Abstract

Cophasing a multiple-aperture optical telescope (MAOT) or optical interferometer requires the knowledge of the tips/tilts and of the differential pistons on its subapertures. In this paper we demonstrate in the case of a point source object that a single focal-plane image is sufficient for MAOT cophasing. Adopting a least-square approach allows us to derive an analytic estimator of the subaperture aberrations, provided that these are small enough (typically for closed-loop operation) and that the pupil is diluted noncentrosymmetric. We then provide the validation of this estimator by simulations as well as a performance comparison with a more conventional iterative algorithm of phase retrieval. Finally, we present the experimental validation of both estimators on a laboratory test bench; our results, especially subnanometric repeatability, demonstrate that focal-plane sensors are appropriate for the cophasing of phased array telescopes.

© 2008 Optical Society of America

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2006 (1)

F. Cassaing, B. Sorrente, L. Mugnier, G. Rousset, V. Michau, I. Mocoeur, and F. Baron, “BRISE: a multipurpose bench for cophasing sensors,” Proc. SPIE 6268, 626834 (2006).

2005 (1)

J. Idier, L. M. Mugnier, and A. Blanc, “Statistical behavior of joint least square estimation in the phase diversity context,” IEEE Trans. Image Process. 14, 2107-2116 (2005).
[CrossRef] [PubMed]

2004 (1)

C. V. M. Fridlund, “The DARWIN mission,” Adv. Space Res. 34, 613-617 (2004).
[CrossRef]

2003 (1)

A. Blanc, T. Fusco, M. Hartung, L. M. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations. Application of the phase diversity technique,” Astron. Astrophys. 399, 373-383 (2003).
[CrossRef]

2001 (2)

R. A. Gonsalves, “Small-phase solution to the phase retrieval problem,” Opt. Lett. 26, 684-685 (2001).
[CrossRef]

F. Cassaing, “Optical path difference sensors,” C.R. Acad. Sci. Paris, Ser. IV 21, 87-98 (2001).

2000 (2)

M. G. Löfdahl and G. B. Scharmer, “A predictor approach for closed-loop phase diversity wavefront sensing,” Proc. SPIE 4013, 737-748 (2000).
[CrossRef]

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” Proc. SPIE 4091, 48-63 (2000).
[CrossRef]

1998 (2)

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase-diversity-based blind deconvolution,” Proc. SPIE 3353, 994-1005 (1998).
[CrossRef]

M. G. Löfdahl, A. L. Duncan, and G. B. Scharmer, “Fast phase diversity wavefront sensor for mirror control,” Proc. SPIE 3353, 952-963 (1998).
[CrossRef]

1994 (1)

1993 (2)

1992 (1)

1989 (1)

1988 (1)

1982 (2)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829-832 (1982).

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758-2769 (1982).
[CrossRef] [PubMed]

1981 (1)

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735-738 (1981).
[CrossRef]

1976 (2)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Acton, D. S.

Alis, C.

M. Mesrine, E. Thomas, S. Garin, P. Blanc, C. Alis, F. Cassaing, and D. Laubier, “High resolution Earth observation from Geostationary orbit by optical aperture synthesis,” in Sixth International Conference on Space Optics, Proceedings of ESA/CNES ISCO 2006, A. Wilson, ed., CDROM, ESA SP-621 (2006).

Barillot, M.

F. Cassaing, F. Baron, E. Schmidt, S. Hofer, L. M. Mugnier, M. Barillot, G. Rousset, T. Stuffler, and Y. Salvadé, “DARWIN Fringe Sensor (DWARF): Concept study,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539 (ESA, 2003), pp. 389-392.

E. Schmidt, F. Cassaing, S. Hofer, M. Barillot, F. Baron, L. M. Mugnier, G. Rousset, and T. Stuffler, “DARWIN Fringe Sensor (DWARF): breadboard development,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539, (ESA, 2003), pp. 575-577.

Baron, F.

F. Cassaing, B. Sorrente, L. Mugnier, G. Rousset, V. Michau, I. Mocoeur, and F. Baron, “BRISE: a multipurpose bench for cophasing sensors,” Proc. SPIE 6268, 626834 (2006).

F. Cassaing, F. Baron, E. Schmidt, S. Hofer, L. M. Mugnier, M. Barillot, G. Rousset, T. Stuffler, and Y. Salvadé, “DARWIN Fringe Sensor (DWARF): Concept study,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539 (ESA, 2003), pp. 389-392.

E. Schmidt, F. Cassaing, S. Hofer, M. Barillot, F. Baron, L. M. Mugnier, G. Rousset, and T. Stuffler, “DARWIN Fringe Sensor (DWARF): breadboard development,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539, (ESA, 2003), pp. 575-577.

