Analysis of azimuthal phase mask coronagraphs

Transcription

1 Aalysis of azimuthal phase mask coroagraphs Fraçois Héault Istitut de Plaétologie et d Astrophysique de Greoble Uiversité Greoble-Alpes, Cetre Natioal de la Recherche Scietifique B.P. 53, 384 Greoble Frace ABSTRACT I this paper is preseted a aalytical study of the azimuthal phase-mask coroagraph curretly evisioed for detectig ad characterizig extra-solar plaets. Special emphasis is put o the physical ad geometrical iterpretatio of the mathematical developmet. Two ecessary coditios are defied for achievig full extictio i the pupil plae of the coroagraph, statig that the complex amplitude geerated by the phase mask should have zero average, o the oe had, ad its Fourier coefficiets should oly be eve, o the other had. Examples of such phase fuctios are reviewed, icludig optical vortices, four-quadrat phase masks, ad azimuthal cosie phase fuctios. Hits for buildig more sophisticated fuctios are also give. Fially, a simplified expressio of light leaks due to mask imperfectio is proposed. Keywords: Coroagraphy, Phase mask coroagraph, Fourier optics INTRODUCTION With the recet commissioig of groud istrumets such as SPHERE [] or GPI [] ad future space observatories like WFIRST-AFTA [3], coroagraphy should probably become a efficiet tool for characterizig the atmospheres of extra-solar plaets i the forthcomig years, especially whe operated from space. Coroagraphic istrumets are themselves divided ito three or four families, oe of which beig Phase-mask coroagraphs (PMC) whose priciple is depicted i Figure. It basically cosists i settig trasparet phase plates at the telescope focal plae or aother optically cojugated plae. Such plates are spreadig the optical beam oto a secodary pupil image, where a aperture stop (amed Lyot stop) blocks the diffracted starlight. Very deep extictio ratios ca the be achieved i the fial image plae, revealig the presece of fait plaets orbitig aroud the cetral star. Historically, the first phase fuctio proposed by Roddier ad Roddier was axis symmetric [4], but could ot achieve perfect extictio over the full Lyot stop without apodizig the etrace pupil [5]. Nowadays the most popular phase masks are the Four-quadrat phase mask (FQPM) ad vortex phase mask (VPM), whose phase fuctios oly deped o a azimuthal coordiate θ. Extesive literature has already bee published o their fudametal properties [6-9], maufacturig techologies [- ], ad experimetal ad observatioal achievemets [-3]. I particular, it has bee demostrated by usig pure mathematical aalysis [7-9, 4-5] that they produce full cacellatio of the electric field over the whole surface of the Lyot stop, which is the basic requiremet for this type of istrumets. I spite of their rigor ad clarity, these demostratios essetially based o Hakel trasforms may suffer from a lack of physical or geometrical iterpretatio. Moreover, they are essetially focused at the FQPM or VPM coroagraphs, though it ca be cojectured that they are applicable to other types of azimuthal phase masks. The mai goal of this paper is to exted the aalysis to other phase masks ad to provide egieers with practical requiremets for achievig deep extictios with PMCs. This is realized i two steps: firstly, a alterative mathematical developmet of complex amplitudes formed ito the Lyot stop plae is described i sectio, ad allows defiig two ecessary coditios for a azimuthal phase mask to achieve full extictio. Examples of applicatios are the give i sectio 3, icludig a review of various azimuthal phase fuctios ( 3.), hits for buildig more sophisticated fuctios ( 3.), ad a simplified expressio of the light leaks iside Lyot stop due to mask imperfectio, suitable to system desig ad toleracig ( 3.3). A brief coclusio is draw i sectio 4. The most laborious mathematical developmets are give i a appedix.

2 MATHEMATICAL DEVELOPMENT. Basic hypotheses ad coordiate systems It is assumed i this study that the telescope pupil is illumiated with a moochromatic referece wavefrot of waveumber k = π/λ with λ the wavelegth of the electromagetic field. Therefore the coditios of full extictio are oly valid for that sigle wavelegth. Moreover the followig hypotheses are made: - The iput wavefrot is flat (i.e. origiatig from a poit, uresolved source) ad does ot exhibit phase aberratios. - It is spatially uiform over a circular pupil with o cetral obscuratio, or secodary mirror support legs. - The phase mask is assumed to have ifiite width. - The aalytical study is restricted to itesity distributios formed by the phase mask iside the Lyot stop plae. Other importat performace metrics attached to the fial coroagraph image plae, such as Ier workig agles (IWA) or throughput (radial trasmissio curve of a off-axis plaet) are ot cosidered i the preset paper. For the sake of simplicity, the magificatio factor betwee the iput pupil ad Lyot stop plaes is take equal to uity, ad all focusig ad collimatig optics have a idetical focal legth deoted F. The employed coordiate systems are idicated i Figure ad cosist i three referece frames R(OXYZ), R (O X Y Z) ad R (O X Y Z), where Z is the optical axis of the system: - R is attached to the exit pupil plae of the telescope OXY, with O the pupil cetre. Poits P i that plae are deoted either by their Cartesia coordiates (x,y) or by their polar coordiates (ρ,θ). - R is attached to the telescope image plae O X Y where the phase mask is located ad cetred at poit O alog the optical axis. Poits P i the O X Y plae are deoted by Cartesia coordiates (x,y ) or polar coordiates (ρ,θ ). - R is attached to the Lyot stop plae O X Y, with O the stop cetre ad poits P are deoted by Cartesia coordiates (x,y ) or polar coordiates (ρ,θ ).. Fourier optics aalytic developmet The theoretical frame of this study is restricted to first-order Gaussia optics ad Frauhofer scalar diffractio. I that case, the diffracted complex amplitude at ay poit M located i the image plae O X Y is proportioal to the Fourier trasform of the pupil trasmissio fuctio, which writes i Cartesia coordiates: + + ( x y ) = K B ( x, y) exp[ iπ ( xx + yy ) F] A, D λ dxdy, () where K is a ormalizatio factor ad the pupil trasmissio fuctio is a circular pillbox fuctio of radius R ad diameter D defied as: B D (x,y) = whe ρ = x + y R = D/, ad: B D (x,y) = whe ρ > R. From ow, we shall ot use Cartesia coordiates ay loger ad express Eq. i polar coordiates: A π + ( ρ θ ) = K B ( ρ, θ ) exp[ iπρρ cos( θ θ ) λf ], ρdρ. () D

