Imaging and nulling properties of sparse-aperture Fizeau interferometers

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1 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Fraçois Héault Istitut de Plaétologie et d Astrophysique de Greoble Uiversité Joseph Fourier, Cetre Natioal de la Recherche Scietifique.P. 53, 3841 Greoble Frace ASTRACT I this commuicatio are preseted rigorous ad approximate aalytical expressios of the Poit Spread Fuctio (PSF) ad Field of View (FoV) achievable by multi-aperture Fizeau iterferometers, either of the imagig or ullig types. The described formalism ca be helpful for dimesioig future space missios i search of habitable extra-solar plaets. Herei the characteristics of PSF ad FoV are derived from simple aalytical expressios that are further computed umerically i order to evidece the critical role of pupil re-imagig alog the iterferometer arms. The formalism is also well suited to simulatig pseudo-images geerated by a ullig Fizeau iterferometer, ad umerical computatios demostrate that it is oly efficiet for very short baselies. Fially, two differet desigs improvig the ullig capacities of such exoplaet observig istrumets are briefly preseted ad discussed. Keywords: Fourier optics, Phased telescope array, Fizeau iterferometer, Nullig iterferometry 1 INTRDUCTIN Sice the historical writigs of Fizeau [1], Stépha [2] ad Michelso [3], multi-aperture optical systems ad their imagig properties have bee the subject of extesive literature, leadig amog other to the curretly admitted distictio betwee Fizeau ad Michelso types of iterferometer. It is ofte cosidered today that the major differece betwee both cocepts relies o the fact that the former obeys to a golde rule statig that the output pupil of the iterferometer must be a scaled replica of its etrace pupil [4-5], while the latter does ot. Some ew multiaperture cocepts, however, have emerged durig the three last decades, such as space bore, ifrared ullig iterferometers or telescopes dedicated to the search of extra-solar plaets [6-7]. I two recet publicatios was described a simple Fourier optics formalism allowig to derive the basic bject-image relatioships of such systems [8] ad approximate expressios of their Poit Spread Fuctios (PSF) ad achievable Field of View (FoV) [9]. The purpose of the preset commuicatio is to complete ad simplify this formalism agai (sectio 2) ad to provide a deeper aalysis of sparse-aperture Fizeau iterferometers. Examples of applicatios are described i sectio 3, where is illustrated the critical role of pupil re-imagig alog the iterferometer arms. Fially, two alterative desigs improvig the ullig ad pseudo-imagig capacities of such istrumet evisioed for future space missios searchig for habitable extra-solar plaets are preseted ad discussed i sectio 4. 2 GENERAL RELATINSHIPS Here the formalism described i Refs. [8-9] is briefly summarized i sectio 2.1, before derivig the theoretical relatioships applicable to the PSF ( 2.2), maximal achievable field of view ( 2.3). Furthermore, a approximatio regardig the maximal achievable FoV i Ref. [9] has bee clarified. 2.1 bject-image relatioship Let us cosider a optical system desiged either for high-agular resolutio imagig or ullig iterferometry purpose, beig composed of N collectig apertures ad N recombiig apertures. Figure 1 depicts the three mai coordiate systems beig used here after: they are a o-sky agular coordiates system (U,V), a etrace pupil referece frame (,X,Y,Z) where Z is the mai optical axis, ad a exit pupil referece frame (,X,Y,Z). Let us further assume that:

2 1) For all idices comprised betwee 1 ad N, the th collectig aperture of ceter P is optically cojugated with its associated combiig aperture of ceter P without ay pupil aberratio. 2) All collectig apertures have a idetical diameter D ad cosequetly all recombiig apertures share the same diameter D. Practically, it meas that all collectig telescopes ad optical trais coveyig the beams from the etrace to the exit apertures are idetical, which is most ofte the case i curret iterferometer facilities, either of Fizeau or Michelso types. The geeric optical layout of the iterferometer is depicted i Figure 2. Let us fially defie the followig parameters (bold characters deotig vectors): S s (s ) Ω, dω PSF T (s) K A uit vector of directio cosies (u,v,1) directed at ay poit i the sky (correspodig to ay poit M i the image plae), where agular coordiates u ad v are cosidered as first-order quatities A uit vector of directio cosies (u,v,1) poited at a give sky object (or a elemetary agular area of it) The agular brightess distributio of a exteded sky object The total observed FoV i terms of solid agle, ad its differetiatig elemet The PSF of a idividual collectig telescope, beig projected back oto the sky. For a uobstructed pupil of diameter D, this would be the classical Airy distributio equal to 2J 1 (ρ)/ρ 2, where ρ = k D s /2 ad J 1 is the type-j essel fuctio at the first order The waveumber 2/λ of the electro-magetic field assumed to be moochromatic, ad λ is its wavelegth a The amplitude trasmissio factor of the th iterferometer arm (1 N) ϕ A phase-shift itroduced alog the th iterferometer arm for ptical Path Differeces (PD) compesatio or ullig purposes (1 N) P A vector defiig the ceter P of the th sub-pupil i the etrace pupil plae P (1 N) The maximal baselie betwee ay couple (, ) of telescopes (1 ad N) P Correspodigly, a vector defiig the ceter P of the th sub-pupil i the exit pupil plae P (1 N) The maximal baselie betwee ay couple (, ) of exit sub-apertures (1 ad N) m The optical compressio factor of the system, equal to m = D /D = F C /F where F ad F C respectively are the focal legth of the collectig telescopes ad of the relay optics (see Figure 2). Hece accordig to Refs. [1] ad [9] the expressio of the image I(s) formed by the multi-aperture optical system ad projected back oto the sky writes i a first-order approximatio: I( s) = ( s ) PSF ( s - s T s Ω = 1 ) N a [ φ ] exp[ ikξ( s, s) ] 2 exp i dω, (1a) with fuctio ξ(s,s) beig a extra PD term: ξ( s, s ) s P s' P' / m =. (1b) This very geeral bject-image relatioship ca oly be reduced to covolutio products if certai coditios are fulfilled, which were extesively discussed i Ref. [9]. Herei the followig sectios oly deal with some cosequeces o the PSF ad effective FoV accessible by the whole system. 2.2 Poit Spread Fuctio We ca defie a geeralized PSF of the optical system by simply replacig the object brightess fuctio (s ) with the impulse Dirac distributio δ(s-s ) i Eq. 1a: PSF ( s, s G N T ( s - s ) a exp, = 1 [ ] [ ( )] 2 iφ exp ikξ s ) = PSF s. (2) PSF G (s,s ) presets the particularity of costatly varyig with the agular locatio s of the sky object i the istrumet FoV, hece differig sigificatly from the familiar, ivariat PSF of Fourier optics. Cosequetly, the otios of

3 ptical Trasfer Fuctio (TF) ad Modulatio Trasfer Fuctio (MTF) caot by applied i classical sese ad are ot discussed further. 2.3 Maximal achievable Field of View We may ituitively defie a maximal achievable Field of View of the multi-aperture optical system as the image that would be formed uder the followig hypotheses: o No physical diaphragm of ay kid (stops, mirrors edge or cetral obscuratio) is take ito accout. o The sky object is uiformly bright over a 2-steradia solid agle, hece (s ) = 1. o The optical system is free from aberratios, thus PSF T (s) is assumed to be the classical Airy fuctio. Uder such assumptios the maximal achievable FoV deduced from Eq. 1a is: FoV( s) = PSF ( s - s T s Ω = 1 ) N a [ φ ] exp[ ikξ( s, s )] 2 exp i dω. (3) Eq. 3 is still a complicated mathematical expressio that caot be simplified aalytically ad is requirig extesive computig times whe calculated umerically. A additioal heuristic simplificatio preseted i Ref. [9] cosisted i assumig that PSF T (s) ca be approximated to the Dirac distributio δ(s). I that case the itegral of Eq. 3 ca be reduced to: N = 1 [ ] [ ( )] 2 iφ exp ik s P ' P' FoV( s ) = a m. (4) exp / This is a simplified ad codesed expressio of the theoretical achievable FoV. However Eq. 4 remais a approximate relatioship that should oly be used with the greatest care, because it is oly valid for whe idividual telescopes are of large aperture size, or they are close oe to the other. I other cases Eq. 3 has to be computed accurately. V -sky agular coordiates Sky object u v U s s Y P 1 Etrace pupil plae X D Y P 3 Exit pupil plae P 2 P 3 P 4 D P 4 P 1 P 2 s s X F Y M Detectio plae X Z M Figure 1: Sketch of the used referece frames o-sky (U,V), o the etrace pupil plae (,X,Y) ad o the exit pupil plae (,X,Y ). The coordiate system (,X,Y ) attached to the image plae is optically cojugated with the (U,V) referece frame.