F. Baron, “Définition et test d'un capteur de cophasage sur télescope multipupilles: application à la détection d'exoplanètes et à l'observation de la Terre,” Ph.D. thesis (Ecole Doctorale d'Astronomie et d'Astrophysique d'Ile de France, 2005).

B. Sorrente, F. Cassaing, F. Baron, C. Coudrain, B. Fleury, F. Mendez, V. Michau, L. Mugnier, G. Rousset, L. Rousset-Rouvière, and M.-T. Velluet, “Multiple-aperture optical telescopes: cophasing sensor test-bed,” in Proceedings of the 5th International Conference on Space Optics, B.Warmbein, ed., Vol. SP-554 (ESA, 2004), pp. 479-484.

Benson, L.

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” Proc. SPIE 4091, 48-63 (2000).
[CrossRef]

Blanc, A.

J. Idier, L. M. Mugnier, and A. Blanc, “Statistical behavior of joint least square estimation in the phase diversity context,” IEEE Trans. Image Process. 14, 2107-2116 (2005).
[CrossRef] [PubMed]

A. Blanc, T. Fusco, M. Hartung, L. M. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations. Application of the phase diversity technique,” Astron. Astrophys. 399, 373-383 (2003).
[CrossRef]

A. Blanc, “Identification de réponse impulsionnelle et restauration d'images: apports de la diversité de phase,” Ph.D. thesis (Université Paris XI Orsay, 2002).

L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elvesier, 2006), Vol. 20, pp. 1-76.

Blanc, P.

M. Mesrine, E. Thomas, S. Garin, P. Blanc, C. Alis, F. Cassaing, and D. Laubier, “High resolution Earth observation from Geostationary orbit by optical aperture synthesis,” in Sixth International Conference on Space Optics, Proceedings of ESA/CNES ISCO 2006, A. Wilson, ed., CDROM, ESA SP-621 (2006).

Cassaing, F.

F. Cassaing, B. Sorrente, L. Mugnier, G. Rousset, V. Michau, I. Mocoeur, and F. Baron, “BRISE: a multipurpose bench for cophasing sensors,” Proc. SPIE 6268, 626834 (2006).

F. Cassaing, “Optical path difference sensors,” C.R. Acad. Sci. Paris, Ser. IV 21, 87-98 (2001).

I. Mocoeur, L. M. Mugnier, and F. Cassaing, “Cophasage de télescopes multi pupilles sur scènes étendues par diversité de phase temps-réel,” presented at la 26ième Colloque sur le Traitement du Signal et des Images, September 11-14, 2007, Troyes, France. Available from the authors: mugnier@onera.fr.

M. Mesrine, E. Thomas, S. Garin, P. Blanc, C. Alis, F. Cassaing, and D. Laubier, “High resolution Earth observation from Geostationary orbit by optical aperture synthesis,” in Sixth International Conference on Space Optics, Proceedings of ESA/CNES ISCO 2006, A. Wilson, ed., CDROM, ESA SP-621 (2006).

E. Schmidt, F. Cassaing, S. Hofer, M. Barillot, F. Baron, L. M. Mugnier, G. Rousset, and T. Stuffler, “DARWIN Fringe Sensor (DWARF): breadboard development,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539, (ESA, 2003), pp. 575-577.

B. Sorrente, F. Cassaing, F. Baron, C. Coudrain, B. Fleury, F. Mendez, V. Michau, L. Mugnier, G. Rousset, L. Rousset-Rouvière, and M.-T. Velluet, “Multiple-aperture optical telescopes: cophasing sensor test-bed,” in Proceedings of the 5th International Conference on Space Optics, B.Warmbein, ed., Vol. SP-554 (ESA, 2004), pp. 479-484.

F. Cassaing, “Analyse d'un instrument à synthèse d'ouverture optique: méthodes de cophasage et imagerie à haute résolution angulaire,” Ph.D. thesis (Université Paris XI Orsay, 1997).

F. Cassaing, F. Baron, E. Schmidt, S. Hofer, L. M. Mugnier, M. Barillot, G. Rousset, T. Stuffler, and Y. Salvadé, “DARWIN Fringe Sensor (DWARF): Concept study,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539 (ESA, 2003), pp. 389-392.

Cederquist, J. N.

Chan, T.

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase-diversity-based blind deconvolution,” Proc. SPIE 3353, 994-1005 (1998).
[CrossRef]

Coudrain, C.

B. Sorrente, F. Cassaing, F. Baron, C. Coudrain, B. Fleury, F. Mendez, V. Michau, L. Mugnier, G. Rousset, L. Rousset-Rouvière, and M.-T. Velluet, “Multiple-aperture optical telescopes: cophasing sensor test-bed,” in Proceedings of the 5th International Conference on Space Optics, B.Warmbein, ed., Vol. SP-554 (ESA, 2004), pp. 479-484.

