PSF and field of view characteristics of imaging and nulling interferometers

Transcription

1 PSF ad field of view characteristics of imagig ad ullig iterferometers Fraçois Héault UMR 6525 CRS H. Fizeau US, CA, Aveue icolas Coperic, Grasse Frace ASTRACT I this commuicatio are preseted some complemets to a recet paper etitled Simple Fourier optics formalism for high agular resolutio systems ad ullig iterferometry [1], dealig with imagig ad ullig capacities of a few types of multi-aperture optical systems. Herei the characteristics of such systems i terms of Poit Spread Fuctio (PSF) ad Field of View (FoV) are derived from simple aalytical expressios that are further evaluated umerically for various cofiguratios. We cosider successively the geeral cases of Fizeau ad Michelso iterferometers, ad those of a moolithic pupil, ullig telescope, of a ullig, Sheared-Pupil Telescope (SPT), ad of a sparse aperture, Axially Combied Iterferometer (ACI). The aalytical formalism also allows establishig the exact bject-image relatioships applicable to ullig PSTs or ACIs that are plaed for future space missios searchig for habitable extra-solar plaets. Keywords: Fourier optics, Phased telescope array, ullig iterferometry, Achromatic phase shifter 1 ITRUCTI Sice the historical writigs of Fizeau [2], Stépha [3] ad Michelso [4], 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. We use to cosider today that the major differece betwee both cocepts relies o the fact that the former obeys a golde rule statig that the output pupil of the iterferometer must be a scaled replica of its etrace pupil [5-6], while the latter does ot. Some ew multi-aperture cocepts, however, have emerged durig the three last decades, such as spacebore, ifrared ullig iterferometers or telescopes dedicated to the search of extra-solar plaets [7-8] or visible hypertelescopes havig usurpassed imagig capacities [9]. I a recet paper [1] was described a simple first-order, Fourier optics formalism allowig to derive the basic bject-image relatioships of the here above systems as covolutio products suitable for fast ad accurate calculatio. The purpose of this commuicatio is to complete the previous paper, summarizig ad simplifyig agai the formalism i sectio 2, ad providig additioal examples i sectio 3: herei are i tur cosidered the geeral cases of Fizeau ad Michelso iterferometers, of a moolithic, ullig Sheared-Pupil Telescope (SPT), ad of a sparse aperture, Axially Combied Iterferometer (ACI). Moreover, some imagig properties of the ullig SPTs or ACIs evisaged for future space missios searchig for habitable extra-solar plaets are further addressed i sectio 4. 2 ASIC RELATISHIPS Here the formalism described i ref. [1] is briefly summarized (ad also slightly simplified) i sectio 2.1, before derivig some theoretical relatioships applicable to the Poit Spread Fuctio (PSF) ad maximal achievable Field of View (FoV) of geeral multi-aperture optical systems (sectios 2.2 ad 2.3).

2 2.1 Geeral formalism Let us cosider a optical system desiged either for high-agular resolutio imagig or ullig iterferometry purpose, beig composed of collectig apertures ad recombiig apertures. The four attached coordiate systems are depicted i Figure 1, although oly three of them will be used herei: a o-sky agular coordiates system (U,V), the etrace pupil referece frame (,X,Y,Z) where Z is the mai optical axis, ad the exit pupil referece frame (,X,Y,Z). Let us further assume that: 1) For all idices comprised betwee 1 ad, the th collectig aperture of ceter P is optically cojugated with its associated recombiig aperture of ceter P without ay pupilar aberratio. 2) All collectig apertures have a idetical diameter ad cosequetly all recombiig apertures share the same diameter. Practically, it meas that all collectig telescopes ad optical trais coveyig the beams from etrace to exit apertures are idetical, which is most ofte the case for the presetly built iterferometric facilities, either of Fizeau or Michelso types (see the geeric optical layout of Figure 2). V -sky agular coordiates Sky object u v U s P 2 P 3 s Y P 1 P 4 Etrace pupil plae X P 4 P 1 Y P 3 P 2 s s Exit pupil plae X F Y M etectio plae X Z M Figure 1: 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 ot used i this paper. Let us fially defie the followig parameters (bold characters deotig vectors): s s (s ) Ω, dω PSF T (s) k λ a ϕ m A uit vector of directio cosies (1,u,v) 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 (1,u,v ) 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 o-sky. For a uobstructed pupil of diameter, it would be equal to 2J 1 (ρ)/ρ 2, where ρ = k s /2 ad J 1 is the type-j essel fuctio at the first order The wave umber 2π/λ of the electro-magetic field, assumed to be moochromatic The wavelegth of the electro-magetic field The amplitude trasmissio factor of the th iterferometer arm A phase-shift itroduced alog the th iterferometer arm for ptical Path iffereces (P) compesatio or ullig purpose The optical compressio factor of the system, equal to m = / = F C /F where F ad F C respectively are the focal legth of the collectig telescope ad of the relay optics (see Figure 2)