3 For a circular pupil the diffracted complex amplitude is axis symmetric ad does ot deped o θ. Eq. ca thus be rewritte as: + π ( ρ ) = K ( ρ)exp[ π ρρ cosθ λ ] θ ρ ρ = π ˆ B ( ρ D i F d d R B D ) A, (3a) where the fuctio ( ) B ˆ D ρ is defied as: J ( πrρ λf ) B D ( ρ ) = πr ρ λf ˆ, (3b) ˆ ρ ad J is the type-j Bessel fuctio at the first order. Classically the square modulus of B D( ) is the well-kow Airy fuctio ad the poit-spread fuctio (PSF) of the telescope. Iside the coroagraph, the complex amplitude, A ρ with the complex trasmissio A ( ρ θ ) formed at the Lyot stop plae O X Y is obtaied by multiplyig ( ) of the phase mask i the image plae, the takig the iverse Fourier trasform of the result. Deotig φ ( ρ θ ) phase mask fuctio ad assumig ifiite width, it yields: A π + ( ρ θ ) = K Bˆ ( ρ )exp[ φ ( ρ, θ )] exp[ π ρ ρ cos( θ θ D i i ) λf], the, ρ dρ. (4) where K is the ormalizatio factor. Sice our study is limited to the case of azimuthal phase mask coroagraphs, the phase mask fuctio oly depeds o the agular coordiate θ. Isertig Eq. 3b ito Eq. 4, chagig the itegratio variable ρ to ν with ν = ρ λf, ad adjustig K so that A ( O ) = i the absece of phase mask (i.e. φ ( θ ) = ) fially gives a itegral expressio of the complex amplitude distributio i the Lyot stop plae: A Settig codesed form: π + ( ρ θ ) = R exp[ iφ ( θ )] J ( πrν ) [ iπνρ ( θ θ )] dν exp cos,. (5) β = θ θ i Eq. 5 ad defiig the reduced coordiate a ρ cos β R A π θ ( ρ θ ) = R exp[ iφ ( β + θ )] F( a) dβ with fuctio F(a) beig defied as: ( a) = J ( πrν ) exp[ iπraν ] θ = allows rewritig it i a more,, (6a) + F dν. (6b) The geometrical iterpretatio of the parameter a i Eqs. 6 is illustrated i Figure 3: at a give poit P of polar A, is the azimuthal itegral of the complex coordiates (ρ,θ ) i the Lyot stop plae, the complex amplitude ( ρ θ ) trasmissio of the phase mask weighted by fuctio F(a). The coditio a = ρ cos β R whatever the agle β i a [, π] iterval defies a disk of radius R that correspods to the telescope pupil. As demostrated ito the Appedix, the aalytical expressio of the weightig fuctio F(a) is foud to be: F F ( a) = + i πr a ( a) = πr a a a for a <, ad: (7a) for a >. Isertig Eqs. 7 ito 6a gives a expressio of the complex amplitude i the Lyot stop plae as fuctio of the origial variables (ρ,θ ): (7b)

4 A ad: ( ) ( ) ρ cos θ θ ρ, θ = + i exp[ iφ ( θ )] for ( θ θ ) R π ( ) θ,π R ρ cos θ θ A [ ] a ρ cos, (8a) ( ) ( ) ρ cos θ θ ρ, θ = exp[ iφ ( θ )] for ( θ θ ) > R π ρ cos ( θ θ ) R a > θ [,π ] ρ cos. (8b) The previous equatios are used i the two ext sub-sectios i order to evaluate the itesity distributio ( ρ, θ ) = ( ρ, θ ) A I i the whole Lyot stop plae. They are illustrated geometrically i Figure 4 where the parameter a = ρ cos β R correspods to the ratio O H R. For ay poit P i the Lyot stop plae, the poit H will ru two full laps aroud the red dashed circles of the Figure whe β rages from to π. Depedig o the locatio of P those circles may itersect the cotour of the telescope pupil or ot, hece two cases are distiguished: - Whe poit P is located iside the image of the telescope pupil (Figure 4-), the expressio of the itesity I ρ, θ oly ivolves Eq. 8a ad is established i.3. ( ) - Whe it is located outside of the pupil image (Figure 4-) both Eqs.8 must be used. This case is treated i.4. Aother illustratio of the complex fuctio F(a) is give i image Figure 5, where its real ad imagiary parts are plotted as fuctio of the agle β for differet values of the ρ R ratio..3 Itesity distributio iside pupil image For all poits P of polar coordiates (ρ,θ ) located iside the image of the telescope pupil, the coditio ρ cos β R holds ad the complex amplitude i the Lyot stop plae is described by Eq. 8a. Usig agai the agular itegratio variable β = θ θ ad defiig the parameter = R ρ c allows rewritig it as: θ + π cosβ, dβ π θ c cos β ( ρ θ ) = + i exp[ iφ ( β + θ )] A (9) Let us ow make the two followig assumptios: ca be exteded to the [,+ ] rage ad made periodic with a period of π, evetually exhibitig discotiuities at θ = π, where is ay positive or egative iteger. The the itegratio bouds i Eq. 9 ca be chaged from [ θ, θ +π] to [ π,+π], which will be more coveiet for the ext calculatios. ) Although iitially defied over a [,π] iterval, the phase mask fuctio φ ( θ ) ) The the complex mask trasmissio [ iφ ( θ )] exp is also periodic ad ca be decomposed i Fourier series of coefficiets a, with iteger idex spaig from to +. Hece it may be writte as: [ iφ ( θ )] = a exp[ i θ ] + = exp (a)