Duncan, A. L.

M. G. Löfdahl, A. L. Duncan, and G. B. Scharmer, “Fast phase diversity wavefront sensor for mirror control,” Proc. SPIE 3353, 952-963 (1998).
[CrossRef]

R. L. Kendrick, D. S. Acton, and A. L. Duncan, “Phase-diversity wave-front sensor for imaging systems,” Appl. Opt. 33, 6533-6546 (1994).
[CrossRef] [PubMed]

Fienup, J. R.

Fleury, B.

B. Sorrente, F. Cassaing, F. Baron, C. Coudrain, B. Fleury, F. Mendez, V. Michau, L. Mugnier, G. Rousset, L. Rousset-Rouvière, and M.-T. Velluet, “Multiple-aperture optical telescopes: cophasing sensor test-bed,” in Proceedings of the 5th International Conference on Space Optics, B.Warmbein, ed., Vol. SP-554 (ESA, 2004), pp. 479-484.

Fridlund, C. V. M.

C. V. M. Fridlund, “The DARWIN mission,” Adv. Space Res. 34, 613-617 (2004).
[CrossRef]

Fusco, T.

A. Blanc, T. Fusco, M. Hartung, L. M. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations. Application of the phase diversity technique,” Astron. Astrophys. 399, 373-383 (2003).
[CrossRef]

Garin, S.

M. Mesrine, E. Thomas, S. Garin, P. Blanc, C. Alis, F. Cassaing, and D. Laubier, “High resolution Earth observation from Geostationary orbit by optical aperture synthesis,” in Sixth International Conference on Space Optics, Proceedings of ESA/CNES ISCO 2006, A. Wilson, ed., CDROM, ESA SP-621 (2006).

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Gonsalves, R. A.

Hartung, M.

A. Blanc, T. Fusco, M. Hartung, L. M. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations. Application of the phase diversity technique,” Astron. Astrophys. 399, 373-383 (2003).
[CrossRef]

Hofer, S.

F. Cassaing, F. Baron, E. Schmidt, S. Hofer, L. M. Mugnier, M. Barillot, G. Rousset, T. Stuffler, and Y. Salvadé, “DARWIN Fringe Sensor (DWARF): Concept study,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539 (ESA, 2003), pp. 389-392.

E. Schmidt, F. Cassaing, S. Hofer, M. Barillot, F. Baron, L. M. Mugnier, G. Rousset, and T. Stuffler, “DARWIN Fringe Sensor (DWARF): breadboard development,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539, (ESA, 2003), pp. 575-577.

Idier, J.

J. Idier, L. M. Mugnier, and A. Blanc, “Statistical behavior of joint least square estimation in the phase diversity context,” IEEE Trans. Image Process. 14, 2107-2116 (2005).
[CrossRef] [PubMed]

L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elvesier, 2006), Vol. 20, pp. 1-76.

Kendrick, R. L.

Kryskowski, D.

Laubier, D.

M. Mesrine, E. Thomas, S. Garin, P. Blanc, C. Alis, F. Cassaing, and D. Laubier, “High resolution Earth observation from Geostationary orbit by optical aperture synthesis,” in Sixth International Conference on Space Optics, Proceedings of ESA/CNES ISCO 2006, A. Wilson, ed., CDROM, ESA SP-621 (2006).

Löfdahl, M. G.

M. G. Löfdahl and G. B. Scharmer, “A predictor approach for closed-loop phase diversity wavefront sensing,” Proc. SPIE 4013, 737-748 (2000).
[CrossRef]

M. G. Löfdahl, A. L. Duncan, and G. B. Scharmer, “Fast phase diversity wavefront sensor for mirror control,” Proc. SPIE 3353, 952-963 (1998).
[CrossRef]

Marron, J. C.

Mendez, F.

B. Sorrente, F. Cassaing, F. Baron, C. Coudrain, B. Fleury, F. Mendez, V. Michau, L. Mugnier, G. Rousset, L. Rousset-Rouvière, and M.-T. Velluet, “Multiple-aperture optical telescopes: cophasing sensor test-bed,” in Proceedings of the 5th International Conference on Space Optics, B.Warmbein, ed., Vol. SP-554 (ESA, 2004), pp. 479-484.

Mesrine, M.

M. Mesrine, E. Thomas, S. Garin, P. Blanc, C. Alis, F. Cassaing, and D. Laubier, “High resolution Earth observation from Geostationary orbit by optical aperture synthesis,” in Sixth International Conference on Space Optics, Proceedings of ESA/CNES ISCO 2006, A. Wilson, ed., CDROM, ESA SP-621 (2006).

Michau, V.