3 P 1 P 2 (P) Telescope 1 F C Telescope 2 Relay optics 1 Relay optics 2 APS 1 APS 2 Metrology beam 1 Metrology beam 2 Combiig optics F (P ) Covergig optics ivergig optics Fold mirror eamsplitter Acromatic Phase Shifter Focal plae Z Figure 2: Schematic optical layout of a multi-aperture, high agular resolutio iterferometer. Hece accordig to Ref. [1] the expressio of the image I(s) formed by the multi-aperture optical system ad projected back o-sky writes i a first-order approximatio: I( s) = ( s ) PSF ( s - s T s Ω = 1 ) a [ φ ] exp[ ik ( s P s' P' / m) ] exp i dω. (1) This very geeral bject-image relatioship ca oly be reduced to covolutio products if certai coditios are fulfilled, which are extesively discussed i Ref. [1]. Herei the followig sectios oly deal with other cosequeces o the PSFs ad effective Field of View accessible by the whole system Poit Spread Fuctios We ca defie the geeralized PSF of the optical system as simply beig the weightig factor of the object brightess fuctio (s ) uder the bi-dimesioal itegral of Eq. (1) that is: PSF [ ] [ ( )] 2 iφ exp ik s P s' P' G (, s) PSFT ( s - s) a exp / = 1 s = m, (2)

4 ad 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 (i that case, otios of ptical Trasfer Fuctio ad Modulatio Trasfer Fuctio caot by applied i classical sese as discussed i Ref. [1]). It may also be oted that for the sake of illustratio, the latter expressio ca be modified i order to have the PSF always cetered o the optical axis of the system, whatever the real object positio s : PSF [ ] [ ] [ ( )] 2 iφ exp ik s'p' / m exp iks P ' P' G ( s, s ) PSFT ( s) a exp / = Maximal achievable Field of View = m. (3) We may ituitively defie a maximal achievable Field of View for the multi-aperture optical system as the image that would be formed uder the followig hypotheses: o o physical diaphragm of ay kid (stops, mirrors edge or cetral hole) is take ito accout. o The sky object is uiformly bright over a 2π-steradia solid agle, hece (s ) = 1. o The optical system is of ifiite size ad free from aberratios, thus diffractio ca be eglected ad PSF T (s) is assumed to be the irac distributio δ(s). Uder such assumptios the maximal achievable FoV is deduced from Eq. (1), whose itegral reduces to: = 1 [ ] [ ( )] 2 iφ exp ik s P ' P' FoV( s ) = a m. (4) exp / 3 APPLICATIS I this sectio are provided several examples of applicatios of the here above formulas. We first review the major differece betwee a Fizeau ( 3.1) ad Michelso ( 3.2) iterferometer, before studyig the cases of ullig telescopes ( 3.3) ad axially combied iterferometers ( 3.4). The maximal diameter of a idividual telescope is assumed to be 5 m ad the umerical values of F ad F C are respectively equal to F = 50 m ad F C = 10 mm, leadig to a optical compressio factor m = 1/500. All other used umerical parameters are summarized i Table 1 for the various cosidered cases. Computatios are carried out at a wavelegth λ = 10 µm, which is a typical figure for ullig iterferometry i the mid-ifrared bad. Table 1: umerical values of mai physical parameters for all simulated cases. Case umber of etrace pupils umber of exit pupils (m) (m) (mm) (mm) Maximal FoV (arcsec) 1 Sectio Fizeau iterferometer Ifiite 3.1 Michelso iterferometer SRT Sheared-Pupil Telescope Masked SPT Masked SPT ACI Computed usig approximate relatio (9).

