Cophasing a Phased-Array Telescope by Real-Time Phase Diversity Method
Estimation des Défauts D’Alignement d’un Instrument Multipupille par Diversité de Phase Temps-Réel
OPEN ACCESS
Optical interferometry allows to reach the resolution of a large instrument by coupling several subapertures of lower diameter. However, this method requires a very accurate control of the optical path, especially the differential aberrations between the apertures (so-called “cophasing” operation). In this context, focal-plane sensing, based on aberration retrieval from focal-plane intensity measurements, appears to be an advantageous solution with a very simple opto-mechanical setup.
When the aperture function is known, differential aberrations can be determined with a single focal image of an unresolved source source using a non centro-symmetric pupil. However, in a general case (extended object or any aperture configuration), the sole focal-plane image is not sufficient to retrieve piston and tip/tilt without ambiguity. The Phase Diversity method introduced in optic by Gonsalves [Gonsalves (1982)] is then usually applied, using at least a second
image differing by a known phase, conventionally a small defocus. This method is now routinely used for the calibration of monolithic instruments. However, the corresponding algorithms are iterative and consequently time-consuming. The objective of this paper is to present a new algorithm which is analytic and consequently well-suited for real-time cophasing on extended object. To do that, we first begin to explain the principle of Phase Diversity. During these last20 years, different methods have been developped to retrieve both the aberrations â and the observed object ô [Mugnier et al. (2006)]. The conventional processing is based on the joint a posteriori estimation of (â; ô) that are most compatible with the measurements by using statistical information on the data [Blanc (2002)]. The corresponding criterion Eq. (8) and Eq. (10) is convenienly minimized by an iterative conjugated gradient method ; but although this classical iterative estimator is optimal in term of performance, it is time consuming since it requires at least ten iterations to converge, each iteration representing 2Nd FFT to compute (where Nd is the number of images acquired by diversity).
In order to have an analytic Phase Diverity algorithm, we express the classical criterion under the small phase assumption. Then :
– we approximate the denominator of Eq. (11) at a = 0 ;
– we linearize the expression of its numerator, obtaining an affine expression according to the aberrations.
The new expression of the criterion Eq. (13) is then quadratic and the derivation of its gradient with respect to the aberrations lead to a unique solution as shown Eq. (15).
The resulting algorithm is much faster than the iterative, requiring only Nd FFT. In order to sudy its properties, various simulations are made. It appears that our new algorithm can estimate piston aberrations ≤|π/2| rad RMS with an error of λ/60 (figure [3]). Furthermore, it can be used to restore the observed object as well for a signal-to-noise ratio > 10 (figure [6]). In conclusion, our results demonstrate that the analytic algorithm is appropriate for the cophasing of phased array telescopes in closed-loop.
Résumé
Le cophasage d’un instrument multipupille nécessite la mesure des aberrations spécifiques que sont les pistons et basculements différentiels entre les sous-pupilles. La Diversité de Phase est une méthode qui s’avère être bien adaptée pour estimer de telles aberrations, notamment sur objet étendu ; toutefois, les algorithmes associés requièrent des temps de calcul importants et sont incompatibles avec les besoins temps-réel d’un système fonctionnant en boucle fermée. Dans cet article, nous démontrons qu’un estimateur analytique peut être obtenu dans l’hypothèse des faibles phases aberrantes grâce à une approximation judicieuse qui rend le critère à minimiser quadratique. Nous validons ensuite cet estimateur par simulation et comparons ses performances à celles obtenues avec un algorithme itératif conventionnel. Les résultats obtenus démontrent qu’il est tout à fait possible de fermer une boucle de cophasage à faible flux et par la même occasion de restaurer l’objet observé dans un but d’imagerie.
Interferometry, Inverse problems, Optical transfer functions, Phase Diversity, Wave-front sensing.
Mots clés
Interférométrie, Problèmes inverses, Fonction de transfert optique, Diversité de Phase, Analyse de front d’onde.
[Baron et al. (2008)] F. BARON, I. MOCOEUR, F. CASSAING, et L. MUGNIER, « Unambiguous phase retrieval as a cophasing sensor for phased array telescopes. » J. Opt. Soc. Am. A, Vol. 20 # 5, p. 1000-1015, 2008.
[Blanc et al. (2003)] A. BLANC, L. M. MUGNIER, et J. IDIER, «Marginal estimation of aberrations and image Restoration by use of phase diversity.» J. Opt. Soc. Am. A, Vol. 20 # 6, p. 1035-1045, 2003.