F. Cassaing, B. Sorrente, L. Mugnier, G. Rousset, V. Michau, I. Mocoeur, and F. Baron, “BRISE: a multipurpose bench for cophasing sensors,” Proc. SPIE 6268, 626834 (2006).

B. Sorrente, F. Cassaing, F. Baron, C. Coudrain, B. Fleury, F. Mendez, V. Michau, L. Mugnier, G. Rousset, L. Rousset-Rouvière, and M.-T. Velluet, “Multiple-aperture optical telescopes: cophasing sensor test-bed,” in Proceedings of the 5th International Conference on Space Optics, B.Warmbein, ed., Vol. SP-554 (ESA, 2004), pp. 479-484.

Mocoeur, I.

F. Cassaing, B. Sorrente, L. Mugnier, G. Rousset, V. Michau, I. Mocoeur, and F. Baron, “BRISE: a multipurpose bench for cophasing sensors,” Proc. SPIE 6268, 626834 (2006).

I. Mocoeur, L. M. Mugnier, and F. Cassaing, “Cophasage de télescopes multi pupilles sur scènes étendues par diversité de phase temps-réel,” presented at la 26ième Colloque sur le Traitement du Signal et des Images, September 11-14, 2007, Troyes, France. Available from the authors: mugnier@onera.fr.

Mugnier, L.

F. Cassaing, B. Sorrente, L. Mugnier, G. Rousset, V. Michau, I. Mocoeur, and F. Baron, “BRISE: a multipurpose bench for cophasing sensors,” Proc. SPIE 6268, 626834 (2006).

B. Sorrente, F. Cassaing, F. Baron, C. Coudrain, B. Fleury, F. Mendez, V. Michau, L. Mugnier, G. Rousset, L. Rousset-Rouvière, and M.-T. Velluet, “Multiple-aperture optical telescopes: cophasing sensor test-bed,” in Proceedings of the 5th International Conference on Space Optics, B.Warmbein, ed., Vol. SP-554 (ESA, 2004), pp. 479-484.

Mugnier, L. M.

J. Idier, L. M. Mugnier, and A. Blanc, “Statistical behavior of joint least square estimation in the phase diversity context,” IEEE Trans. Image Process. 14, 2107-2116 (2005).
[CrossRef] [PubMed]

A. Blanc, T. Fusco, M. Hartung, L. M. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations. Application of the phase diversity technique,” Astron. Astrophys. 399, 373-383 (2003).
[CrossRef]

L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elvesier, 2006), Vol. 20, pp. 1-76.

I. Mocoeur, L. M. Mugnier, and F. Cassaing, “Cophasage de télescopes multi pupilles sur scènes étendues par diversité de phase temps-réel,” presented at la 26ième Colloque sur le Traitement du Signal et des Images, September 11-14, 2007, Troyes, France. Available from the authors: mugnier@onera.fr.

F. Cassaing, F. Baron, E. Schmidt, S. Hofer, L. M. Mugnier, M. Barillot, G. Rousset, T. Stuffler, and Y. Salvadé, “DARWIN Fringe Sensor (DWARF): Concept study,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539 (ESA, 2003), pp. 389-392.

E. Schmidt, F. Cassaing, S. Hofer, M. Barillot, F. Baron, L. M. Mugnier, G. Rousset, and T. Stuffler, “DARWIN Fringe Sensor (DWARF): breadboard development,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539, (ESA, 2003), pp. 575-577.

Noll, R. J.

Paxman, R. G.

Plemmons, R.

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase-diversity-based blind deconvolution,” Proc. SPIE 3353, 994-1005 (1998).
[CrossRef]

Robinson, S. R.

Rousset, G.

F. Cassaing, B. Sorrente, L. Mugnier, G. Rousset, V. Michau, I. Mocoeur, and F. Baron, “BRISE: a multipurpose bench for cophasing sensors,” Proc. SPIE 6268, 626834 (2006).

A. Blanc, T. Fusco, M. Hartung, L. M. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations. Application of the phase diversity technique,” Astron. Astrophys. 399, 373-383 (2003).
[CrossRef]

B. Sorrente, F. Cassaing, F. Baron, C. Coudrain, B. Fleury, F. Mendez, V. Michau, L. Mugnier, G. Rousset, L. Rousset-Rouvière, and M.-T. Velluet, “Multiple-aperture optical telescopes: cophasing sensor test-bed,” in Proceedings of the 5th International Conference on Space Optics, B.Warmbein, ed., Vol. SP-554 (ESA, 2004), pp. 479-484.

E. Schmidt, F. Cassaing, S. Hofer, M. Barillot, F. Baron, L. M. Mugnier, G. Rousset, and T. Stuffler, “DARWIN Fringe Sensor (DWARF): breadboard development,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539, (ESA, 2003), pp. 575-577.