5 3.1 Fizeau iterferometer A geeric optical scheme of stellar iterferometer is show i Figure 2, either of the Fizeau or Michelso types. Amog may other defiitios utilized by differet authors, let us retai the followig, which is illustrated i Figure 3: Iput sub-pupils Y utput sub-pupils Y Fizeau iterferometer X X Y Y Michelso iterferometer X X Figure 3: Iput ad output pupils cofiguratios for Fizeau (top) ad Michelso iterferometers (bottom). o 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 ) (5) also beig kow as the Pupil i = Pupil out coditio [5] or golde rule of iterferometry [6]. o I oppositio the Michelso iterferometer does ot respect this golde rule: we the may have P = m P. with ay value of m = / differig from m, or eve o homothetic relatioship betwee etrace ad exit apertures at all (e.g. exit sub-apertures are ofte aliged alog a sigle axis eve i the cases of o-liear iterferometric arrays). Alteratively, oe may talk about a pupil desificatio process. The historical Michelso s 20 feet iterferometer o Mout Wilso was perhaps the first example of desified optical system. I the case of Fizeau iterferometers, isertig Eq. (5) ito Eq. (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: [ PSF ( s) F( s) ] ( ) I( s) = T s, (6a) where fuctio: [ ] [ ] 2 s) a exp φ exp i k sp F( = = 1 i (6b) is amed Far-field Frige Fuctio (FFF) i Ref. [1] because it describes the iterferece patter that would be observed i the image plae if all sub-pupils were reduced to a pihole 1. It follows from Eq. (6a) that Fizeau 1 It may be oticed that this FFF defiitio is somewhat related to the dirty map used by Högbom ito his CLEA algorithm [10].

6 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. ow isertig coditio (5) ito Eqs. (3) ad (4) allows retrievig two basic properties of Fizeau iterferometers: 1) The geeralized PSF of the optical system PSF G (s,s ) does ot deped ay loger o vector s, hece the PSF is ivariat over the whole iterferometer Field of View. 2) The maximal achievable FoV of the Fizeau iterferometer ca be expressed as the very simple relatioship: [ ] 2 FoV( s ) = a exp i, (7) φ = 1 implyig that the FoV is uiform, takig a costat value that oly depeds o the amplitude trasmissio factors a ad phase-shifts ϕ. For a classical imagig iterferometer beig perfectly cophased (ϕ = 0. whatever is ), FoV(s) is uiformly equal to 1, which is i agreemet with the golde rule of iterferometry [5-6]. However this golde rule may be of dramatic cosequece whe ullig, Fizeau-type iterferometers are cosidered, sice the phase-shifts ϕ must be chose so as to cacel the light origiatig from the cetral star: this has the cosequece that all PSFs are ivariat ad equal to zero at their theoretical ceters, hece ay bright or fait puctual object (icludig plaets) located aywhere i the FoV should be ulled as well as the paret star, ad therefore ot detectable (i that case, the priciple of eergy coservatio suggests that most of the optical power will be reflected to the metrology sesors represeted i Figure 2). To illustrate this poit, let us cosider the case of a eight-telescopes Fizeau iterferometer i both imagig ad ullig modes, followig the squared arragemet sketched o the left pael of Figure 4 ad usig the umerical parameters of Table 1. It ca be verified that the PSFs are effectively ivariat i the whole achievable FoV, ad that the latter is uiformly bright i imagig mode ad dark i ullig mode. It may the be cocluded that searchig for extra-solar plaets requires a deliberate violatio of the famous Pupil i = Pupil out coditio. Iput pupils Y PSF (FoV ceter) PSF (half FoV) Maximal achievable FoV X 5 arcsec Y 0 π 0 π π X 0 π 0 5 arcsec Figure 4: Ivariat PSFs ad achievable FoV of a eight-telescopes Fizeau iterferometer i both imagig (first row) ad ullig versios (bottom row). Achromatic phase-shifts ϕ for the ullig versio are idicated o left bottom pael.

