phase, and final econstucted WFEs as well as thei diffeence maps with espect to oiginal WFEs fo compaison pupose. 0 R + [a] Refeence WFE [b] Cossed OTF tem (modulus) [c] Cossed OTF tem (wapped WFE) [d] Retieved WFE (afte unwapping) [e] Diffeence Map Fig.4. Case of an off-axis TI sensing atmospheic petubations ( 0 = 25 mm). [a] Refeence WFE PTV = 6.811 λ; RMS = 1.497 λ with λ = 0.6328 µm. [d] Reconstucted WFE PTV = 6.781 λ; RMS = 1.495 λ. [e] Bi-dimensional diffeencemap PTV = 0.159 λ; RMS = 0.008 λ. Gey-levels ae scaled to PTV values.
[a] Refeence WFE [b] Cossed OTF tem (modulus) [c] Retieved WFE (afte unwapping) [d] Diffeence Map Fig.5. Case of high-spatial fequency polishing defects. [a] Refeence WFE PTV = 1.258 λ; RMS = 0.200 λ with λ = 0.6328 µm. [c] Reconstucted WFE PTV = 1.182 λ; RMS = 0.195 λ. [d] Bi-dimensional diffeence-map PTV = 0.347 λ; RMS = 0.018 λ. Gey-levels ae scaled to PTV values. 2.2 Phase-shifting Telescope-Intefeomete Two majo diffeences of the phase-shifting TI with espect to the off-axis vesion ae indicated on the ight side of Fig. 1: fist, the efeence pupil has been e-cented on the optical axis Z of the main telescope 1, and second, the whole efeence suface can be moved along Z of known optical path quantities, coesponding to vaious phase shifts denoted φ. The wave amplitude in the pupil plane now wites: A( P) = BR (P)exp[ i k (P)] + B (P) exp[ iφ] (8) Following the same easoning than in section 2.1, and again employing the afoe mentioned Delta appoximation leads to a simplified expession of the OTF in the pupil plane, associated to a given phase-shift φ : C (P) OTF (P) + C OTF (P) + C B ( P) exp[ i k ( P) + iφ] + C BR (P) exp[ i k (P) iφ] (9) φ R R Giving to φ successive values of 0, π/2, π and -π/2, a simple linea combination of the complex OTFs allows to etieve the oiginal phase, and theefoe the Wavefont Eo ( P ). (P) exp[ i k (P)] [C (P) + i C (P) C (P) i C (P)]/4C (10) BR 0 π/2 π -π/2 Hence fou diffeent PSFs must be acquied hee, wheeas only one is necessay fo the off-axis TI. Although this seems to be a sevee dawback, advantages and limitations of both types of TIs ae addessed in the next section. Numeical simulations pesented in Fig. 6 illustate the whole measuement sequence of the phase-shifting TI. The values of used paametes ae simila to those of section 2.1, excepting Fied s adius 0 that is hee equal to 50 mm. 3.1 Eo analysis 3. PERFORMANCE DISCUSSION AND LIMITATIONS An extensive and in-depth analysis of all types of systematic eos o andom noises that can affect TIs pefomance is beyond the scope of this communication. In view of building o pototyping such devices, this wok will be necessay howeve, and was in fact aleady stated in Ref. [15]. Table 1 povides a peliminay (and non exhaustive) list of eos, togethe with a summay of the main conclusions at this stage of the study. Two kinds of measuement uncetainties wee distinguished, which ae bias o systematic eos, on the one hand, and andom noises, on the othe hand. Fo what concens bias eos, the majo contibutos poved to be the Delta appoximation in Eqs. (7) and (9), and the useful spectal bandwidth and angula size (o magnitude) of the obseved sky-object. Random eos, on thei side, ae fully dominated by the Signal-to-Noise Ratio (SNR) of the employed detecto aay, and moe specially by photon noise. In paticula, an analytical elationship between the SNR and subsequent WFE uncetainty δ ( P ) could be established: δ ( P ) < 1/(k SNR C) = S R /(k SNR S ) (11) 1 Howeve this condition is not absolutely necessay: the efeence pupil aea could be de-cented with espect to the optical axis, povided that it is embedded in the main pupil aea as in the optical configuations pesented in section 4.