F. Cassaing, F. Baron, E. Schmidt, S. Hofer, L. M. Mugnier, M. Barillot, G. Rousset, T. Stuffler, and Y. Salvadé, “DARWIN Fringe Sensor (DWARF): Concept study,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539 (ESA, 2003), pp. 389-392.

Rousset-Rouvière, L.

B. Sorrente, F. Cassaing, F. Baron, C. Coudrain, B. Fleury, F. Mendez, V. Michau, L. Mugnier, G. Rousset, L. Rousset-Rouvière, and M.-T. Velluet, “Multiple-aperture optical telescopes: cophasing sensor test-bed,” in Proceedings of the 5th International Conference on Space Optics, B.Warmbein, ed., Vol. SP-554 (ESA, 2004), pp. 479-484.

Salvadé, Y.

F. Cassaing, F. Baron, E. Schmidt, S. Hofer, L. M. Mugnier, M. Barillot, G. Rousset, T. Stuffler, and Y. Salvadé, “DARWIN Fringe Sensor (DWARF): Concept study,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539 (ESA, 2003), pp. 389-392.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Scharmer, G. B.

M. G. Löfdahl and G. B. Scharmer, “A predictor approach for closed-loop phase diversity wavefront sensing,” Proc. SPIE 4013, 737-748 (2000).
[CrossRef]

M. G. Löfdahl, A. L. Duncan, and G. B. Scharmer, “Fast phase diversity wavefront sensor for mirror control,” Proc. SPIE 3353, 952-963 (1998).
[CrossRef]

G. B. Scharmer, “Object-independent fast phase-diversity,” in High Resolution Solar Physics: Theory, Observations and Techniques, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds., Vol. 183 of Astron. Soc. Pacific Conf. Series (1999), pp. 330-341.

Schmidt, E.

E. Schmidt, F. Cassaing, S. Hofer, M. Barillot, F. Baron, L. M. Mugnier, G. Rousset, and T. Stuffler, “DARWIN Fringe Sensor (DWARF): breadboard development,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539, (ESA, 2003), pp. 575-577.

F. Cassaing, F. Baron, E. Schmidt, S. Hofer, L. M. Mugnier, M. Barillot, G. Rousset, T. Stuffler, and Y. Salvadé, “DARWIN Fringe Sensor (DWARF): Concept study,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539 (ESA, 2003), pp. 389-392.

Schultz, T. J.

Schulz, T. J.

Seldin, J. H.

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” Proc. SPIE 4091, 48-63 (2000).
[CrossRef]

J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble space telescope characterized by using phase retrieval algorithms,” Appl. Opt. 32, 1747-1768 (1993).
[CrossRef] [PubMed]

Sorrente, B.

F. Cassaing, B. Sorrente, L. Mugnier, G. Rousset, V. Michau, I. Mocoeur, and F. Baron, “BRISE: a multipurpose bench for cophasing sensors,” Proc. SPIE 6268, 626834 (2006).

B. Sorrente, F. Cassaing, F. Baron, C. Coudrain, B. Fleury, F. Mendez, V. Michau, L. Mugnier, G. Rousset, L. Rousset-Rouvière, and M.-T. Velluet, “Multiple-aperture optical telescopes: cophasing sensor test-bed,” in Proceedings of the 5th International Conference on Space Optics, B.Warmbein, ed., Vol. SP-554 (ESA, 2004), pp. 479-484.

Stone, R. E.

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” Proc. SPIE 4091, 48-63 (2000).
[CrossRef]

Stuffler, T.

F. Cassaing, F. Baron, E. Schmidt, S. Hofer, L. M. Mugnier, M. Barillot, G. Rousset, T. Stuffler, and Y. Salvadé, “DARWIN Fringe Sensor (DWARF): Concept study,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539 (ESA, 2003), pp. 389-392.

E. Schmidt, F. Cassaing, S. Hofer, M. Barillot, F. Baron, L. M. Mugnier, G. Rousset, and T. Stuffler, “DARWIN Fringe Sensor (DWARF): breadboard development,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539, (ESA, 2003), pp. 575-577.

Thomas, E.

M. Mesrine, E. Thomas, S. Garin, P. Blanc, C. Alis, F. Cassaing, and D. Laubier, “High resolution Earth observation from Geostationary orbit by optical aperture synthesis,” in Sixth International Conference on Space Optics, Proceedings of ESA/CNES ISCO 2006, A. Wilson, ed., CDROM, ESA SP-621 (2006).

Velluet, M.-T.

B. Sorrente, F. Cassaing, F. Baron, C. Coudrain, B. Fleury, F. Mendez, V. Michau, L. Mugnier, G. Rousset, L. Rousset-Rouvière, and M.-T. Velluet, “Multiple-aperture optical telescopes: cophasing sensor test-bed,” in Proceedings of the 5th International Conference on Space Optics, B.Warmbein, ed., Vol. SP-554 (ESA, 2004), pp. 479-484.