7 3.2 Michelso iterferometer As metioed i the previous subsectio, the essetial characteristic of a Michelso iterferometer is that the golde rule is o loger beig respected. If the overall geometry of the telescope array is preserved, oe may write however: P = m P., with: m = / m, (1 ) (8) The iequality betwee m ad m has a kow cosequece o the maximal achievable FoV that will here be cosidered as a saity check of the whole preseted formalism. Accordig to Tallo-osc ad Tallo [11] ad usig the here employed otatios, the maximal polychromatic FoV of a Michelso iterferometer is approximately: FoV 2 λ δλ 1 ( 1 m' m), (9) where δλ is the effective spectral badwidth ad the maximal baselie betwee ay couple of telescopes. It must be oticed that as expected this maximal FoV becomes ifiite (at least theoretically) for a Fizeau iterferometer where by defiitio m = m. I the case of co-axial recombiatio (m = 0, see 3.4), Eq. (9) is also i agreemet with Haiff s FoV defiitio beig equal to the product of the spatial ad spectral resolutios λ 2 / δλ [12]. ut it is also possible to evaluate the polychromatic FoV by direct umerical itegratio of Eq. (4), i.e.: FoV δλ ( s ) = FoV ( s) (λ) dλ (λ) dλ, (10) δλ λ with fuctio δλ (λ) stadig for the eergetic profile of the selected spectral bad 1. ow cosiderig a simple twotelescopes Michelso iterferometer i imagig ad ullig cofiguratios, whose umerical parameters are provided i the secod row of Table 1 ad a spectral badwidth δλ = 5 µm, we ca see i Figure 5 a good agreemet betwee the theoretical ad umerical evaluatios of the maximal achievable FoV. The major iterest of Eq. (10) is that it ca be accurately ad quickly evaluated whatever the total umber of telescopes ad geometrical arragemet ad phaseshifts of their etrace ad exit pupils. δλ δλ δλ PSF (FoV ceter) PSF (half FoV) Polychromatic FoV 5 arcsec Figure 5: Variable PSFs ad achievable FoV of a two-telescopes Michelso iterferometer i imagig (first row) ad ullig modes (bottom row). ashed lies o the right paels idicate FoV limits. 1 The same kid of umerical itegratio could also be performed o Eqs (2-3), allowig oe to evaluate polychromatic PSFs varyig i the whole iterferometer FoV.