5 π + π where by defiitio: a = exp[ ( θ )] a π iφ is the mea value, ad: (b) = π + π π exp [ iφ ( θ )] exp( i θ ) whe. Followig both previous assumptios, ad after ivertig the cotiuous ad discrete sums, the complex amplitude A, may be expressed as fuctio of the coefficiets a : ( ρ θ ) A ( ρ θ ) = a + i exp[ i( β + θ )] or uder a more codesed form: (c) + θ + π cos β, dβ () π = θ c cos β A π with fuctios g (c) beig defied as: ( ρ θ ) = a ( a + ig ( c) ) exp[ i θ ] + =,, (a) g + π ( c) = exp[ i β ] π cos β c cos β dβ. (b) From the geometrical iterpretatio of Figure 4-, this itegral is equal to twice its value over the itegratio rage [,+π]: + π cosβ dβ c cos β ( c) = cos( β ) g. (3) θ β Settig the ew variable = i Eq. 3 the leads to: g + π π ( c) = cos( π ) cos( θ ) si( π ) si( θ ) π siθ c si θ + π π siθ c si θ. (4) Notig that the first term i Eq. 4 is ull sice the fuctio to be itegrated betwee π/ ad +π/ is odd, the fial writes as: expressio of ( c) g g ( c) = i ( ) + π siθ [ ] si( θ ) π c si θ. (5) That last expressio leads us to distiguish the cases whe is either eve or odd: Regroupig Eqs. a, ad 5 ad after a few simplificatios, the expressios of the complex amplitude ito the telescope pupil plae fially are: whe = p is a eve iteger, ad: a A p (, θ ) = π ρ (6a)

6 A i + ( ρ, θ ) = A ( ρ, θ ) ( ) a exp[ i ( p + ) θ ] si( ( p ) θ ) + p + π p= π c si θ + π siθ p (6b) whe = p+ is a odd iteger. These expressios allow defiig two sufficiet coditios for a azimuthal phase to geerate full extictio iside the Lyot stop of a coroagraph. They are: fuctio φ ( θ ) C. The averaged complex amplitude [ iφ ( θ )] C. The decompositio of [ iφ ( θ )] a p+ are ull). exp is ull over a periodic iterval of π (i.e. a = ). exp ito Fourier series shall oly comprise eve terms (all odd Fourier coefficiet These sufficiet coditios call for the followig commets:. C ad C are ecessary but ot sufficiet coditios, sice it caot be excluded that a give combiatio of A, i Eq. a. odd coefficiets a p+ ad fuctios g p+ (c) produces a ull amplitude ( ρ θ ). The ecessary coditio C aloe oly esures ull diffracted itesity at the Lyot stop cetre O, as has bee poited out i Ref. [8]. 3. Startig from a give phase fuctio φ ( θ ) periodic fuctios φ ( pθ ) that satisfies C, it shall be possible to build a family of π/p geeratig full extictio over the whole telescope pupil. This last property will be illustrated by some examples give i sectio 3. It may fially be oted that whe coditio C is ot respected, the geerated itesity iside the Lyot stop is uiform ad deduced from Eq. 6a: (, ) a π I p θ = ρ. (7) 4 This light leak iside Lyot stop is further discussed i sectio Eergy outside the Lyot stop The case whe poit P is located outside of the Lyot stop is a bit less iterestig ad is oly treated here to check cosistecy with the results of Refs. [8] ad [5]. The mathematical developmet is basically the same as i the previous g c i Eq. 3 is somewhat more complicated sice the itegratio domai has sub-sectio, but here the expressio of ( ) to be divided ito two differet sectios where either Eq. 8a or 8b shall be itroduced ito Eq. 6a, depedig o the umerical value of the ratio c = R ρ (c for poits P located iside the telescope pupil ad c < otherwise). Usig the geometrical iterpretatio of Figure 4- for defiig the itegratio bouds, otig that the itegral sums are g c becomes: equal to twice their values over a sigle loop o the red circles ad takig all symmetries ito accout, ( ) g arccosc ( c) = 4 cos( β ) dβ + cos( β ) cosβ cos β c π arccosc arccosc c cosβ cos β dβ. (8) Restrictig us to the case whe coditio C is respected, it ca be show that the secod term of Eq. 8 is equal to zero, by use of the same mathematical argumets as i the previous sub-sectio. Moreover, oly a VPM of topological charge exp i reduces to the vortex itself. = p is cosidered here. The the Fourier decompositio of the fuctio [ φ ( θ )] The applyig coditio C, combiig Eqs. c, ad 8, otig that cos β o the cosidered itegratio domai ad omittig the complex phase term i θ, the expressio of the complex amplitude outside of the telescope pupil simplifies as:

7 A arccosc cos β, dβ π cos β c ( θ ) = cos( β ) ρ. (9) The latter ca further be reduced to a polyomial expasio of the term c by use of three successive tricks: ) Sice = p is a eve iteger, develop cos(β) ito eve powers of cos β followig the geeral relatio cos p b k k= k ( β ) cos β = where the coefficiets b k are kow real umbers. ) Iside the itegrals, set the ew variables expressio of the terms c k : A p π k ( ρ, θ ) = bk [ ( c ) si θ ] π k= u = si β c, the = arcsi u θ leadig to a polyomial. () 3) Fially use the recurrece relatioships S π itegrals after biomial developmet of [ ( c ) θ ] si k = si θ = S ad = π i Eq.. S to evaluate the The previous procedure has bee applied to a VPM of topological charges m =, 4 ad 6 for validatio purpose. A, ad of their mai parameters are compiled i Table. Although obtaied i a Expressios of the amplitudes ( ρ θ ) m much less elegat maer, these expressios are rigorously similar to those give ito the Appedix B of Ref. [5]. Plots of the amplitudes A ( ρ, θ ) ad resultig itesities ( ρ, θ ) = A ( ρ, θ ) reproduced i Figure 6. I outside of the telescope pupil are Table : Expressios of amplitudes radiated outside of the telescope pupil for a VPM of topological charges m =, 4 ad 6, as fuctio of parameter c = R ρ. Vortex topological p = m/ c k coefficiets S k itegrals charge m = p ( k p) ( k p) ad + π ad π 4 4 +, 8 ad +8 π, π 4 ad 3π 6 6 3, +8, 48 ad +3 π, π 4, 3π 6 ad 5π 3 ( ρ θ ) A, c 4 c + 3c 4 3c c + c 6 3 EXAMPLES OF APPLICATIONS I this sectio are firstly recalled some elemetary phase masks fuctios employed i coroagraphy, such as vortex, azimuthal cosie, ad regularly segmeted fuctios ( 3.). The way those fuctios ca be combied together is briefly discussed i 3.. Fially, a empirical formula relatig phase mask errors to resultat light leaks iside the Lyot stop is defied i Elemetary phase masks fuctios

