Imaging and nulling properties of sparse-aperture Fizeau interferometers Imaging and nulling properties of sparse-aperture Fizeau interferometers

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1 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Fraçois Héault Istitut de Plaétologie et d Astrophysique de Greoble, Uiversité Joseph Fourier, CNRS, B.P. 53, Greoble Frace Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui

2 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Geeral layout of a Fizeau iterferometer B P 1 P 2 (P) Telescope 1 F C Telescope 2 Relay optics 1 APS 1 APS 2 beam 1 B beam 2 Relay optics 2 Covergig optics Divergig optics Fold mirror Beamsplitter Acromatic Phase Shifter Combiig optics (P ) F Focal plae All telescopes assumed to be idetical All exit pupils optically cojugated with etrace pupils Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui

3 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers V -sky agular coordiates Sky object u v U P 2 P 3 s Y Iput pupils defied by vectors P utput pupils defied by vectors P 1 N s B P 1 P 4 D Etrace pupil plae X D P 4 P 1 Y B Coordiates systems P 3 P 2 s s Exit pupil plae X F M Y M Detectio plae X Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui

4 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Image of a sky object projected oto the sky Most geeral bject-image relatioship : I( s) = s T = 1 Ω ( s ) PSF ( s - s ) N a exp [ iφ ] exp[ ikξ( s, s) ] 2 dω with ξ ( s, s ) = s P s' P' / m PSF T (s) : PSF of a idividual collectig telescope, projected back oto the sky a : amplitude trasmissio factor of the th telescope ϕ : phase-shift alog the th iterferometer arm for cophasig or ullig purpose k = 2/λ : waveumber of moochromatic electro-magetic field m : optical compressio factor betwee telescopes ad their relay optics Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui

5 First-order approximatio Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Golde rule of Fizeau iterferometers Necessary coditio : P = m P bject-image relatioship [ PSF ( s) F( s) ] ( ) I( s) = T s Ivariat PSF over the Field of view (FoV) Pupil I Pupil ut N = PSFT ( s) a exp = 1 F( s) [ ] [ ] 2 iφ exp i k sp Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui

6 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Examples of applicatio Effect of pupil aberratios Deviatios from the golde rule : ohomothetic exit pupil combiers Simulatio of images i ullig mode Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui

7 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Effect of pupil aberratios Develop all vectors at 2 d -order, e.g. : si u u ' x ' ( 1+ dz F' ) ' ' s = cos u si v v = y ( 1+ dz F' ) cos u cos v 1 u 2 v Isert ito the PD : ξ Use geeral bject-image relatio : No loger a covolutio operator! ' P ' ' ' ' ' (, s ) dz dz m + u x + v y ( u x + v y )( 1+ dz F' ) s I( s) = s 0 T = 1 Ω ( s ) PSF ( s - s ) N 0 dz a dz 2 2 ' 2 2 ( u + v ) 2 dz ( u + v ) 2m 0 exp 0 [ iφ ] exp[ ikξ( s, s) ] ' 2 dω m B B Flat wavefrot Flat wavefrot P 1 P 1 dz 1 dz 1 (P) Iput pupils dz 2 P 2 Collectig telescopes utput pupils (P ) dz 2 P 2 Multi-axial combier (exit pupil plae) F F Beam compressors X Focal plae Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui

8 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Pupil aberratio: 2 telescopes Achievable FoV PSF at FoV cetre PSF at 0.3 arcsec PSF at 0.6 arcsec Aberrated pupil Case 2 Aberrated pupil Case 1 0 No pupil aberratio Ivariat PSF Curved friges, chagig PSF Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui

9 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Pupil aberratio: 8 telescopes Achievable FoV PSF at FoV cetre PSF at 0.3 arcsec PSF at 0.6 arcsec Aberrated pupil Case 2 Aberrated pupil Case No pupil aberratio Ivariat PSF Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui

10 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Differet combiig schemes From collectig telescopes From collectig telescopes B B APS 1 APS 2 beam 1 Axial combier F Frige tracker (P ) Focal plae beam 2 beams 1-4 beams 2-3 Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui B F (P ) Focal plae Crossedcubes uller

11 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Axially Combied Iterferometer (ACI) Telescope 1 Relay optics 1 P 1 beam 1 Axial beam combier F B Telescope 2 APS 1 APS 2 Frige tracker (P ) Focal plae beam 2 Ι( s) = P 2 Relay optics 2 PSF Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui T (P) ( s) * Co-axial combiatio by meas of a balaced set of beamsplitters Extiguishes all diffracted light origiatig from cetral star ly leaves plaet photos Specific bject-image relatioship N = 1 a exp [ iφ ] exp[ iksp ] 2 ( s)

12 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Crossed-cubes uller (CCN) See the Cheapest uller i the World : e talk o Thursday afteroo, 2 posters o Wedesday afteroo Cube 1 Cube 2 Y Focusig optics X " Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui

13 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Pseudo-images of a compaio 20 m 10 mm Star Plaet Fake sky objects Telescope array Phase-shifts maps Star Plaet photos oly Plaet Axial combier Maximal desificatio Fizeau iterferometer Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui

14 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Coclusio A simple Fourier optics formalism allows fast evaluatio of ullig Fizeau iterferometers performace Icludig PSF, achievable Field of view (FoV), ullig maps ad pseudo-images Whatever is the telescope umber N I presece of optical defects (pupil aberratios) Also applicable to o-fizeau iterferometers Simple formalism, o actual Fourier or Fresel trasforms required (gai i accuracy ad computig time) Simulatios show that that the most efficiet ullig schemes should use axial combiers or multi-axial combiers with maximal desificatio (e.g. Crossed-cubes uller) Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui

15 Imagig ad ullig properties of sparse-aperture Fizeau iterferometers Previous publicatios Fibered ullig telescope for extra-solar coroagraphy, ptics Letters vol. 34, 7, p (2009) Computig extictio maps of star ullig iterferometers, ptics Express vol. 16, (2008) Fie art of computig ullig iterferometer maps, Proceedigs of the SPIE 7013, 70131X (2008) Simple Fourier optics formalism for high agular resolutio systems ad ullig iterferometry, JSA A vol. 27, p (2010) PSF ad field of view characteristics of imagig ad ullig iterferometers, Proceedigs of the SPIE vol. 7734, (2010) Imagig power of multi-fibered ullig telescopes for extra-solar plaet characterizatio, Proceedigs of the SPIE vol. 8151, 81510A (2011) Cheapest uller i the world: crossed beamsplitter cubes, Proceedigs of the SPIE vol. 9146, this coferece Perhaps oe sythesis paper some day Coferece ptical ad Ifrared Iterferometry IV Motréal, 23 jui