[Blanc (2002)] A. BLANC, Identification de réponse impulsionnelle et restauration d’images : apports de la diversité de phase. Thèse de doctorat, Université Paris XI Orsay, 2002.
[Conan et al. (1998)] J. M. CONAN, L. M. MUGNIER, T. FUSCO, V. MICHAU, et G. ROUSSET, «Myopic Deconvolution of Adaptive Optics Images by use of Object and Point Spread Function Power Spectra. » Appl. Opt. Vol. 37 # 21, p. 4614-4622, 1998.
[Fridlund (2004)] C. V. M. FRIDLUND, « The Darwin mission. » Adv. in Sp. R.,Vol. 34, p. 613-617.
[Ghiglia et Pritt (1998)] D. C. GHIGLIA, et M. D. PRITT, 1998. Two- Dimensional Phase Unwrapping: Theory, Algorithms, and Software. Wiley-Interscience, 2004.
[Gilmozzi et Spyromilio (2007)] R. GILMOZZI, et J. SPYROMILIO, «The European Extremely Large Telescope (E-ELT).» ESO Messenger, Vol. 127, p. 11-19, 2007.
[Gonsalves (1982)] R. A. GONSALVES, « Phase retrieval and diversity in adaptive optics.», Opt. Eng. Vol. 21 # 5, p. 829-832, 1982.
[Gratadour et al. (2006)] D. GRATADOUR, D. ROUAN, L. M. MUGNIER, T. FUSCO,Y. CLENET, E. GENDRON, et F. LACOMBE, « Near-IR AO dissection of the core of NGC 1068 with NaCo . » Astron. Astrophys. Vol. 446 # 3, p. 813-825, 2006.
[Harvey et Ftaclas (1990)] J. E. HARVEY, et C. FTACLAS, « Fundamental limitations on off-axis performance of phased telescope arrays. » Advanced technology optical telescopes IV, (L. D. BARR, ed.), SPIE proceedings series, vol. 1236, p. 390-405, Soc. Photo-Opt. Instrum. Eng, 1990.
[Idier et al. (2005)] J. IDIER, L. MUGNIER, et A. BLANC, 2005. « Statistical behavior of joint least square estimation in the phase diversity context. », IEEE Trans. Image Processing, vol. 14 #12, p. 2107-2116.
[Mocoeur et al. (2006)] I. MOCOEUR, F. CASSAING, F. BARON, S. HOFER, et H. THIELE, «DARWIN fringe sensor : experimental results on the BRISE bench. », Advances in Stellar Interferometry (J. D. MONNIER, M. SCHOLLER, et W.C. DANCHI, eds.), SPIE proceedings series, vol. 6268, Soc. Photo-Opt. Instrum. Eng, 2006.
[Mugnier et al. (2006)] L. M. MUGNIER, A. BLANC, et J. IDIER, « Phase Diversity: a Technique for Wave-Front Sensing and for Diffraction-Limited Imaging.», Advances in Imaging and Electron Physics, (P. HAWKES, ed.), vol. 141, chapitre 1, p .1-76, Elsevier, 2006.
[Mugnier et al. (2005)] L. MUGNIER, F. CASSAING, G. ROUSSET, F. BARON, V. MICHAU, I. MOCOEUR, B. SORRENTE, et M.-T. VELLUET, « Continuous High-Resolution Earth Observation with Multiple Aperture Optical Telescopes.», Proceedings of the OPTRO 2005 International Symposium, Assoc. Aéro. et Astro. de France, 2005.
[Mugnier et al. (2004)] L. MUGNIER, F. CASSAING, B. SORRENTE, F. BARON, M.-T. VELLUET, V. MICHAU, et G. ROUSSET, « Multiple-Aperture Optical Telescopes: some key issues for Earth observation from a GEO orbit. », 5th International Conference On Space Optics, Vol. SP-554, p. 181-187, CNES/ESA, 2004.
[Noll (1976)] R. J. NOLL, « Zernike polynomials and atmospheric turbulence.», J. Opt. Soc. Am., Vol. 66 #3, p. 207-211, 1976.
[Seldin et al., (2006)] J. H. SELDIN, R. G. PAXMAN, V. G. ZARIFIS, L. BENSON, et R. E. STONE, « Closed-loop wavefront sensing for a sparse-aperture multitelescope array using broadband phase diversity.», Imaging Technology and Telescopes (J. W. BILBRO, J. B. BRECKINRIDGE, R. A. CARRERAS, S. R. CZYZAK, M. J. ECKART, R. D. FIETE, et P. S. IDELL, eds.), SPIE proceedings series, vol. 4091, Soc. Photo-Opt. Instrum. Eng, 2000.