8 3.. Vortex fuctios Nowadays the most popular azimuthal phase fuctio is probably the vortex fuctio that was firstly itroduced i the field of coroagraphy by Swartzlader [7] ad is also commoly used for particles trappig [9]. It is a very simple liear phase ramp writte as: ( θ ) = θ φ m, () where m is a positive or egative iteger amed topological charge. It follows that the Fourier decompositio of the associated complex amplitude exp [ iφ ( θ )] reduces to the sole o-zero coefficiet a m =. Therefore, vortex complex amplitudes ca be cosidered as the basis for all other amplitude distributios geeratig extictios iside the Lyot stop of the coroagraph, as poited out i Refs. [8] ad [5]. Obviously, this statemet is ot applicable to the φ ( θ ) phase fuctio itself. The major properties of optical vortex PMCs of topological charges m =,, 3 ad 4 are illustrated i Figure 7- to 7-4 respectively. For each case, three-dimesioal plots of the phase fuctios φ ( θ ) are displayed o the left colum. The. The modules of the Fourier coefficiets a associated to the fuctios are scaled i terms of waves, i.e. φ ( θ ) π complex amplitude [ iφ ( θ )] exp are illustrated by the histograms of the cetral colum. They have bee computed umerically by direct applicatio of Eqs.. Fially, the right colum of the figure exhibits grey-scale maps of the itesity distributio I ( ρ, θ ) geerated i the Lyot stop plae. They have bee computed usig a double Matrix Fourier trasform (MFT) algorithm described i Ref. [] with iput pupil samplig of 496 x 496 embedded ito 89 x 89 arrays to prevet aliasig effects. The itesity maps are show i logarithmic scale for better visualizatio of the cetral extictio area. They reveal the clear superiority of vortices of eve topological charges (m = ad 4) with respect to odd charges (m = ad 3) for achievig deep extictio ito the telescope pupil area. These results are i agreemet with the ecessary coditios C ad C of sub-sectio.3, because eve phase fuctios respect both of them, while odd fuctios respect coditio C oly. It ca be oticed however that cetral extictio teds to improve with icreasig vortex charges, thus it might be cojectured that odd charges m = p + could also be suitable to coroagraphs for icreasig odd vortex charges. 3.. Azimuthal cosie-modulated fuctios The azimuthal cosie-modulated (ACM) phase fuctio is already kow for possible applicatios i laser beam shapig ad particles maipulatio [-] ad has recetly bee proposed for coroagraphic applicatios [8] [3]. The mai differece with respect to vortex or regularly segmeted coroagraphs is that they exhibit a sigle idetermiate poit at O istead of oe or several discotiuity lies. Mathematically, the ACM fuctio is defied as: ( θ ) φ ( θ ) = z m cos k, () where k is a iteger umber amed as the fuctio frequecy, ad z m is the m th zero of the type-j Bessel fuctio J (z). The geeral properties of the ACM fuctios are illustrated i Figure 8. As for vortex case, the histograms i the cetral colum of the figure show the modules of the Fourier coefficiets a whe k =,, 3 ad 4. Their expressio ca be ito Eq. c ad usig the itegral defiitio of type-j Bessel fuctios, deduced from the case k = by isertig φ ( θ ) fially leadig to a ( i) J ( z ) =. Therefore the ecessary coditio C is fulfilled sice the coefficiet a is m always equal to zero, while coditio C is respected for all eve itegers k. It ca be verified i Figure 8 that the a coefficiets are idetical to their theoretical values, ad that full extictio iside the Lyot stop is oly achieved whe k = ad 4. As illustrated by their radial ad azimuthal profiles show i Figure 9, the itesity distributios produced i the Lyot stop plae are o loger axis symmetric, due to the presece of differet o ull Fourier coefficiets. More details about the ACM phase masks will be foud i the sectio 3. of Ref. [8], where they have bee compared to the VPM i terms of achievable extictio rate, ier workig agle ad throughput. It was cocluded that despite of slightly degraded performace, ACM fuctios represet a iterestig alterative to the VPM, especially because they may be easier to produce as phase plates or by usig deformable mirrors.