8 3.3 ullig telescopes Several desigs of ullig telescopes have bee proposed by differet authors i the past years [8] [13-14]. The basic idea is to adapt some of the best plaed techologies for sparse apertures ullig iterferometry to the case of a sigle space telescope of moderate diameter, i order to characterize the exo-zodiacal clouds of a umber of target extra-solar systems as well as their Jupiter-like plaets, o the oe had, ad to validate the evisaged data reductio ad pseudoimagig processes, o the other had. elow are reviewed some of these cocepts i light of the preseted formalism Moolithic ullig telescope A very simple idea to tur a afocal, moolithic telescope ito a ullig istrumet could be to direct the output compressed beam to a Mach-Zehder Iterferometer (MZI) as depicted i the left pael of Figure 6, where a couple of Achromatic Phase Shifters (APS) made of wedged dispersive plates are itroducig a achromatic, half-wave P differece betwee both iterferometer arms (ϕ - ϕ = π). This techique should fail, however, whe the telescope iput pupil (P) is optically cojugated with the MZI output pupil (P ), sice the whole system fially obeys to the golde rule of Fizeau iterferometers, ad the destructive iterferece patter should the spread out over the whole FoV as discussed i 3.1. Hece alterative cocepts have to be foud. From moolithic telescope From moolithic telescope APS 1 APS 2 APS 1 APS 2 ad P compesatio Metrology beam Metrology beams (P ) (P ) F F Z Focal plae Z Focal plae Figure 6: Recombiatio schemes of a moolithic ullig telescope (left) ad of a Sheared-Pupil Telescope (right) Super-Resolvig Telescope (SRT) A tetative cocept of super-resolvig telescope (SRT) with multiple exit apertures has bee previously described i Refs. [1] ad [14]. It cosists i oe sigle afocal collectig telescope, optically feedig a umber of off-axis, parallel exit beams by meas of cascaded beamsplitters. The beams are further expaded ad multi-axially recombied i the image plae, thus eablig PSF cores to be much thier tha those geerated by the moolithic telescope itself. Although this system actually overcomes Rayleigh s diffractio limit ad Abbe s sie law, it has bee demostrated that it evertheless does ot provide ay real super-resolutio capacity, sice its basic bject-image relatioship writes [1]: [ PSF ( s) ( )] =, with far-field frige fuctio: [ ] [ ] 2 I( s) F( s) T s F( s ) = a exp iφ exp - i k s'p' / m. (11) Hece all high spatial frequecy iformatio of the object (s) has already bee filtered out by the moolithic telescope aperture before the icidet optical power is fially cocetrated o the thied PSF cores. This is illustrated by the umerical simulatios of Figure 8 showig typical PSF ad FoV characteristics of a SRT, computed from relatioships (2), (4) ad (10) with the umerical parameters provided i Table 1 (the distributio of phase-shifts ϕ i the exit aperture plae is the same as depicted i the bottom left pael of Figure 4). It ca be oted that the PSFs are composed of = 1

9 a group of sharp diffractio lobes cofied ito a circular agular area correspodig to the telescope PSF, ad that the peaks amplitudes are varyig over the whole FoV. The moochromatic FoV itself appears as a ifiite regular grid of thi bright spots, which reduces to the four cetral oes o a spectral badwidth δλ/λ = 50%. We coclude that the worst drawback of the SRT probably cosists i such a extremely restricted FoV Sheared-Pupil Telescope (SPT) The Sheared-Pupil Telescope (SPT) cocept is ideed aother example of desified optical system. It may basically be uderstood as a variat of the moolithic ullig telescope preseted i sectio 3.3.1, where the output arms of the MZI are laterally shifted oe with respect to the other 1 as show o the right pael of Figure 6. Geeralizig this priciple to ay telescope umber the geerates a set of sheared sub-apertures i the output pupil plae that ca iterfere together (see Figure 7). Here two mai types of SPT ca be distiguished, depedig o the presece of a Lyot stop i the output pupil plae (P ): curret sectio deals with the case of a umasked SPT (o Lyot stop i the exit pupil plae) while the case of a masked SPT (icludig the Lyot stop) is cosidered i sectio Iput pupil plae utput pupil plae Umasked output sub-pupils Y Y X Moolithic pupil telescope Y Y X Masked output sub-pupils X X Figure 7: Two possible etrace ad exit pupil arragemets for the Sheared-Pupil Telescope ( = 4). PSF ad FoV behaviors for the umasked SPT are illustrated o the secod row of Figure 8 i the case of a ullig Agel cross cofiguratio ( = 4, ϕ 1 = 0, ϕ 2 = π, ϕ 3 = 0, ϕ 4 = π). Here agai the PSF is varyig over the whole FoV, ad a oticeable star leakage appears at the FoV cetre. The accessible moochromatic ad polychromatic FoVs ow look much more exteded tha those provided by the desified SRT. It must be highlighted, however, that the both systems are closely related, sice they are govered by the same fudametal bject-image relatioship that is Eq. (11) Masked SPT With respect to the previous desig, the masked SPT icorporates a additioal Lyot stop placed at the exit pupil plae (P ) ad vigettig each sub-pupil so that they are all reduced to a commo circular area of diameter as depicted o the bottom pael of Figure 7. This has the effect of selectig delimited, sheared sub-areas of diameter o the etrace pupil of the moolithic telescope (usually its primary mirror) ad makig them iterfere together i the image plae. For this reaso the imagig properties of masked SPTs will be similar to those of the Axially Combied Iterferometer (ACI) preseted i the ext sectio. Imagig properties of both masked SPT ad ACI cocepts are further addressed i sectio 4. 1 This lateral shift ca be itroduced usig other iterferometer types, e.g. a Michelso equipped with cube-corers as i Ref. [8].