4 Telescope #1 Telescope #2 L1 F F C L2 L3 M1 eam collectig optics eam lauchig optics L4 L5 L6 M2 Sigle mode optical fiber Multi-axial combiig optics L7 F X Figure 2: Schematic optical layout of a multi-aperture iterferometer, icludig achromatic phase shifters (APS) used i ullig mode. Z 3 APPLICATIN T FIZEAU INTERFERMETERS I this sectio are discussed the typical bject-image relatioship of multi-aperture Fizeau iterferometers ( 3.1) ad the ifluece of pupil re-imagig aberratios alog the iterferometer arms o their performace ( 3.2). The limitig capacity of this type of iterferometers for pseudo-imagig plaets i ullig mode will be addressed i aother sectio ( 4.3). This sectio is supported by a set of umerical simulatios whose umerical parameters are summarized i Table 1 for all cosidered cases. The umerical values of F ad F C are respectively equal to F = 5 m ad F C = 1 mm, leadig to a optical compressio factor m = 1/5. All computatios were carried out at the wavelegth λ = 1 µm, which is a typical umber for ullig iterferometers workig i the mid-ifrared bad.

5 Case Table 1: Numerical values of mai physical parameters for all simulated cases. Number of etrace ad exit pupils (m) D (m) (mm) D (mm) Sectio Ideal Fizeau iterferometer Pupil aberratios 2 ad Imagig capacity i ullig mode Theoretical relatioships Classically, Fizeau iterferometers preset the uique property that their output pupil is the scaled replica of their etrace pupil: all the etrace ad exit sub-apertures as well as their relative arragemet are perfectly homothetic. Mathematically this coditio implies that: P = m P, (1 N), (5) also beig kow as the Pupil i = Pupil out coditio [4] or golde rule of iterferometry [5]. I that case, isertig Eq. 5 ito Eqs. 1 readily leads to the familiar covolutio product of Fourier optics betwee the object (s) ad its image I(s) formed by the multi-aperture istrumet: N where fuctio: [ ] [ ] 2 F( s) = a exp φ exp i k sp = 1 [ PSF ( s) F( s) ] ( ) I( s) = T s, (6a) i (6b) was amed Far-field Frige Fuctio (FFF) i Ref. [8] because it describes the iterferece patter that would be observed i the image plae if all sub-pupils were reduced to a pihole. It follows from Eq. 6a that Fizeau iterferometers possess the atural ability to form real images of a observed sky object, beig evetually disturbed by shifted replicas of the same object geerated by the FFF. Isertig ow the coditio 5 ito Eqs. 2 ad 4 allows retrievig two basic properties of Fizeau iterferometers: 1) The geeralized PSF of the optical system does ot deped ay loger o vector s, hece the PSF is ivariat over the whole iterferometer FoV, ad equal to PSF T (s) F(s). 2) The maximal achievable FoV of the Fizeau iterferometer ca be approximated by the very simple relatioship: N = 1 [ ] 2 FoV( s ) = a exp i. (7) φ Hece the FoV should be uiform, takig a costat value that oly depeds o the amplitude trasmissio factors a ad phase-shifts ϕ of the iterferometer. For a classical imagig istrumet beig perfectly co-phased (ϕ =. whatever is ), FoV(s) is uiformly equal to 1, which is i agreemet with the golde rule of iterferometry. Eq. 7 suggests however that this golde rule may be of dramatic cosequece whe ullig iterferometers are cosidered, sice the phase-shifts ϕ geerated by their Achromatic Phase Shifters (APS) must be chose so as to cacel the light origiatig from the cetral star. This has the cosequece that the ivariat PSF is equal to zero at its theoretical ceter as illustrated i Figure 3, where the case of a eight-telescope Fizeau iterferometer is cosidered i both imagig ad ullig modes (the squared arragemet of the array is sketched o the left pael of the Figure, usig umerical parameters o the first row of Table 1). Effectively, the computed PSF shows a dark ceter i ullig mode, ad followig Eq. 7 oe may expect the FoV to be uiformly dark everywhere, therefore makig udetectable ay extra-solar plaet orbitig aroud its paret star. ut here the rigorous expressio of the FoV must be evaluated from Eq. 3, turig ito the followig expressio whe coditio 5 is fulfilled: FoV( s) = PSF ( s - s T s Ω = 1 ) N a [ ] [ φ ] exp ikp ( s s ) exp i dω. (8) 2