[a] Tansmission map of the phase shifting TI [b] Acquisition of a single PSF [c] MTF of a single computed OTF [d] MTF of the fou combined OTFs [e] Refeence WFE [f] Wapped WFE [g] Retieved WFE [h] Diffeence Map (afte unwapping) Fig. 6. Case of a phase shifting TI sensing atmospheic petubations ( 0 = 50 mm). [a] Full tansmission map of the TI, including the cental efeence pupil. [b] Acquisition of one single PSF. [c] MTF deived fom one single acquied OTF. [d] MTF deduced fom the fou combined OTFs. [e] Refeence WFE PTV = 3.395 λ; RMS = 0.943 λ with λ = 0.6328 µm. [g] Reconstucted WFE PTV = 3.281 λ; RMS = 0.937 λ. [h] Bi-dimensional diffeence-map PTV = 0.251 λ; RMS = 0.017 λ. Gey-levels ae scaled to PTV values. Table 1. Non-exhaustive list of potential eos affecting Telescope-Intefeometes and elevant lessons leaned. TYPES OF ERRORS Systematic o Bias Eos Intinsic eo due to the Delta appoximation Maximal spectal bandwidth Angula adius of the obseved sky-object Diffeential WFE between efeence and main pupils Accuacy of phase steps (fo phase-shift TIs) Random Eos Detection noises including: Photon o shot noise Read-out Noise (RON) Dak cuent Atmospheic tubulence (fo long integation times) Scintillation effect LESSONS LEARNED Can be maintained well below diffaction limit when S << S R, but at the expense of inceased photon noise. About 0.4 and 20 % fo off-axis and phase-shift TIs espectively. Coesponds to sta magnitude anging fom 2 to 12, depending on TI type and dimensions. Negligible when efeence pupil is diffaction-limited [12]. Can be calibated and coected [16-17]. Mostly govened by photon noise see Eq. (11) and section 4.2. Majo contibuto Negligible Negligible Method fails to etieve WFEs. Should only be used on shot integation times [15]. Not studied so fa.
Relationship (11) epesents indeed a fundamental fomula fo the dimensioning and pefomance assessment of any type of TI. Moeove, it shows that the attainable measuement accuacy δ ( P ) is linked to the spatial esolution R of the device: the latte is indeed equal to the atio S R /S, which is the invese of the TI contast atio C (see section 2). Hence Eq. (11) may be ewitten as: δ ( P ) < R /(k SNR) (12) It can be concluded that fo a given SNR of the detecto, the highe is the desied spatial esolution, the wose the measuement eos will be. Convesely, low WFE spatial esolutions impove the intinsic measuement accuacy of the TI system. Hee a balance must be defined unambiguously between both the equied spatial esolution and allowable measuement eo: it can be felt intuitively that if the WFE to be estimated is mainly composed of low spatial fequency defects (e.g. pistons eos of a lage segmented telescope, tip/tilt, o focus), lage values of (and thus C) can be chosen, and the global accuacy δ ( P ) will be moe favoable. An othe impotant conclusion, howeve, is the fact that in ode to pefom seeing measuements in AO egime, the adius of the efeence pupil should neve exceed the actual Fied s adius 0. Theefoe two golden ules may be applied when dimensioning a TI: Select the highest R satisfying condition < 0, Use Eqs. (11) o (12) in ode to optimize the global measuement accuacy. Running compute codes descibed in Ref. [15], TIs measuement eos wee also estimated as a function of spectal bandwidth and angula adius of a taget sta. Fo that pupose, diametes of the main and efeence pupils of both TIs (eithe off-axis o phase-shifting) wee espectively set to D = 2R = 5 m and d = 2 = 1 m (the latte assumption implies excellent seeing conditions). Mean wavelength was λ = 0.5 µm and the pimay mio of the telescope was composed of seven o six hexagonal facets of 0.8 m side, each being affected with piston eos anging fom -λ/2 to +λ/2. SNRs wee computed fo an integation time τ = 10 msec in ode to stay compatible with an AO egime. Numeical esults ae plotted in Fig. 7. They show that Both designs adically diffe fo what concens maximal spectal bandwidths. Off-axis TIs ae found to be vey sensitive to wavelength, since they cannot affod spectal anges highe than 0.4 %. Convesely, phase shift TIs look much moe convenient, since the maximal bandwidth is aound 20 %. Maximal angula sizes of the obseved sta ae aound 20 mas, which coespond to magnitude 2. Photon noise dominates the off-axis TI whateve the sta magnitude, while it govens phase shift TI above magnitude +1. Consequently, contibution of photon noise is so impotant that heavy numeical simulations such as those pefomed in this section become useless, and a ealistic estimation of the TIs measuement accuacy can be obtained using the sole elationship (11). Thus all calculations of section 4.2 will be pefomed that way. Measuement Eo (Waves) 0.15 0.12 0.09 0.06 0.03 0.00 PTV Off-axis TI RMS 1 10 100 1000 PTV Spectal Bandwidth (nm) RMS Phase shift TI Measuement Eo (Waves RMS) 0.1000 0.0100 0.0010 0.0001 Photon Noise Off-axis TI Phase-shift TI 0 6 12 19 25 Angula Radius (mas) Bias Eos Phase-shift TI Off-axis TI Fig. 7. Left, measuement accuacy as function of spectal bandwidth (black lines: phase-shifting TI; gay lines: off-axis TI; solid lines: PTV values; dashed lines: RMS values). Right, measuement accuacy as function of angula adius of the obseved sky-object (solid lines: RMS bias eo; dashed lines: photon noise contibution). The cuves show undeniable supeioity of phase-shifting TIs with espect to off-axis TIs.