10 Moochromatic PSF (FoV ceter) Moochromatic PSF (1/4 FoV) Moochromatic FoV Polychromatic FoV ACI = 20 m Masked SPT = 1 m Masked SPT = 0.5 m Sheared-Pupil Telescope Super-Resolvig Telescope 5 arcsec Figure 8: PSF ad field of view characteristics of super-resolvig telescope (first row), sheared-pupil telescopes (middle rows) ad the axially combied iterferometer (bottom row).

11 The third ad fourth rows i Figure 8 display the variable PSFs ad achievable moochromatic ad polychromatic FoVs of a ullig, masked SPT costituted of eight etrace sup-pupils with the same phase-shifts as for the SRT case. The two major differeces with respect to the umasked SPT are as follows: o stellar leakage is apparet at the FoV cetre, which is a much more favorable coditio to reach a deep ullig of the cetral star. Ufortuately, the latter advatage is somewhat couterbalaced by the exteded area of the ulled FoV where ay object (icludig plaets) will suffer from a low throughput. As a example, yellow circles i Figure 8 are materializig ier regios where throughput is always iferior to 0.25, correspodig to darkeed areas of 4 ad 1.5 arcsec for baselies equal to 0.5 ad 1 m respectively. ly the secod case seems acceptable for plaet searchig, however it implies that the useful sectio of the collectig telescope is reduced to 3 m (see Table 1). Hece good sky coverage seems oly attaiable at the price of some losses i radiometric efficiecy. e may thus expect masked SPTs to be preferred i terms of rejectio rate, ad umasked SPTs for what cocers radiometric performace (i tur improvig sigal-to-oise ratio ad itegratio time). The poit is further discussed i sectio 4, after havig examied oe last, classical multi-aperture optical system that is the Axially Combied Iterferometer (ACI). 3.4 Axially Combied Iterferometer (ACI) The axially combied iterferometer may be cosidered as a special case of Michelso iterferometer where all output sub-pupils are merged together (i.e. m = 0 followig the herei preseted formalism). Give a iterferometer costituted of separated telescopes, this coditio ca be realized by meas of a arragemet of beamsplitters such as represeted i Figure 9. A umber of ACIs has already bee desiged ad built for imagig purpose [15-16], ad they ca readily be tured ito ullig istrumets by meas of a APS device as i racewell s origial cocept [7]. The specific bject-image relatioship of the ACI has bee demostrated ad extesively discussed i Ref. [1], writig: = PSF ( s) [ F( s) ( )], with far-field frige fuctio: [ ] [ ] 2 I( s) T s F( s) = a exp φ exp i k sp = 1 i (12) It must be emphasized that due to the presece of the Lyot stop, Eq. (12) is also fully applicable to the masked SPT, hece both types of system share the same imagig properties that are further discussed i sectio 4. PSF ad field of view characteristics of a ullig ACI are illustrated o the bottom row of Figure 8, showig total extictio at the FoV cetre, high throughput o the costructive iterferece peaks, ad a very thi polychromatic agular FoV varyig as the iverse of the telescope baselie as predicted by relatio (9). 4 IMAGIG PRPERTIES F ULLIG SPT A ACI I this sectio are fially provided some prelimiary ad qualitative iterpretatios of the imagig properties of ullig SPTs ad ACIs. It has bee metioed i sectios 3.3 ad 3.4 that bject-image relatioships applicable to the umasked SPT, o the oe had, ad to both masked SPTs ad ACIs, o the other had, are defied by Eqs. (11) ad (12) respectively. The latter relatioship is very remarkable, sice it implies that the observed sky-object is masked by the FFF of the iterferometric array before diffractio from the sigle pupil of the telescope occurs. This is perhaps the fudametal reaso why the masked SPT ad ACI desigs are so appropriate for ullig, because they allow i priciple to cacel all the light origiatig from a bright cetral star regardless of diffractio effects. Aother importat cosequece of Eq. (12) 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 iterferometer arms. I practice however, this coditio should still require the use of spatial or modal wavefrot error filterig devices i the image plae of the iterferometer, as is foresee for all curret projects.