6 From Eq. 8 FoV(s) is clearly a costat umber that ca be evaluated umerically from the parameters of Table 1, ad is empirically foud to ted toward uity as the /D ratio becomes larger. Thus i that case, we coclude that a ullig Fizeau iterferometer do ot fudametally differ from its imagig versio, as will be cofirmed by the umerical simulatios preseted i sectio 4.3. Iput pupils Y PSF (3D view) PSF (logarithmic gray-scale map) Nullig mode Imagig mode Y D X D X 1 arcsec Figure 3: Ivariat PSF formed by a eight-telescope Fizeau iterferometer i both imagig (top) ad ullig modes (bottom). The achromatic phase-shifts ϕ of the ullig versio are idicated o the left bottom pael. 3.2 Derivig pupil imagig requiremets So far was assumed that the agular orietatios (u,v ) ad (u,v) i vectors s ad s respectively, are small quatities. I the frame of a first-order approximatio, this leads to the defiitio of the geeral bject-image relatioships of Eqs. 1, ad further to its applicatio to the special case of Fizeau iterferometers i Eqs. 6 [8]. ut ideed, the geeral relatios 1-3 ca still be employed whatever are the expressios of the scalar products s P ad s P. It allows, for example, to study the effects of pupil re-imagig aberratios o the iterferometer performace, ad first of all of their axial shifts alog differet arms. Let us cosider a sparse-aperture iterferometer whose etrace ad exit pupils are affected are sufferig from axial shifts dz ad dz (1 N) with respect to their omial plaes P ad P. All the employed otatios are idicated i Figure 4 for the case = 2 sub-apertures. Etrace pupil shifts dz may origiate from positioig errors of the collectig telescopes (here assumed to be the stop apertures as o the left side of the Figure), while exit pupil shifts dz will be geerated by focusig errors i pupil cojugatio optics (show by gray areas i Figure 2). The both vectors s ad P shall be developed at the secod-order: si u u s = cos u si v v (9b), ad: 2 2 cos u cos v 1 u 2 v 2 Similarly, vectors s ad P ca be writte i this secod-order approximatio: x ' P = y. (9b) dz