Vogel, C. R.

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase-diversity-based blind deconvolution,” Proc. SPIE 3353, 994-1005 (1998).
[CrossRef]

Wackerman, C. C.

Walker, J. G.

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735-738 (1981).
[CrossRef]

Zarifis, V. G.

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” Proc. SPIE 4091, 48-63 (2000).
[CrossRef]

Adv. Space Res. (1)

C. V. M. Fridlund, “The DARWIN mission,” Adv. Space Res. 34, 613-617 (2004).
[CrossRef]

Appl. Opt. (4)

Astron. Astrophys. (1)

A. Blanc, T. Fusco, M. Hartung, L. M. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations. Application of the phase diversity technique,” Astron. Astrophys. 399, 373-383 (2003).
[CrossRef]

C.R. Acad. Sci. Paris, Ser. IV (1)

F. Cassaing, “Optical path difference sensors,” C.R. Acad. Sci. Paris, Ser. IV 21, 87-98 (2001).

IEEE Trans. Image Process. (1)

J. Idier, L. M. Mugnier, and A. Blanc, “Statistical behavior of joint least square estimation in the phase diversity context,” IEEE Trans. Image Process. 14, 2107-2116 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735-738 (1981).
[CrossRef]

Opt. Eng. (Bellingham) (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829-832 (1982).

Opt. Lett. (1)

Optik (Stuttgart) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Proc. SPIE (5)

M. G. Löfdahl and G. B. Scharmer, “A predictor approach for closed-loop phase diversity wavefront sensing,” Proc. SPIE 4013, 737-748 (2000).
[CrossRef]

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase-diversity-based blind deconvolution,” Proc. SPIE 3353, 994-1005 (1998).
[CrossRef]

M. G. Löfdahl, A. L. Duncan, and G. B. Scharmer, “Fast phase diversity wavefront sensor for mirror control,” Proc. SPIE 3353, 952-963 (1998).
[CrossRef]

F. Cassaing, B. Sorrente, L. Mugnier, G. Rousset, V. Michau, I. Mocoeur, and F. Baron, “BRISE: a multipurpose bench for cophasing sensors,” Proc. SPIE 6268, 626834 (2006).

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” Proc. SPIE 4091, 48-63 (2000).
[CrossRef]

Other (17)

F. Cassaing, “Analyse d'un instrument à synthèse d'ouverture optique: méthodes de cophasage et imagerie à haute résolution angulaire,” Ph.D. thesis (Université Paris XI Orsay, 1997).

F. Cassaing, F. Baron, E. Schmidt, S. Hofer, L. M. Mugnier, M. Barillot, G. Rousset, T. Stuffler, and Y. Salvadé, “DARWIN Fringe Sensor (DWARF): Concept study,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539 (ESA, 2003), pp. 389-392.

E. Schmidt, F. Cassaing, S. Hofer, M. Barillot, F. Baron, L. M. Mugnier, G. Rousset, and T. Stuffler, “DARWIN Fringe Sensor (DWARF): breadboard development,” in Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, M.Fridlund, T.Henning, and H.Lacoste, eds., Vol. SP-539, (ESA, 2003), pp. 575-577.

B. Sorrente, F. Cassaing, F. Baron, C. Coudrain, B. Fleury, F. Mendez, V. Michau, L. Mugnier, G. Rousset, L. Rousset-Rouvière, and M.-T. Velluet, “Multiple-aperture optical telescopes: cophasing sensor test-bed,” in Proceedings of the 5th International Conference on Space Optics, B.Warmbein, ed., Vol. SP-554 (ESA, 2004), pp. 479-484.

F. Baron, “Définition et test d'un capteur de cophasage sur télescope multipupilles: application à la détection d'exoplanètes et à l'observation de la Terre,” Ph.D. thesis (Ecole Doctorale d'Astronomie et d'Astrophysique d'Ile de France, 2005).

I. Mocoeur, L. M. Mugnier, and F. Cassaing, “Cophasage de télescopes multi pupilles sur scènes étendues par diversité de phase temps-réel,” presented at la 26ième Colloque sur le Traitement du Signal et des Images, September 11-14, 2007, Troyes, France. Available from the authors: mugnier@onera.fr.

G. B. Scharmer, “Object-independent fast phase-diversity,” in High Resolution Solar Physics: Theory, Observations and Techniques, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds., Vol. 183 of Astron. Soc. Pacific Conf. Series (1999), pp. 330-341.

A. Blanc, “Identification de réponse impulsionnelle et restauration d'images: apports de la diversité de phase,” Ph.D. thesis (Université Paris XI Orsay, 2002).