12 The imagig properties of ullig SPTs ad ACIs are illustrated i Figure 10 for various types ad cofiguratios summarized i Table 1. We cosider successively the umasked SPT, a masked SPT with etrace baselies = 0.5 m ad = 1 m, ad a ullig ACI with = 10 m ad = 20 m. All computatios are performed with λ = 10 µm ad a maximal diameter of the idividual telescopes equal to 5-m. For each case the moochromatic FoV of the istrumet is displayed as a gray-scale map (yellow circle idicatig cetral areas where plaet throughput remais lower tha 0.25) ad 3 view. Rightmost colums reproduce images formed by all cosidered systems from a fake sky object whose brightess distributio is show o the left bottom pael of Figure 10 (here we aim at ullig the cetral area ad isolatig the outermost bright star). From ullig SPTs dow to ACIs, the followig treds have bee be oted: 1) The ullig umasked SPT presets the advatages of a high throughput ad best cocetratio of eergy origiatig from the off-axis star, however there remais a residual leakage at the 5 % level from cetral objects. 2) The masked SPT provides full extictio of the cetral area, but its radiometric efficiecy is foud to be very low (< 1 %) for small baselies ( = 0.5 m), which cofirms the coclusios of Loger baselies should be preferred despite of the decreased sub-pupils area ad loss i radiometric efficiecy. 3) The ullig ACI behaves as a masked SPT with high throughput advatage for very short baselies (e.g. = 10 m where the idividual telescope pupils are coected side by side). I this case the ulled FoV is sigificatly arrower tha the SPT FoV ad the cetral star has bee fully caceled, but a 40 % leakage from the cetral rig is preset. For much loger baselies ( > 20 m), the FoV becomes so small that the ullig capacity of the istrumet is defiitively lost ad the direct image is the same as would be observed with a idividual, moolithic telescope of 5-m diameter. P 1 P 2 (P) Telescope 1 Telescope 2 Relay optics 1 Relay optics 2 APS 1 APS 2 Metrology beam 1 Metrology beam 2 Axial beam combier Frige tracker (P ) F Z Focal plae Figure 9: Schematic optical layout of a axially combied iterferometer.

13 Moochromatic FoV Sky image Sky object ACI ACI = 20 m = 10 m 2 arcsec Masked SPT = 1 m Masked SPT = 0.5 m Sheared-Pupil Telescope Figure 10: Imagig properties of ullig SPTs ad ACIs. First row: ullig, umasked SPT. Secod ad third rows: ullig masked SPT with = 0.5 m ad = 1 m. Fourth ad fifth rows: ullig ACI with = 10 m ad = 20 m. Left colums: gray-scale maps ad 3 views of moochromatic FoV. Right colums: gray-scale maps ad 3 views of image projected back o-sky. Last row is showig from left to right: origial sky object, PSF of a idividual telescope, ad image of the sky object see through the idividual telescope (gray-scale maps ad 3 view). ote that the latter have bee ormalized to uity regardless of the actual throughput.