7 si u u s = cos u si v v (9c), ad: 2 2 cos u cos v 1 u 2 v 2 ' P ' = x ' ' y ' ( 1+ dz F' ) ' ( 1+ dz F' ) ' dz, (9d) where F stads for the focal legth of combiig optics. The the PD fuctio ξ(s,s) defied i Eq. 1b becomes: ξ ' ' ' ' (, s ) dz dz m + u x + v y ( u x + v y )( 1 dz F' ) m s ' 2 2 ( u + v ) 2 dz ( u + v ) 2m dz. (1) The previous expressio of ξ(s,s) ca be decomposed ito three differet types of terms, where agles u, v, u ad v have idetical power umbers, 1 ad 2: ' 1) The ull-order term dz dz m is a costat PD, or pisto error that is usually compesated for by meas of a delay lie iserted withi the optical trai (ot show i Figure 2), hece it ca be eglected. 2) A first-order term with respect to u, v, u ad v is directly related to the geometry of the iput ad output apertures of the iterferometer. I the Fizeau case the golde rule of Eq. 5 applies, so that x = m x ad y ' ' ' = m y, therefore the relevat PD terms cacel. ut there remais a residual PD (u x v y ) dz F' demostratig a violatio of the golde rule ad cosequetly reducig the maximal FoV achievable by the iterferometer. This term will also be eglected, however, sice herei we are maily iterested i higher-order effects. 3) Fially, there exists secod-order terms proportioal to the logitudial pupil shifts ad to the square power of agles u, v, u ad v. This term is resposible for iterferece patters distortios, as will be illustrated by the followig umerical simulatios. The cosequeces of pure secod-order sub-pupil aberratio are illustrated i Figure 5 ad Figure 6 for the cases whe N = 2 ad N = 8 telescopes respectively. Numerical simulatios were carried out, based o the physical parameter values give i the secod row of Table 1. All the etrace sub-pupil shifts dz were set to zero, while the effects of exit pupil shifts beig equal to dz = mm (ideal case, o pupil aberratios), dz =.5 mm, ad dz =.1 mm were modeled successively. It must be oted that the two last cases correspod to equivalet displacemets of the collectig telescopes of 12.5 ad 25 m respectively. These simulatios are providig the followig outputs: + (P ) X (P) F dz 2 Flat wavefrot P 1 dz 2 P 2 F eam compressors Z Flat wavefrot P 1 dz 1 P 2 Z dz 1 Collectig telescopes Multi-axial combier (exit pupil plae) Focal plae Figure 4: Illustratio of sub-aperture axial shifts dz 1 ad dz 2 ear the etrace pupil plae P (left side), ad dz 1 ad dz 2 ear the exit pupil plae P (right side).

8 2.5 arcsec Achievable FoV 2.5 arcsec PSF at.6 arcsec PSF at.3 arcsec PSF at FoV cetre No pupil aberratio Aberrated pupil Case 1 Aberrated pupil Case 2 Figure 5: Illustratig the effects of secod-order sub-pupils aberratio for the case N = 2 telescopes. The sup-pupils arragemet (icludig phase-shifts) is show o the upper left corer. I the first row are two gray-scale maps of the achievable FoV for differet axial shifts dz. From the secod to the last rows are displayed the iterferometer PSFs at differet FoV locatios. The left colum correspods to the ideal case whe there is o pupil aberratio (it ca be see that the PSF shape is ivariat), to be compared with the cetral ad right colums where pupil aberratio is preset.

9 2.5 arcsec Achievable FoV 2.5 arcsec PSF at.6 arcsec PSF at.3 arcsec PSF at FoV cetre No pupil aberratio Aberrated pupil Case 1 Aberrated pupil Case 2 Figure 6: Same graphics presetatio tha i Figure 5 for the case N = 8 telescopes. o The PSFs of the iterferometer are readily obtaied by combiig Eqs. 2 ad 1. Their gray-scale maps are displayed i the three last rows of Figure 5 ad Figure 6 at differet FoV locatios. It is see that the frige patters are more ad more distorted as pupil aberratio is icreasig (i.e. straight friges becomig circular). Also of iterest is the fact that the PSF is o loger FoV-ivariat i presece of pupil aberratio, cofirmig that the golde rule of Fizeau iterferometers is ot respected due to secod-order PD terms.