3.2 Application ange Having asseted that photon noise is the dominant cause of eo in a TI system, and knowing a simple elationship fo evaluating it Eq. (11) allows to define moe easily the application ange of the method. The main scope of this section is to define acceptable limits on some citical paametes diving a TI pefomance, namely the main and auxiliay apetue diametes and the magnitude of the obseved sta. Hee one of the main goals is to state if the method is appopiate fo an AO system having the additional capacity of sensing the piston eos affecting the segmented mios of an ELT. Thus the main telescope diamete will be anging fom 10 to 50 metes, while efeence pupil diamete vaies fom 0.1 to 1 mete. The wavelength λ and integation time τ stay always equal to 0.5 µm and 10 msec espectively. We shall conside that photon noise is acceptable as long as it does not geneate WFE uncetainties highe than the Maéchal s citeion, i.e. WFE < 0.075 λ in RMS sense [14]. Some majo conclusions wee deived fom the numeical esults 1, which ae illustated in Fig. 8. The best measuement accuacies coespond to the smallest values of the main telescope diamete. This is indeed a diect consequence of Eq. (11), and the fact that the efeence pupil aea S is limited by Fied s adius 0. Theefoe the best pefomance is attained fo 10 m-class telescope diamete (left side of Fig. 8). The phase-shifting TI takes a clea advantage fom its extended spectal ange: a 10 m-diamete phase shift TI shows pefomance simila to a 50 m-diamete off-axis TI (left side of Fig. 8). An ELT of 30 m-diamete equipped with a phase-shift system should stay diffaction-limited when the efeence pupil diamete d = 2 0 is equal to 0.3 m. A 50 m-diamete ELT would equie that d = 2 0 = 0.4 m, which coesponds to good o vey good seeing conditions (left side of Fig. 8). Depending on atmospheic distubances, the limiting magnitude of a phase-shifting TI should vay between 8 ( 0 = 0.25 m) and 11 ( 0 = 0.5 m). The limiting magnitude of an off-axis TI could neve exceed 4 (ight side of Fig. 8). Hence the phase-shifting TI looks the most pomising. Howeve it still suffes fom an incomplete sky coveage, on the one hand, and only shows its best pefomance when atmospheic seeing is favoable, on the othe hand. Fo those two easons it cannot be consideed as pat of a multi-pupose AO system, but could be advantageously employed in some paticula cicumstances (peiodical co-phasing of eflective facets on an ELT, scientific obsevations of modeate magnitude sky-objects, such as those achieved by planet-finding instuments). Measuement Eo (Waves RMS) 1.00 Off-axis TI, D=10 m Phase-shift TI, D=10 m Off-axis TI, D=30 m Phase-shift TI, D=30 m Off-axis TI, D=50 m Phase-shift TI, D=50 m 0.10 0.01 0.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Refeence Pupil Diamete (m) Measuement Eo (Waves RMS) 0.20 0.15 0.10 0.05 0.00 Off-axis TI, = 0.5 m Diffaction limit Phase-shift TI, = 0.25 m Phase-shift TI, = 0.5 m 4 6 8 10 12 Magnitude (V Band) Fig. 8. Left, measuement accuacy as function of efeence pupil diamete d = 2, fo vaious telescope diametes D = 2R (black lines: phase-shifting TI; gay lines: off-axis TI; dotted lines: D = 10 m; dashed lines: D = 30 m; solid lines: D = 50m). Right, measuement accuacy as function of sta magnitude fo D = 30 m (dashed line: = 0.25 m; solid lines: = 0.5 m). Hee again advantages of the phase-shifting TI clealy appea. 1 Although some esults do not fully comply with those pesented in Ref. [15], majo conclusions emain unchanged.