9 3..3 Regularly segmeted fuctios Historically, the first azimuthal phase fuctios proposed for coroagraphy were regularly segmeted fuctios defied as: ( θ π ) φ ( θ ) = π It k, (3) where fuctio It(z) stads for the iteger part of ay real umber z, ad k is a iteger equal to half the umber of segmets. I particular, the phase kife [4], four-quadrat [6] ad eight-octat [5] coroagraphs are obtaied whe k = ; ad 4 respectively. They are all illustrated i Figure ad Figure, together with the less utilized six-sextat (k = 3). Oce agai, it ca be verified that full extictio is oly achieved for eve k umbers. Here the theoretical expressio of the Fourier coefficiets a is ot particularly simple, but its compliace with the computed histograms has also bee checked. 3. Composite phase fuctios All phase fuctios preseted i the last sub-sectios may be cosidered as basic bricks for buildig more elaborated fuctios, also geeratig full extictio iside the Lyot stop of the coroagraph, such as the composite phase fuctio described i Ref. [6]. The way these fuctios ca be combied for geeratig cetral extictios (i.e. fulfillig coditio C oly) has already bee described i Ref. []. I particular it was demostrated that: a. Cetral ulls geerated by azimuthal phase fuctios are ot affected by axis-symmetric cetral obscuratios. b. I case of agular trucatio (i.e. θ ragig from arbitrary segmets [θ,θ ] istead of [,π]), cetral ulls ca be preserved by proper rescalig of the phase fuctio amplitude. c. Differet types of such phase fuctios ca be juxtaposed o differet agular segmets ad still geerate a cetral ull. For phase mask coroagraphs the itroductio of additioal coditio C implies severe limitatios ad few of the above properties shall remai valid. For example a cetral obscuratio of radius ρ shall tur the itegratio domai [,+ ] i Eq. 6b ito [ρ,+ ], thus makig the expressios of fuctio F(a) i Eqs. 7 ivalid, ad so property a. Similarly, restrictig the itegratio domai [,π] i relatio b will modify the expressios of A ( ρ, θ ) i Eqs. 6, which cotradicts property b. Hece the phase masks should oly be o obstructed disks. It remais possible, however, to defie composite phase fuctios built from the three differet types idetified i sectio 3. ad achievig full extictio ito the Lyot stop. Figure illustrates three examples of such fuctios that are aalytically defied i Table satisfyig coditio C,. The geeral procedure for costructig them cosists i firstly idetifyig a fuctio φ ( θ ) the buildig the fuctio φ ( pθ ) ifiity of composite ullig phase fuctios φ ( θ ) where p is a iteger as explaied i.3. I that way there certaily exists a suitable to coroagraphy, albeit exhibitig differet itesity distributios outside of the Lyot stop as show i the rightmost colum of the figure. Table : Three examples of composite ullig phase fuctios φ ( θ ) illustrated i Figure. Number Aalytic expressio Remark φ θ ) = 4θ + π It( 4θ π ) ( Additio of a charge 4 vortex with a four-quadrat φ ( θ ) = 4θ π θ < π/ ad θ < +π/ Compositio of vortex ad ACM 3 φ ( θ ) = z cos( 4θ ) π/ θ < ad +π/ θ < +π fuctios over four π/ segmets φ ( θ ) = π It( 8θ π ) π θ < π/ ad θ < +π/ Composite eighth-octat φ ( θ ) = π It 8 θ π π π/ θ < ad +π/ θ < +π phase fuctio ( ( ) )

10 3.3 Light leaks iside Lyot stop Although the aalytical formalism of this study is restricted to the case of a aberratio-free telescope, i.e. with o phase errors itroduced ito the pupil plae OXY or i ay itermediate plae located betwee poits O ad O (usually requirig Fresel diffractio aalysis), it turs out that it is usable for itroducig phase defects ito the image plae O X Y. Such aberratios may for istace result from phase mask maufacturig errors ad are usually eglected or treated umerically. I that case however a approximate formula ca be established for evaluatig the effect of radom phase mask errors o the achievable extictio depth iside the Lyot stop. For phase fuctios φ ( θ ) satisfyig both coditios C ad C, it ca be writte from Eq. 7: (, ) a π I p θ = ρ, (4a) 4 where the symbol stads for statistical average, ad phase errors δ φ ( θ ) + π π are itroduced ito the expressio of a : a = exp[ i( φ ( θ ) + δφ ( θ ))] (4b) π θ Assumig weak phase distortios, i.e. φ ( θ ) φ ( ) << + i z, thus leadig to: δ the complex expoetial exp[i z] ca be approximated to + π i a δφ ( θ ) [ iφ ( θ )] exp. (5) π π Applyig Schwarz s iequality allows obtaiig a worst-case estimatio of a : + π a δφ exp i π + π ( θ ) [ φ ( θ )] δφ ( θ ) σ, (6) π where σ is the stadard deviatio of the phase mask errors assumed to be Gaussia. It must be oted that the pessimistic use of Schwarz s iequality is somewhat couterbalaced by the optimistic assumptio of eglectig eergy leaks carried by the other Fourier coefficiets a. A approximate expressio for the averaged itesity distributio iside the Lyot stop is the: ( ρ, θ ) σ 4 π. (7) I p 4 I Figure 3 are illustrated the effects of radom phase errors added to a vortex phase mask of charge m =, both o its Fourier coefficiets a ad itesity distributio geerated i the Lyot stop plae. The plots are show i logarithmic I ρ, θ computed from Eq. 7 i red scale for better ehacig low values. Figure 3-3 ad 4 show plots of ( ) solid lies, compared with the results of umerical simulatios idicated with red dots. Curves are plotted as fuctio of the stadard deviatio σ i liear ad logarithmic scale respectively. It ca be oted that simulatio results are i global agreemet with the approximate formula 7, eve if data do ot match perfectly. The remaiig deviatios most probably origiate from the three employed approximatios (weak phase errors, use of Schwarz s iequality, ad Fourier decompositio reduced to the sigle coefficiet a ), that may also explai o-uiform irradiace iside the Lyot stop show i Figure 3-. Despite of the limitatios of the aalytical model, it ca be cocluded however that: Resultig from the fourth-power law i relatio 7, light leaks are icreasig very rapidly with phase errors. Assumig ecessary extictio ratios of -6 ad - for observig Jupiter or Earth-like extra-solar plaets respectively, phase plates maufacturig requiremets should be ad rad, or λ/79 ad λ/793 equivaletly (all umbers are i RMS terms). p