14 I view of the previous results, it might be expected that the best desigs would either be a masked SPT with reduced etrace sub-pupils or a umasked SPT associated with a robust leakage calibratio procedure, depedig whether deep ullig or high radiometric efficiecy is to be favored. However a rigorous tradeoff betwee both cocepts should obviously ivolve much more criteria such as those summarized i sectio 6 of Ref. [1], ad is clearly beyod the scope of the preset commuicatio. 5 CCLUSI I this commuicatio were reviewed some classical cocepts of multi-aperture, imagig or ullig iterferometers i the perspective of a first-order Fourier optics formalism that has bee recetly preseted i Ref. [1]. Various topics such as fudametal differece betwee Fizeau ad Michelso iterferometers (the golde rule of iterferometry), maximal achievable Field of View i moochromatic ad polychromatic light, or imagig capacities of ullig, moolithic telescopes ad axially combied iterferometers were revisited. bject-image relatioships applicable to all the preseted optical systems have bee provided, discussed ad illustrated with the help of umerical simulatios. It must be emphasized that the mai goal of this paper is ot to select the most promisig optical architecture, but simply to provide the reader with a set of quick computig tools, allowig fast ad accurate calculatio of the poit-spread fuctios, field of view, ad simulated images formed by these complex high agular resolutio systems. REFERECES [1] F. Héault, Simple Fourier optics formalism for high agular resolutio systems ad ullig iterferometry, JSA A vol. 27, p (2010). [2] H. Fizeau, Rapport sur le cocours du Prix ordi de l aée 1867, Comptes Redus des Séaces de l Académie des Scieces vol. 66, p (1868). [3] E. Stépha, Sur l extrême petitesse du diamètre apparet des étoiles fixes, Comptes Redus des Séaces de l Académie des Scieces vol. 78, p (1874). [4] A. A. Michelso, F. G. Pease, Measuremet of the diameter of alpha riois with the iterferometer, Astrophys. J. vol. 53, p (1921). [5] J. M. eckers, Field of view cosideratios for telescope arrays, Proceedigs of the SPIE vol. 628, p (1986). [6] W. A. Traub, Combiig beams from separated telescopes, Applied ptics vol. 25, p (1986). [7] R.. racewell ad R. H. MacPhie, Searchig for o solar plaets, Icarus vol. 38, p (1979). [8]. Meesso, M. Shao,. M. Levie, J. K. Wallace,. T. Liu, E. Seraby, S. C. Uwi, C. A. eichma, ptical plaet discoverer: wow to tur a 1.5m telescope ito a powerful exo-plaetary systems imager, Proceedigs of the SPIE vol. 4860, p (2003). [9] A. Labeyrie, Resolved imagig of extra-solar plaets with future km optical iterferometric arrays, Astroomy ad Astrophysics Supplemet Series vol. 118, p (1996). [10] J. A. Högbom, Aperture sythesis with a o-regular distributio of iterferometer baselies, Astroomy ad Astrophysics Supplemet vol. 15, p (1974). [11] I. Tallo-osc, M. Tallo, Imagig a exteded object with a Michelso iterferometer, i High Resolutio Imagig by Iterferometry II, Proc. ES Cof., p (1991). [12] C. Haiff, A itroductio to the theory of iterferometry, ew Astroomy Reviews vol. 51, p (2007). [13] E. Seraby,. Meesso, Accessig small ier workig agles with a rotatig sub-aperture uller, Proceedigs of IAU Colloquium 200, p (2005). [14] F. Héault, Fibered ullig telescope for extra-solar coroagraphy, ptics Letters vol. 34, p (2009). [15] J. E. aldwi, R. C. oyse, G. Cox, C. A. Haiff, J. Rogers, P. J. Warer,. M. A. Wilso, C.. Mackay, esig ad performace of CAST, Proceedigs of the SPIE vol. 2200, p (1994). [16] S. A. Riehart, T. Armstrog,. J. Frey, J. Kirk,. T. Leisawitz,.. Levito, L. Lobsiger, R. Lyo, A. J. Martio, T. Pauls, L. G. Mudy, E. Sears, The Wide-Field Imagig Iterferometry Testbed I: Progress, results, ad future plas, Proceedigs of the SPIE vol. 5491, p (2004).