3.3 Telescope-Intefeometes tade-off The following citeia wee selected in ode to establish a peliminay tade-off between both types of TIs: spectal bandwidth, limiting magnitude of the taget sta, minimal numbe of equied pixels 1, and hadwae costs and complexity. Elements of answe ae summaized in Table 2. Table 2. Telescope-Intefeometes tade-off. Allowed spectal bandwidth Citeia OFF-AXIS TI PHASE-SHIFTING TI Limiting sta magnitude fo D = 30 m Minimal numbe of pixels Hadwae complexity and costs 0.4 % 4 1024 1024 High, equies manufactuing of dedicated auxiliay telescope and associated delay line [12] Between 20 and 30 % 11 256 256 Modeate, can be implemented with small and simple optical components located at the telescope focal plane (see section 4) The conclusion of this tade-off seems obvious, since the phase-shifting TI is supeio fom any point of view. This leads to finally discad the off-axis TI fom futhe design study even peliminay. Theefoe the following section 4 will only be focused at the pactical implementation of a phase-shifting Telescope-Intefeomete. 4. PRACTICAL IMPLEMENTATION ON A TELESCOPE FACILITY In this section ae descibed two possible optical aangements fo wavefont sensing based on the pinciple of phaseshifting TIs. Those measuement schemes ae new, since peviously poposed implementations in Refs. [12] and [13] wee well founded, but suffeed fom a few pactical dawbacks 2. WFE measuements can indeed be pefomed following two diffeent modes, depending on PSFs acquisition schemes. The latte can eithe be simultaneous o sequential, as descibed in sections 4.1 and 4.2 espectively. In both cases the phase sensing device is located behind the telescope focal plane, within a compact optical layout whee the fou phase-shifts φ = 0, π/2, π and -π/2 ae added to the telescope WFE. It is also assumed that the pimay mio of the telescope includes a efeence segment of high image quality (i.e. diffaction-limited), coesponding to the efeence pupil aea whee phase-shifts have to be intoduced (see Figs. 9 and 10). 4.1 Simultaneous measuements Hee PSFs acquisitions ae ealized simultaneously, duing a full integation time τ = 10 msec. The WFE sensing device incopoates one Collimating Lens (CL1, see Fig. 9), then splits the collimated beam into fou diffeent optical ams by means of thee beam-splittes BS1, BS2 and BS3. Each optical am is composed of the following optics o electonics components: A Phase Plate (denoted PP1, PP2, PP3 and PP4 in Fig. 9) in chage of adding the efeence phase-shift φ to the telescope WFE. The Phase Plate is located at an image plane of the telescope pupil, nea CL1 focal plane. The phase-shift φ is intoduced by means of an equivalent glass thickness in the efeence pupil aea. Depending on the selected mateial and effective spectal bandwidth of the device, it might be necessay o not to use a set of achomatic plates simila to those cuently used in nulling intefeomety [18], instead of one single plate. A Focusing Lens (FL1, FL2, FL3 and FL4) e-imaging the phase-shifted PSF on a detecto aay. The focal length of the FL is adjusted in ode to achieve a cetain magnification atio M between telescope and camea focal planes. The exact value of M depends on the camea pixels size and the equied OTF spatial sampling. A CCD detecto aay (Cameas 1-4 in Fig. 9) finally acquiing the phase-shifted PSF. 1 Since a limited pixels numbe is moe suited to implementation of the algoithms (e.g. FFT) in quasi eal-time. 2 Let us mention fo example the Michelson configuation in Ref. [12], o the altenative design of Ref. [13], section 4.2, whee the efeence pupil was integated into the segmented pimay mio of the main telescope: the majo difficulty was hee to contol the displacement of the lage efeence mio with sufficient pecision.