11 Kowig that phase masks employed i coroagraphy usually have small areas (typically mm ), these specificatios look difficult but feasible for Jupiter-like extra-solar plaets. The case of Earth-like plaets still remais very challegig. At this stage it may also be of iterest to evaluate the effects of local, determiistic phase mask errors o the itesity distributios formed i the fial image plae of the coroagraph. Sice Eq. 7 is o loger relevat, the questio is addressed umerically with the help of the umerical model described i 3..: startig from the complex amplitude A ( ρ, θ ) i the Lyot stop plae, oe third ad last Fourier trasform allows computig irradiace maps i the coroagraph image plae. The results of this simulatio are illustrated i Figure 4. Local phase mask errors of amplitude δφ ±. 4λ were itroduced at radom locatios ito the phase mask plae O X Y. The resultig itesity distributio i the fial image plae (referred to the X ad Y axes, assumig a magificatio factor of oe betwee both image plaes) are show i Figure 4-. I additio to a expected star leakage o the optical axis, it ca be see that there is a correlatio betwee local errors of the phase mask ad residual speckles i the coroagraph image plae, o the oe had, ad that the stregth of these speckles is iversely proportioal to their distace to poit O, o the other had. Such parasitic off-axis images of the star could evetually be mistake with speckles geerated by quasistatic pupil aberratios. The way of disetaglig those two differet types of static speckles is however beyod the scope of the paper. 4 CONCLUSION I this paper was preseted a aalytical study of the azimuthal phase-mask coroagraph, oe of the most promisig istrumetal cocepts evisioed for detectig ad characterizig Jupiter or Earth-like extra-solar plaets i the ear future. Special emphasis was put o the physical ad geometrical iterpretatio of the mathematical developmet. It allowed defiig two ecessary coditios for the phase mask to achieve full extictio i the pupil plae of the coroagraph. They state that the complex amplitude geerated by the azimuthal phase mask should have zero average, o the oe had, ad that its Fourier coefficiets should oly be eve, o the other had. Examples of such phase fuctios have bee reviewed, icludig optical vortices, four-quadrat phase mask, ad the less utilized azimuthal cosie-modulated fuctios. Some hits for buildig more sophisticated fuctios were give, suggestig that there exist huge varieties of phase fuctios satisfyig to the ecessary coditios. Fially, a simplified expressio of the light leaks due to mask imperfectio ad suitable to system desig ad toleracig was proposed. It highlights the extreme difficulty of maufacturig phase masks suitable to the detectio of Earth-like extra-solar plaets. The author would like to thak his colleague P. Rabou for careful readig of the mauscript ad improvemet of Eglish wordig. He also thaks the aoymous reviewers for relevat commets leadig to cosiderable improvemets of the origial mauscript. REFERENCES [] J.-L. Beuzit, M. Feldt, K. Dohle, D. Mouillet et al, SPHERE: a plaet fider istrumet for the VLT, Proceedigs of the SPIE vol. 74, 748 (8). [] B. Macitosh, J. Graham, D. Palmer et al, The Gemii Plaet Imager, Proceedigs of the SPIE vol. 67, 67L (6). [3] F. Zhao, WFIRST-AFTA coroagraph istrumet overview, Proceedigs of the SPIE vol. 943, 943O (4). [4] F. Roddier, C. Roddier, Stellar coroagraph with phase mask, Publicatios of the Astroomical Society of the Pacific vol. 9, p (997). [5] C. Aime, R. Soummer, A. Ferrari, Total coroagraphic extictio of rectagular apertures usig liear prolate apodizatios, Astroomy ad Astrophysics vol. 389, p ().

12 [6] D. Roua, P. Riaud, A. Boccaletti, Y. Cléet, A. Labeyrie, The four-quadrat phase-mask coroagraph. I. Priciple, Publicatios of the Astroomical Society of the Pacific vol., p (). [7] G. A. Swartzlader Jr, Peerig ito darkess with a vortex spatial filter, Optics Letters vol. 6, p (). [8] D. Mawet, P. Riaud, O. Absil, J. Surdej, Aular groove phase mask coroagraph, Astrophysical Joural vol. 633, p. 9- (5). [9] G. A. Swartzlader Jr, The optical vortex coroagraph, J. Opt. A: Pure Appl. Opt. vol., 94 p. -9 (9). [] F. Lemarquis, M. Lequime, G. Albrad, L. Escoubas, J.-J. Simo, J. Baudrad, P. Riaud, D. Roua, A. Boccaletti, P. Baudoz, D. Mawet, Maufacturig of 4-quadrat phase mask for ullig iterferometry i thermal ifrared, Proceedigs of the SPIE vol. 55, p (4). [] D. Mawet, E. Seraby, K. Liewer, C. Haot, S. McEldowey, D. Shemo, N. O Brie, Optical vectorial vortex coroagraphs usig liquid crystal polymers: theory, maufacturig ad laboratory demostratio, Optics Express vol. 7 p (9). [] A. Boccaletti, P. Riaud, P. Baudoz, J. Baudrad; D. Roua, D. Gratadour, F. Lacombe, A.-M. Lagrage, The fourquadrat phase mask coroagraph. IV. First light at the Very Large Telescope, Publicatios of the Astroomical Society of the Pacific vol. 6, p. 6-7 (4). [3] D. Mawet, E. Seraby, K. Liewer, R. Burruss, J. Hickey, D. Shemo, The vector vortex coroagraph: Llaboratory results ad first light at Palomar Observatory, Astrophysical Joural vol. 79,:p (). [4] C. Jekis, Optical vortex coroagraphs o groud-based telescopes, Mo. Not. R. Astro. Soc. 384, (8). [5] A. Carlotti, G. Ricort, C. Aime, Phase mask coroagraphy usig a Mach-Zehder iterferometer, Astroomy ad Astrophysics vol. 54, p (9). [6] I. M. Gel'fad, G. E. Shilov, Geeralized fuctios, Academic Press ( ). [7] M. Abramowitz, I. A. Stegu, Hadbook of mathematical fuctios, Dover Publicatios, INC., New York (97). [8] F. Héault, A. Carlotti, C. Vériaud, Aalysis of ullig phase fuctios suitable to image plae coroagraphy, Proceedigs of the SPIE vol. 99, 996K (6). [9]. A. M. Yao, M. J. Padgett, Orbital agular mometum: origis, behavior ad applicatios, Advaces i Optics ad Photoics vol. 3, p. 6-4 (). [] R. Soummer, L. Pueyo, A. Sivaramakrisha, R. J. Vaderbei, Fast computatio of Lyot-style coroagraph propagatio, Optics Express vol. 5, p (7). [] S. Topuzoski, L. Jaicijevic, Diffractio characteristics of optical elemets desiged as phase layers with cosieprofiled periodicity i the azimuthal directio, JOSA A vol. 8, p (). [] F. Héault, Strehl ratio: a tool for optimizig optical ulls ad sigularities, JOSA A vol. 3, p (5). [3] O. Ma, Q. Cao, F. Hou, Wide-bad coroagraph with siusoidal phase i the agular directio, Optics Express vol., p (). [4] L. Abe, F. Vakili, A. Boccaletti, The achromatic phase kife coroagraph, Astroomy ad Astrophysics vol. 374, p (). [5] N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, L. Abe, Achromatic eight-octat phase-mask coroagraph usig photoic crystal, The Astrophysical Joural vol. 74, p (). [6] F. Hou, Q. Cao, M. Zhu, O. Ma, Wide-bad six-regio phase mask coroagraph, Optics Express vol., p (4).