L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elvesier, 2006), Vol. 20, pp. 1-76.

VLTI website, http://www.eso.org/projects/vlti.

KECK website, http://planetquest.jpl.nasa.gov/Keck/keck_index.html.

OHANA website, http://www.cfht.hawaii.edu/~lai/ohana.html.

DARWIN website, http://sci.esa.int/science-e/www/area/index.cfm?fareaid=28.

TPF website, http://planetquest.jpl.nasa.gov/TPF/tpf_index.html.

JWST website, http://www.jwst.nasa.gov.

M. Mesrine, E. Thomas, S. Garin, P. Blanc, C. Alis, F. Cassaing, and D. Laubier, “High resolution Earth observation from Geostationary orbit by optical aperture synthesis,” in Sixth International Conference on Space Optics, Proceedings of ESA/CNES ISCO 2006, A. Wilson, ed., CDROM, ESA SP-621 (2006).

NAOS website, http://www.eso.org/instruments/naco.

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Figures (13)

Fig. 1
Fig. 1

Piston and tip/tilt on a MAOT.

Fig. 2
Fig. 2

Shapes of the OTF modulus and phase for no aberration (first row), a piston (second row), a tilt (third row) on one aperture.

Fig. 3
Fig. 3

Responses of FUSCHIA and of the iterative estimator to a piston ramp (a) and a tilt ramp (b). The tilt estimation is slightly biased for FUSCHIA in the linear domain due to the approximated nature of the algorithm (c).

Fig. 4
Fig. 4

Error and standard deviation on piston estimates for FUSCHIA and the iterative estimator (no aperture aberrations).

Fig. 5
Fig. 5

Error and standard deviation on tilt estimates for FUSCHIA and the iterative estimator (no aperture aberrations).

Fig. 6
Fig. 6

Error and standard deviation on piston estimates for FUSCHIA and the iterative estimator (aperture aberrations of λ 6 ).

Fig. 7
Fig. 7

Error, standard deviation, and bias on tilt estimates for FUSCHIA and the iterative estimator (aperture aberrations of λ 6 ).

Fig. 8
Fig. 8

Agreement between the experimental (right column) and the direct model (center column) in response to a given perturbation (left column).

Fig. 9
Fig. 9

Bench stability: (a) PSFs obtained with the planar (top) and deformable mirror (bottom), with the corresponding temporal evolution (b) and (c).

Fig. 10
Fig. 10

Linearity of piston estimators: (a) PSFs obtained during a piston scan, from λ c to λ c 2 ; (b) piston estimated with FUSCHIA for different bandwidths; (c) comparison between FUSCHIA and the iterative algorithm for the 40 nm bandwidth.

Fig. 11
Fig. 11

Piston repeatability estimated with FUSCHIA and the iterative algorithm.

Fig. 12
Fig. 12

Linearity of tilt estimators: (a) PSFs obtained during a tilt scan, here illustrated from λ c 3 to + λ c 7 ; (b) tilt estimated with FUSCHIA for different bandwidths; (c) comparison between FUSCHIA and the iterative algorithm for the 40 nm bandwidth.

Fig. 13
Fig. 13

Tilt repeatability estimated with FUSCHIA and the iterative algorithm.

Equations (49)

Equations on this page are rendered with MathJax. Learn more.