X Camea C2 Wave-font sensing device Entance Wave-font Seconday Mio M2 Main Telescope Refeence segment Focal plane Collimating Lens CL1 BS3 BS1 FL2 PP2 BS2 PP1 FL1 BS FL PP Beamsplitte Focusing Lens Phase Plate Camea C1 Z Segmented Pimay Mio M1 PP3 FL3 FL4 PP4 Camea C4 Camea C3 Fig. 9. Schematic view of a phase-shifting TI aangement, designed fo simultaneous PSF/OTF measuements. The majo advantage of this WFS configuation consists in the simultaneity of PSFs measuements, ensuing a eliable WFE econstuction. In etun, the pesence of seveal beam-splittes will decease the signal levels ecoded by each camea of at least 75%, which epesents a stong disadvantage fo astonomical applications. The following configuation eliminates this dawback, since PSF measuements ae ealized sequentially. 4.2 Sequential measuements In this configuation, one single CCD camea is needed fo the fou diffeent PSF acquisitions. Measuements ae pefomed sequentially athe than simultaneously. The full acquisition time must stay equal to 10 msec, thus the elementay integation time (at each diffeent phase-shift φ) will be τ = 2.5 msec. As in pevious section, the WFE sensing device is composed of one Collimating Lens imaging the telescope exit pupil on a efeence flat mio (see Fig. 10). That mio is pieced at a location coesponding to the telescope efeence facet, and a mandel caying a small, flat optical suface is piezoelectically moved along the optical axis, thus geneating the equied phase-shifts φ. A Focusing Lens finally foms consecutive images of the PSF in the plane of the detecto aay, whee they ae ecoded befoe data pocessing. Fom a adiometic point of view, this last configuation appeas as the most favoable, since esulting SNRs should be multiplied by a facto aound two. On the othe hand, any change of the telescope WFE between successive acquisitions may alte the econstuction pocess and then the final achieved accuacy. 5. CONCLUSION This pape povides a synthesis about the pinciples, pefomance and limitations of what I named Telescope- Intefeometes in pevious woks Refs. [12-13] and [15]. The basic idea consists in tansfoming a telescope into a WFE sensing device by giving to sky photons an additional access to the focal plane of the telescope though its exit pupil. This can be achieved in two diffeent ways, namely the off axis and phase-shifting TIs. In the fist case a small and decented efeence telescope is added aside the main pupil, ceating a weakly modulated finge patten in the image plane. In the second configuation, diffeent calibated phase shifts ae applied ove the efeence pupil aea. In both cases the PSFs measued in the focal plane of the telescope cay infomation about the tansmitted WFE, which is etieved via fast and simple algoithms suitable to an adaptive optics opeational mode.
Entance Wave-font Seconday Mio M2 Main Telescope X Refeence segment Focal plane Camea Collimating Lens Wave-font sensing device Focusing Lens Flat Mio Moving efeence facet Z Segmented Pimay Mio M1 Fig. 10. Schematic view of the phase-shifting TI aangement, designed fo sequential PSF/OTF measuements. Heein was evaluated the accuacy of both types of TIs, in tems of noise and systematic eos. It was highlighted that RMS measuement eo is popotional to the geometical aea of the main telescope, and invesely popotional to detecto SNR and efeence pupil aea. The spectal bandwidth λ and adiating popeties of the taget sta wee defined (e.g. λ < 150 nm and magnitude compised between -2 and +11 fo a phase-shifting TI woking in the V band). It was shown that WFE measuement eos ae paticulaly sensitive to photon noise, which apidly govens the achieved accuacy fo telescope diametes highe than 10 m. Nevetheless, TI method seems to be applicable to adaptive optics systems on telescope diametes anging fom 10 to 50 m (i.e. ELTs), depending on the seeing conditions and magnitude of the obseved stas. Also, phase-shifting TIs wee found clealy supeio to thei off-axis vesion assuming the same geometical chaacteistics. This led to ule out the off-axis TI method at the end of a shot tade-off, and only tentative designs based on phase-shifting TI pinciple wee pesented in the last section, showing two pomising wavefont sensing configuations. Othe attactive theoetical studies emain to be undetaken aound TI matte, such as application to scintillation measuements, compaison with diffeent types of WFS and paticulaly those depicted in Refs. [10-11] o impovements of the method by implementing a deconvolution pocess in ode to get id of the Delta appoximation. Howeve, pioity should be given now to the pactical ealization of an expeiment o a TI pototype, whose esults would validate the theoy, pedicted pefomance and eos estimation. This seems to be a mandatoy step befoe a lage-scale implementation of a phase-shifting TI can be envisaged on an existing telescope facility.
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