10 o The maximal achievable FoV is estimated by combiig Eqs. 3 ad 1. It ca be visualized i the first rows of Figure 5 ad Figure 6 by the dark cetral spot, sice this is a ullig iterferometer. The agular radius of this dark area (turig ito a bright oe for imagig Fizeau iterferometers) is iversely proportioal to the amout of pupil aberratio, ad looks suitable to defiig more quatitative optical requiremets o the pupil reimagig quality of the whole system. e remarkable achievemet of the preseted formalism ad computatioal method is that o actual Fourier or Fresel trasform algorithms are required explicitly, i oppositio with other proposed methods [1]. Hece a appreciable gai i computig time ad accuracy shall result. Fially, it is worth metioig that: The here above aalytical developmet for pupil re-imagig imperfectios ca be applied with o difficulty to other types of iterferometers differig for the basic Fizeau layout. Further, Eqs. 1-3 are also well suited to itroducig variatios of the optical compressio factor m o ay type of iterferometer 1 (for the Fizeau scheme this implies other deviatios from its golde rule). Hece the whole set of Eqs. 1-3 ad 1 together provide us with a robust model eablig rapid ad efficiet evaluatio of iterferometer performace at system level. This approach is fully complemetary with optical desig methods ivolvig diverse geometrical aberratios such as described i Ref. [11]. 4 EYND CLASSICAL FIZEAU INTERFERMETER Two possible evolutios of ullig Fizeau iterferometers hutig for habitable extra-solar plaets i the future decades are briefly evoked below, both beig based o somehow deliberate violatios of the golde rule applicable to Fizeau iterferometers. They are the Axially Combied Iterferometer ( 4.1), ad the crossed-cubes ullig iterferometer ( 4.2). Some of their ullig ad pseudo-imagig properties are compared with those of classical Fizeau iterferometers i sub-sectio Axially Combied Iterferometer (ACI) The axially combied iterferometer may be cosidered as a special case of Michelso 2 iterferometer where all output sub-pupils are merged together (i.e. P = for 1 N i Eqs. 1-4). Give a iterferometer costituted of N separated telescopes, this coditio ca be realized by meas of a arragemet of 3N/2 cascaded beamsplitters such as represeted i Figure 7. A umber of ACIs have already bee desiged ad experimeted for imagig purpose [12-13], ad they ca readily be tured ito ullig istrumets by meas of APS devices as i racewell s origial cocept [6]. The specific bject-image relatioship of the ACI has bee demostrated ad discussed i Ref. [8], writig as: [ F( s) ( )] I( s) = PSFT ( s) s, (11) where the far-field frige fuctio F(s) has the same aalytical expressio tha i Eq. 6b. The latter relatioship is quite remarkable, sice it implies that the observed sky-object is masked by the FFF of the iterferometer array before diffractio from the sigle pupil of the telescope occurs. This is perhaps the fudametal reaso why the ACI desig is so appealig for ullig iterferometry, because it allows i priciple to cacel all the light origiatig from a bright cetral star, regardless of diffractio effects. Aother importat cosequece of Eq. 11 is that deep ullig should be feasible eve with imperfect optics (i.e. PSF T (s) differig from a ideal Airy distributio), provided that the defects of all telescopes ad relay optics are idetical alog the N iterferometer arms. I practice however, this coditio should still require the use of spatial or modal wavefrot filterig devices located at the image plae of the system, as was foresee for all the studied projects. 1 r of its combier focal legth F, should it be made of differet segmets. 2 For a discussio about the differeces betwee Fizeau ad Michelso iterferometers, see e. g. Ref. [9], 3.

11 From collectig telescopes APS 1 APS 2 Metrology beam 1 Metrology beam 2 Axial beam combier Frige tracker (P ) F Z Focal plae Figure 7: Schematic optical layout of a axially combied iterferometer (from Ref. [9]). 4.2 Crossed-cubes ullig iterferometer A itermediate cofiguratio betwee the ACI ad the usual Fizeau layout cosists i deliberately breakig the golde rule. I that case Eq. 5 is ot valid ad P = m P with m m as for the historical Michelso s 2 iterferometer o Mout Wilso [3] that had two desified exit sub-pupils. A good example for such cofiguratios is the Crossed-cubes ullig iterferometer (CCN) that is the subject of aother commuicatio i this coferece [14], to which the reader is ivited to refer. e remarkable property of the CCN is that it authorizes pre-determied violatios of the homothetic rule, sice its etrace ad exit baselies ad are liked together by the relatio (see Figure 8): ( 1) 2 ( 1 taθ) = A ( λ) ' = A, (3) where A is the hypoteuse of the cube,θ the refracted agle, ad (λ) the refractive idex of the cube material. I particular, A ca be sized so as to achieve maximal desificatio of the output beams, as illustrated o the right side of the Figure. θ θ A θ A θ Figure 8: Adjustmet of the CCN exit baselie as fuctio of the etrace baselie ad cube parameters A ad θ.

