# American Institute of Mathematical Sciences

November  2011, 5(4): 893-913. doi: 10.3934/ipi.2011.5.893

## Cumulative wavefront reconstructor for the Shack-Hartmann sensor

 1 Industrial Mathematics Institute, Johannes Kepler University Linz, A-4040 Linz, Austria, Austria 2 Industrial Mathematics Institute, Johannes Kepler University Linz A-4040 Linz 3 MathConsult GmbH, Altenbergerstrae 69, A-4040 Linz, Austria

Received  November 2010 Revised  September 2011 Published  November 2011

We present a new direct algorithm aiming at the reconstruction of the optical wavefront from the Shack-Hartmann sensor measurements in Single Conjugate Adaptive Optics (SCAO) systems. The objective of an adaptive optics system designed for a large telescope can be only achieved if the wavefront reconstruction is sufficiently fast. Our scheme does not contain any explicit regularization for the reconstruction process but is still able to provide a good quality of reconstruction. The analysis of quality is given for three varying parameters: the diameter of the telescope, the number of subapertures and the level of photon noise. It has been shown both analytically and numerically that the quality of the reconstruciton, measured by the Strehl ratio, is reasonable for the small photon noise level and increases with the increasing number of subapertures for the same telescope size. The impact of the photon noise on the reconstruction gets higher with the increasing telescope diameter. The computational complexity of the method is linear in the number of unkowns. Counting all summation and multiplication steps the scaling factor is $14$. Moreover, due to its simple structure, the cumulative reconstructor is pipelinable and parallelizable, which makes the effective computation even faster.
Citation: Mariya Zhariy, Andreas Neubauer, Matthias Rosensteiner, Ronny Ramlau. Cumulative wavefront reconstructor for the Shack-Hartmann sensor. Inverse Problems and Imaging, 2011, 5 (4) : 893-913. doi: 10.3934/ipi.2011.5.893
##### References:
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##### References:
 [1] J. M. Beckers, Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics, in "Proc. European Southern Observatory Conf. and Workshop on Very Large Telescopes and Their Instrumentation'' (ed. M. H. Ulrich), Vol. 30, (1988), 693-703. [2] M. A. Davison, The ill-conditioned nature of the limited angle tomography problem, SIAM J. Appl. Math., 43 (1983), 428-448. doi: 10.1137/0143028. [3] B. L. Ellerbroek, Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques, J. Opt. Soc. Am., 19 (2002), 1803-1816. doi: 10.1364/JOSAA.19.001803. [4] B. L. Ellerbroek and C. R. Vogel, Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes, in "Proc. SPIE vol 5169-23," Adaptive Optics System Technologies II, (2003), 206-217. doi: 10.1117/12.506580. [5] B. L. Ellerbroek and C. R. Vogel, Inverse problems in astronomical adaptive optics, Inverse Problems, 25 (2009), 063001, 37 pp. [6] H. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'' Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996. [7] D. L. Fried, Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements, J. Opt. Soc. Am., 67 (1977), 370-375. doi: 10.1364/JOSA.67.000370. [8] L. Gilles, Order $N$ sparse minimum-variance open-loop reconstructor for extreme adaptive optics, Opt. Lett., 28 (2003), 1927-1929. doi: 10.1364/OL.28.001927. [9] L. Gilles, Closed-loop stability and performance analysis of least-squares and minimum-variance control algorithms for multi-conjugate adaptive optics, Appl. Opt., 44 (2004), 993-1002. doi: 10.1364/AO.44.000993. [10] L. Gilles, C. R. Vogel and B. L. Ellebroek, Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction, J. Opt. Soc. Am. A, 19 (2002), 1817-1822. doi: 10.1364/JOSAA.19.001817. [11] J. Herrmann, Least-squares wave front errors of minimum norm, J. Opt. Soc. Am., 70 (1980), 28-35. doi: 10.1364/JOSA.70.000028. [12] A. Neubauer, On the ill-posedness and convergence of the Shack-Hartmann based wavefront reconstruction, J. Inv. Ill-Posed Problems, 18 (2010), 551-576. doi: 10.1515/JIIP.2010.025. [13] L. A. Poyneer, D. T. Gavel and J. M. Brase, Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform, J. Opt. Soc. Am. A, 19 (2002), 2100-2111. doi: 10.1364/JOSAA.19.002100. [14] L. A. Poyneer and J.-P. Véran, Optimal modal Fourier transform wave-front control, J. Opt. Soc. Am. A, 22 (2005), 1515-1526. doi: 10.1364/JOSAA.22.001515. [15] F. Roddier, "Adaptive Optics in Astronomy,'' Cambridge University Press, Cambridge, New York, 1999. doi: 10.1017/CBO9780511525179. [16] M. C. Roggemann and B. M. Welsh, "Imaging Through Turbulence,'' CRC Press, Boca Raton, Florida, 1996. [17] E. Thiébaut and M. Tallon, Fast minimum variance wavefront reconstruction for extremely large telescopes, J. Opt. Soc. Am. A, 27 (2010), 1046-1059. doi: 10.1364/JOSAA.27.001046. [18] C. R. Vogel and Q. Yang, Multigrid algorithm for least-squares wavefront reconstruction, Applied Optics, 45 (2006), 705-715. doi: 10.1364/AO.45.000705. [19] Q. Yang, C. R. Vogel and B. L. Ellerbroek, Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography, Applied Optics, 45 (2006), 5281-5293. doi: 10.1364/AO.45.005281.
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