13 APPENDIX. CALCULATION OF FUNCTION F(a) Let us first make use of the itegral expressio of the J Bessel fuctio that is: π i ( z) = exp[ i ( θ + z cosθ )] π J. (A) Isertig Eq. A ito 6b ad permutig the summatio sigs readily leads to: π i F ( a) = exp[ iθ ] Hˆ ( a, θ ), (A) π + where: ˆ ( a, θ ) = exp[ iπrν ( a + cosθ )] H dν. (A3) The fuctio H ˆ ( a,θ ) i Eq. A3 actually appears as the Fourier trasform of the Heaviside distributio H(z) at z = R( a + cosθ ). Kowig that the Fourier trasform of H(z) is equal to δ ( z) i π z, where δ(z) is the impulse Dirac distributio, F(a) ca be rewritte as: F π ( a) δ ( R( a + cosθ )) exp[ iθ ] i = F 4π 4π R ( a) + F ( a) [ iθ ] i exp = 4π + 4π R a + cosθ π with: ( a) = δ ( R( a cosθ )) [ iθ ] ad: ( a) + exp π (A4) F (A5a) π [ iθ ] exp = a + cosθ F. (A5b) The aalytical expressios of F (a) ad F (a) are the evaluated separately. Aalytical expressio of F (a) Let us firstly rewrite F (a) as: π ( a) = Re[ F ( a) ] + Im[ F ( a) ] = δ ( R( a + cosθ )) cosθ + i δ ( R( a cosθ )) π + siθ F (A6) with Re[ ] ad Im[ ] respectively deotig the real ad imagiary parts of a complex umber. We shall the make use of the geeral relatioship [6]: ( f ( θ )) = N δ ( θ θ ) f ( ) = θ δ, (A7)

14 where f'(θ) is ay real fuctio of the variable θ, θ with N its roots over the itegratio rage, ad f' (θ) the derivative of f'(θ). For f ( θ ) = R( a + cosθ ) the umber of roots is depeds o the value of a ad is either equal to or : whe a, N =, a θ = π arccos ad = π + arccos a θ., whe a >, N = ad both fuctios δ ( ( a + cosθ )) R ad F (a) are ull. si arccos a Whe a, combiig Eqs A6 ad A7 ad usig the elemetary relatio ( ) Re a [ F ( a) ] = ad: Im[ ( )] a = R Aalytical expressio of F (a) a Here also F (a) may be rewritte as: F π ( a) = Re[ F ( a) ] + i Im[ F ( a) ] = + i t i [ ( )] = a fially leads to: F (A8) π cosθ siθ a + cosθ a + cosθ. (A9) Chagig the variable = cosθ Im F a readily shows that it is equal to zero, therefore the expressio of F (a) reduces to its real part that is, after elemetary arithmetic ad chagig itegratio bouds: + π ( a) = Re[ F ( a) ] = π a a π cosθ F. (A) The from Ref. [7], sectio , the expressios of F (a) are: whe a : ( a) whe a > : F ( a) Fial expressio of F(a) ( + a) ( + a) a taθ π log a = = a taθ + a F, (Aa) = π a arcta a ( a + ) + π + π π π taθ a = π a a Combiig Eqs. A4, A6, A8 ad A fially leads to the aalytical expressio of F(a) that is: F F ( a) = + i πr a ( a) = πr a a a π. (Ab) whe a <, ad: (Aa) whe a >. (Ab) It should be oted that similar expressios ca be derived from Ref. [7], sectios.4.37 ad.4.38, but their validity rages are restricted to < a < ad a >, respectively for Eqs. Aa ad Ab.

15 Pupil plae Image mask plae Lyot stop plae Coroagraphic plae T T O O O O T T Telescope Collimatig les Focusig les Figure : Basic priciple of a phase-mask coroagraph.

16 Y P x = ρ cos θ y = ρ si θ O ρ θ X Y M x = ρ cos θ y = ρ si θ R F ρ θ X O Exit pupil plae Z Image mask plae Y M x = ρ cos θ y = ρ si θ ρ θ Y X x = ρ cos θ P y = ρ si θ O F O ρ θ X Image mask plae Z R L Lyot stop plae Figure : Coordiate frames ad scietific otatios.

17 Y M H β P θ ρ θ O R X Figure 3: Geometrical iterpretatio of the parameter a = ρ cos β R. For a give poit P of polar A ρ, θ is defied as coordiates (ρ,θ ) i the Lyot stop plae O X Y, the itegratio domai of ( ) θ π or θ β π θ (Eq. 6a). It ca be represeted as a poit M rotatig by the agle θ +β alog a dashed-lie circle of radius ρ. The quatity a = ρ cos β R is equal to the ratio O H R, with H the projectio of P o segmet O M. At a give agle β the iequality a refers to a area of ifiite legth ad width R (show i light blue). As β rages from to π those areas itersect at the blue cetral disk of radius R, which is the geometrical image of the telescope pupil. Similarly, the coditio a > applies to poits P located outside of the pupil.