12 4.3 Imagig properties of ullig iterferometer I this sectio is fially provided a qualitative iterpretatio of the imagig properties of ullig iterferometers. It has bee metioed i sectios 3.1 ad 4.1 that the bject-image relatioships applicable to Fizeau ad axially combied iterferometers are defied by Eqs. 6a ad 11 respectively. ut it is also of iterest to cosider the itermediate case of ullig Michelso iterferometers (such as the CCN, see 4.2), for which oly Eq. 1a is applicable. The ullig properties of all those istrumets employig differet combiatio schemes are illustrated i Figure 9. Here is studied the case of a futuristic 24-telescope ullig iterferometer havig a checkerboard distributio of its phase-shifts ϕ = or as idicated o the pael (c). All computatios are carried out usig the geometrical parameters summarized i Table 1 at the same cetral wavelegth λ = 1 µm. We cosider successively a ullig ACI with a maximal etrace baselie = 2 m ad a ull exit baselie = mm, the the case whe the idividual exit sub-pupils have joiig edges, correspodig to the maximal achievable desificatio ( = 8 mm), ad fially a ullig Fizeau iterferometer obeyig to the golde rule ( = 4 mm). For each case is displayed a gray-scale map of the formed image of a fictitious astroomical scee, composed of a bright cetral star ad a off-axis compaio of 5 % relative lumiosity. Here we obviously aim at ullig the cetral star ad isolatig its compaio, but the results preseted o paels (d), (e) ad (f) of Figure 9 evidece clear differeces betwee the three cosidered combiig schemes: 1) ly the ullig ACI demostrates a full extictio of the cetral star, as illustrated o Figure 9-d. The produced image appears as a lumious halo fully origiatig from the compaio (ad cetered o it), elarged by the diffractio lobe of PSF T (s) as predicted by Eq ) Eve for maximal desificatio, the ullig Michelso iterferometer caot achieve deep extictio of the cetral star: here the brightest cetral lobe still origiates from the compaio, but parasitic images of the star are apparet at the FoV corers. However these replicas remai faiter tha the observed compaio (here by a factor of 56 %), ad it may be assumed that the searched plaet ca readily be isolated from them. 3) Fially, the image produced by the ullig, homothetic Fizeau iterferometer looks quite the same as would be observed with a costructive versio of it: here the mai differece betwee ullig ad imagig modes is that the astroomical scee has bee shifted agularly, the cetral star ad its compaio havig apparetly bee swapped without ay oticeable cotrast ehacemet. Hece it is cocluded that the ullig capacity of this istrumet has bee lost defiitively. 5 CNCLUSIN I this paper were reviewed some classical cocepts of multi-aperture, imagig ad ullig iterferometers i the perspective of a first-order Fourier optics formalism. Various topics were revisited ad discussed, such as the golde rule of iterferometry, maximal achievable Field of View, performace degradatio due to pupil aberratio, ad the actual imagig capacities of sparse-aperture ullig iterferometers. The bject-image relatioships applicable to the preseted optical systems have also bee illustrated with the help of umerical simulatios. The coclusios of this study are that the most suitable combiig schemes for ullig purpose seems to be the axially combied iterferometer or a CCN-like desig with joiig exit pupil edges. It has also bee cofirmed that respectig the classical golde rule of imagig iterferometry severely hampers the ullig capacity. Fially, this paper provides the reader with a set of quick computig tools, ot requirig ay Fourier or Fresel trasform, allowig fast ad accurate calculatio of the poitspread fuctio, field of view ad imagig capacity of these complex high agular resolutio systems; i presece of certai types of istrumetal defects.

13 2.5 arcsec 2.5 arcsec (a) (d) 2 m 2.5 arcsec (b) (e) 1 mm 2.5 arcsec (c) (f) Figure 9: Imagig properties of a 24-telescope ullig iterferometer equipped with differet combiig optics. A fake sky object made of oe cetral star ad its half-power compaio is show i pael (a). View (b) depicts the geometry of the iterferometer etrace pupil. I (c) are illustrated the checkerboard distributio of the phase-shifts ϕ ad the geometrical arragemet of the exit sub-pupils for the maximal desificatio case. Paels (d), (e) ad (f) show gray-scale maps of the obtaied images, respectively for the cases of axial combiatio, desified Michelso, ad homothetic Fizeau iterferometers.

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