p = n = 1 N T p n δ u n ,
Π ( u ) = { 1 for 0 u R 0 elsewhere .
p n ( u ) = Π ( u ) exp [ j k = 1 k max a k n Z k ( u ) ] ,
s ( u ) = ( p p ) ( u ) = u R 2 p ( u ) p * ( u + u ) ,
s = n = 1 N T n = 1 N T ( p n δ u n ) ( p n δ u n ) = n = 1 N T n = 1 N T ( p n p n ) δ u n u n .
s n = 1 N T n = 1 N T Λ exp ( j k = 1 3 α n n k Z k ) δ u n u n ,
α n n k a k n + ϵ k a k n ,
Z k ( u ) 1 2 u S ( u ) { Z k ( u ) ϵ k Z k ( u + u ) } d u ,
L ( a ) = u D 1 2 σ 2 s ( a , u ) i ̃ ( u ) 2 ,
( k , n ) L a k n ( a ̂ ) = 0 .
L a k n = 1 σ 2 u D R { ( s i ̃ ) s * a k n } .
L ( a ) = u D \ D 0 1 2 σ 2 s ( a , u ) i ̃ ( u ) 2 .
L a k n = 1 σ 2 u D \ D 0 R { ( s i ̃ ) s * a k n } .
s k n = n = 1 n n N T { [ Λ exp ( j k = 1 3 α n n k Z k ) ] δ u n u n + [ Λ exp ( j k = 1 3 α n n k Z k ) ] δ u n u n } .
s * a k n = j n = 1 n n N T { [ Z k Λ exp ( j k = 1 3 α n n k Z k ) ] δ u n u n + [ ϵ k Z k Λ exp ( j k = 1 3 α n n k Z k ) ] δ u n u n } ,
s s * a k n = j n = 1 n n N T ( Z k Λ 2 δ u n u n + ϵ k Z k Λ 2 δ u n u n ) .
L a k n = 1 σ 2 u D \ D 0 R { i ̃ ( u ) s * a k n ( u ) } .
( k , n ) , L a k n ( α k n ̂ ) = 0 .
u D \ D 0 I { i ̃ n = 1 n n N T [ Z k Λ exp ( j k = 1 3 α n n k ̂ Z k ) ] δ u n u n + [ ϵ k Z k Λ exp ( j k = 1 3 α n n k ̂ Z k ) ] δ u n u n } ( u ) = 0 .
n , u D \ D 0 I { i ̃ n = 1 n n N T [ Z k Λ exp ( j k = 1 3 α n n k ̂ Z k ) ] δ u n u n } ( u ) = 0 .
β n k = n = 1 n n N T α n n k
a k = 1 N T n = 1 N T a k n .
β n 1 = N T ( a 1 n a 1 ) .
a 1 n = a 1 + β n 1 N T .
n , n = 1 n n N T I { exp ( j α n n 1 ̂ ) u D n n i ̃ ( u ) Λ ( u u n + u n ) } = 0 .
n , n n Arg [ exp ( j α n n 1 ̂ ) u D n n i ̃ ( u ) Λ ( u u n + u n ) ] = 0 [ mod π ] ,
n , n n α n n 1 ̂ + Arg [ u D n n i ̃ ( u ) Λ ( u u n + u n ) ] = 0 [ mod π ] .
a ̂ 1 n = 1 N T n = 1 n n N T Arg [ u D n n i ̃ ( u ) Λ ( u u n + u n ) ] [ mod 2 π ] .
n , n = 1 n n N T u D n n I { [ Λ 2 Z k ] ( u u n + u n ) [ 1 + j Φ ( u ) j k = 1 3 α ̂ n n k Z k ( u u n + u n ) ] } 0 ,
n , n = 1 n n N T u D n n Φ ( u ) [ Λ 2 Z k ] ( u u n + u n ) n = 1 n n N T k = 1 3 α ̂ n n k [ u D n n [ Λ 2 Z k Z k ] ( u u n + u n ) ] .
n , β n k ̂ = n = 1 n n N T α ̂ n n k n = 1 n n N T u D n n Φ ( u ) [ Λ 2 Z k ] ( u u n + u n ) n = 1 n n N T u D n n [ Λ 2 Z k 2 ] ( u u n + u n ) .
β n k = ( N T 2 ) a k n + N T a k .
n = 1 N T β n k = 2 N T ( N T 1 ) a k .
a ̂ k n = β n k ̂ N T 2 1 2 ( N T 1 ) ( N T 2 ) n = 1 N T β n k ̂ .
σ = λ 2 π γ SNR n pix
SNR = N n pix N n pix + σ d 2 ,
( p n p n ) ( u ) = u R 2 Π ( u ) Π ( u + u ) exp j [ ϕ n ( u ) ϕ n ( u + u ) ] d u .
Φ n n ( u ) = u R 2 Π ( u ) Π ( u + u ) [ ϕ n ( u ) ϕ n ( u + u ) ] d u = u S ( u ) [ ϕ n ( u ) ϕ n ( u + u ) ] d u ,
Λ ( u ) = ( Π Π ) ( u ) = u R 2 Π ( u ) Π ( u + u ) d u = u S ( u ) 1 d u .
( p n p n ) ( u ) = Λ ( u ) exp [ j Φ n n ( u ) ] .
Φ n n ( u ) = u S ( u ) k = 1 3 [ a k n Z k ( u ) a k n Z k ( u + u ) ] d u
= u S ( u ) k = 1 3 { a k n a k n 2 [ Z k ( u ) + Z k ( u + u ) ] + a k n + a k n 2 [ Z k ( u ) Z k ( u + u ) ] } d u
= k = 1 3 ( α n n k + Z k + α n n k Z k + ) ( u ) ,
α n n k ± = a k n ± a k n ,
Z k ± ( u ) = 1 2 u S ( u ) [ Z k ( u ) ± Z k ( u + u ) ] d u .
Z k = Z k + , α n n k = α n n k .
Z k = Z k , α n n k = α n n k + .
s n = 1 N T n = 1 N T Λ exp ( j k = 1 3 α n n k Z k ) δ u n u n .
p = n = 1 N T Π exp ( j k = 1 3 a k n Z k ) δ u n .

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