18 Y β + π P β θ O ρ H θ X R Y R β = arccos ρ P R β = π arccos ρ H H O ρ β θ θ X R R β 4 = arccos ρ R β3 = arccos π ρ Figure 4: Defiitio of β itegratio domais for poits P located iside the telescope pupil () ad outside of it (). For a give poit P of polar coordiates (ρ,θ ), poits H move alog the red dashed circles, ruig two laps whe β rages from to π. Case : H always remais withi the image of the telescope pupil ad the coditio a is respected, the oly Eq. 8a is eeded for computig I ( ρ, θ ). Moreover the itegratio rage ca be restricted to [,π] which stads for a sigle loop o the red circle (same poits H are foud for ay couple of agles β ad β + π). Case : the red circle itersects the cotour of the telescope pupil. The [,π] itegratio rage must be divided ito four subdomais idicated by the agles β i ( i 4). Eq. 8a should be replaced with Eq. 8b whe β β β 3 ad β 4 β β.

19 ρ /R Re[F(a)] Im[F(a)] ρ /R β (rad) Figure 5: Plots of the real () ad imagiary parts () of the weightig fuctio F(a) as fuctio of the agle β, for differet values of the ratio ρ R.

20 Relative amplitude m = m = 4 m = ρ /R.E+ m = m = 4 m = 6 Relative itesity.e-.e-.e-3.e ρ /R Figure 6: Amplitudes () ad itesities () radiated outside of the telescope pupil by VPMs of topological charges m =, 4 ad 6, as fuctio of the ratio ρ R. Itesities are plotted i logarithmic scale.

21 Topological V charge m = Topological V charge m = Topological V3 charge m = Topological V4 charge m = Phase fuctio i waves Fourier coefficiets Lyot stop itesity Figure 7: Illustratig vortex phase fuctios of topological charges m =,, 3 ad 4 (Figs. 7- to 7-4 respectively). For each case are show three-dimesioal plots of the phase fuctio φ ( θ ) (left colum), the modules of the Fourier coefficiets a defied i Eqs. (cetral colum), ad grey-scale maps of the itesity distributio I ( ρ, θ ) geerated i the Lyot stop plae (right colum). The latter are show i logarithmic scale for ehacig the cetral extictio area.

22 .6 Azimuthal period C umber k = Azimuthal period C umber k = Azimuthal period C3 umber k = Azimuthal period C4 umber k = Phase fuctio i waves Fourier coefficiets Lyot stop itesity Figure 8: Illustratig azimuthal cosie phase fuctios. Figs. 8- to 8-4 correspod to the cases of azimuthal period umbers k =,, 3 ad 4 respectively. Plots are similar to those of Figure 7, with white lies idicatig the radial ad azimuthal profiles of Figure 9.

23 Normalized itesity Radial profiles k = k = k = 3 k = Reduced radius ρ /R Azimuthal profiles.4 k = k = k = 3 k = 4 Normalized itesity Reduced agle θ /π Figure 9: Radial ad azimuthal profiles of the itesity distributios geerated by the ACM fuctios of Figure 8 ito the Lyot stop plae. Top: Radial profiles at agle θ = degree. Bottom: Azimuthal profiles at radius ρ =.6 R.

24 Phase Q kife Four-Quadrats Q Six-Sextats Q Eight-Octats Q Phase fuctio i waves Fourier coefficiets Lyot stop itesity Figure : Illustratig the case of regularly segmeted phase fuctios. Figs. 9- to 9-4 respectively show the phase kife, four-quadrats, six-sextats ad eight-octats phase masks. Plots are similar to those of Figure 7. White lies idicate the radial ad azimuthal profiles of Figure.

25 Normalized itesity Radial profiles k = k = k = 3 k = Reduced radius ρ /R Azimuthal profiles.4 k = k = k = 3 k = 4 Normalized itesity Reduced agle θ /π Figure : Radial ad azimuthal profiles of itesity distributios geerated by the segmeted phase fuctios of Figure. Top: Radial profiles at agles θ = 9; 45, 3 ad.5 degrees for k =,, 3 ad 4 respectively. Bottom: Azimuthal profiles at ρ =.6 R.

26 .6 Azimuthal cosie k = C-Q4 + Four-Quadrats Azimuthal C4V4 cosie k = 4 + Vortex charge m = Composite Q66 eight-octats Phase fuctio i waves (3D ad D views) Fourier coefficiets Lyot stop itesity Figure : Illustratig the case of regularly segmeted phase fuctios. Plots are similar to those of Figure 7-9.

27 .E+.E-.E-4.E Fourier coefficiets 3.E-3.E- Lyot stop itesity.e-3.e-3.e-3.e-4.e+.e RMS phase error (waves) 4 RMS phase error (waves) Figure 3: Itroductio of phase errors ito a vortex mask of charge m =. () Fourier coefficiets. () Itesity distributio i Lyot stop plae. Both plots are show i logarithmic scale. (3) ad (4): Curves of the mea irradiace iside the Lyot stop as fuctio of phase errors stadard deviatio, i liear ad logarithmic scale respectively. Numerical simulatio results are idicated by red dots ad the aalytical formula 7 by red solid lies.

28 Phase errors ( λ) 6 λ/d Phase mask errors.8 Normalized itesity λ/d Itesity i coroagraph image plae Figure 4: Itroductio of local phase errors over a vortex mask of charge m =. () Grey-scale map ad three-dimesioal plot of iput phase errors. () Same illustratios of itesity distributio geerated i the coroagraph image plae. Local errors lay iside a disk of diameter.3 mm i the phase mask plae, which is equivalet to 6λ/D.