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 & Imaging, 2011, 5 (4) : 893-913. doi: 10.3934/ipi.2011.5.893
References:
[1]

J. M. Beckers, Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics,, in, (1988), 693.   Google Scholar

[2]

M. A. Davison, The ill-conditioned nature of the limited angle tomography problem,, SIAM J. Appl. Math., 43 (1983), 428.  doi: 10.1137/0143028.  Google Scholar

[3]

B. L. Ellerbroek, Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,, J. Opt. Soc. Am., 19 (2002), 1803.  doi: 10.1364/JOSAA.19.001803.  Google Scholar

[4]

B. L. Ellerbroek and C. R. Vogel, Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,, in, (2003), 5169.  doi: 10.1117/12.506580.  Google Scholar

[5]

B. L. Ellerbroek and C. R. Vogel, Inverse problems in astronomical adaptive optics,, Inverse Problems, 25 (2009).   Google Scholar

[6]

H. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'', Mathematics and its Applications, 375 (1996).   Google Scholar

[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.  doi: 10.1364/JOSA.67.000370.  Google Scholar

[8]

L. Gilles, Order $N$ sparse minimum-variance open-loop reconstructor for extreme adaptive optics,, Opt. Lett., 28 (2003), 1927.  doi: 10.1364/OL.28.001927.  Google Scholar

[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.  doi: 10.1364/AO.44.000993.  Google Scholar

[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.  doi: 10.1364/JOSAA.19.001817.  Google Scholar

[11]

J. Herrmann, Least-squares wave front errors of minimum norm,, J. Opt. Soc. Am., 70 (1980), 28.  doi: 10.1364/JOSA.70.000028.  Google Scholar

[12]

A. Neubauer, On the ill-posedness and convergence of the Shack-Hartmann based wavefront reconstruction,, J. Inv. Ill-Posed Problems, 18 (2010), 551.  doi: 10.1515/JIIP.2010.025.  Google Scholar

[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.  doi: 10.1364/JOSAA.19.002100.  Google Scholar

[14]

L. A. Poyneer and J.-P. Véran, Optimal modal Fourier transform wave-front control,, J. Opt. Soc. Am. A, 22 (2005), 1515.  doi: 10.1364/JOSAA.22.001515.  Google Scholar

[15]

F. Roddier, "Adaptive Optics in Astronomy,'', Cambridge University Press, (1999).  doi: 10.1017/CBO9780511525179.  Google Scholar

[16]

M. C. Roggemann and B. M. Welsh, "Imaging Through Turbulence,'', CRC Press, (1996).   Google Scholar

[17]

E. Thiébaut and M. Tallon, Fast minimum variance wavefront reconstruction for extremely large telescopes,, J. Opt. Soc. Am. A, 27 (2010), 1046.  doi: 10.1364/JOSAA.27.001046.  Google Scholar

[18]

C. R. Vogel and Q. Yang, Multigrid algorithm for least-squares wavefront reconstruction,, Applied Optics, 45 (2006), 705.  doi: 10.1364/AO.45.000705.  Google Scholar

[19]

Q. Yang, C. R. Vogel and B. L. Ellerbroek, Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,, Applied Optics, 45 (2006), 5281.  doi: 10.1364/AO.45.005281.  Google Scholar

show all references

References:
[1]

J. M. Beckers, Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics,, in, (1988), 693.   Google Scholar

[2]

M. A. Davison, The ill-conditioned nature of the limited angle tomography problem,, SIAM J. Appl. Math., 43 (1983), 428.  doi: 10.1137/0143028.  Google Scholar

[3]

B. L. Ellerbroek, Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,, J. Opt. Soc. Am., 19 (2002), 1803.  doi: 10.1364/JOSAA.19.001803.  Google Scholar

[4]

B. L. Ellerbroek and C. R. Vogel, Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,, in, (2003), 5169.  doi: 10.1117/12.506580.  Google Scholar

[5]

B. L. Ellerbroek and C. R. Vogel, Inverse problems in astronomical adaptive optics,, Inverse Problems, 25 (2009).   Google Scholar

[6]

H. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'', Mathematics and its Applications, 375 (1996).   Google Scholar

[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.  doi: 10.1364/JOSA.67.000370.  Google Scholar

[8]

L. Gilles, Order $N$ sparse minimum-variance open-loop reconstructor for extreme adaptive optics,, Opt. Lett., 28 (2003), 1927.  doi: 10.1364/OL.28.001927.  Google Scholar

[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.  doi: 10.1364/AO.44.000993.  Google Scholar

[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.  doi: 10.1364/JOSAA.19.001817.  Google Scholar

[11]

J. Herrmann, Least-squares wave front errors of minimum norm,, J. Opt. Soc. Am., 70 (1980), 28.  doi: 10.1364/JOSA.70.000028.  Google Scholar

[12]

A. Neubauer, On the ill-posedness and convergence of the Shack-Hartmann based wavefront reconstruction,, J. Inv. Ill-Posed Problems, 18 (2010), 551.  doi: 10.1515/JIIP.2010.025.  Google Scholar

[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.  doi: 10.1364/JOSAA.19.002100.  Google Scholar

[14]

L. A. Poyneer and J.-P. Véran, Optimal modal Fourier transform wave-front control,, J. Opt. Soc. Am. A, 22 (2005), 1515.  doi: 10.1364/JOSAA.22.001515.  Google Scholar

[15]

F. Roddier, "Adaptive Optics in Astronomy,'', Cambridge University Press, (1999).  doi: 10.1017/CBO9780511525179.  Google Scholar

[16]

M. C. Roggemann and B. M. Welsh, "Imaging Through Turbulence,'', CRC Press, (1996).   Google Scholar

[17]

E. Thiébaut and M. Tallon, Fast minimum variance wavefront reconstruction for extremely large telescopes,, J. Opt. Soc. Am. A, 27 (2010), 1046.  doi: 10.1364/JOSAA.27.001046.  Google Scholar

[18]

C. R. Vogel and Q. Yang, Multigrid algorithm for least-squares wavefront reconstruction,, Applied Optics, 45 (2006), 705.  doi: 10.1364/AO.45.000705.  Google Scholar

[19]

Q. Yang, C. R. Vogel and B. L. Ellerbroek, Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,, Applied Optics, 45 (2006), 5281.  doi: 10.1364/AO.45.005281.  Google Scholar

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