Abstract

A scalable sparse minimum-variance open-loop wave-front reconstructor for extreme adaptive optics (ExAO) systems is presented. The reconstructor is based on Ellerbroek’s sparse approximation of the wave-front inverse covariance matrix [J. Opt. Soc. Am. A 19, 1803 (2002)]. The baseline of the numerical approach is an iterative conjugate gradient (CG) algorithm for reconstructing a spatially sampled wave front at N grid points on a computational domain of size equal to the telescope’s primary mirror’s diameter D that uses a multigrid (MG) accelerator to speed up convergence efficiently and enhance its robustness. The combined MGCG scheme is order N and requires only two CG iterations to converge to the asymptotic average Strehl ratio (SR) and root-mean-square reconstruction error. The SR and reconstruction squared error are within standard deviation with figures obtained from a previously proposed MGCG fast-Fourier-transform based minimum-variance reconstructor that incorporates the exact wave-front inverse covariance matrix on a computational domain of size equal to 2D. A cost comparison between the present sparse MGCG algorithm and a Cholesky factorization based algorithm that uses a reordering scheme to preserve sparsity indicates that the latter method is still competitive for real-time ExAO wave-front reconstruction for systems with up to N104 degrees of freedom because the update rate of the Cholesky factor is typically several orders of magnitude lower than the temporal sampling rate.

© 2003 Optical Society of America

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References

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  1. M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  2. L. Gilles, C. R. Vogel, and B. L. Ellerbroek, J. Opt. Soc. Am. A 19, 1817 (2002).
    [CrossRef]
  3. L. A. Poyneer, D. T. Gavel, and J. M. Brase, J. Opt. Soc. Am. A 19, 2100 (2002).
    [CrossRef]
  4. L. A. Poyneer, M. Troy, B. Macintosh, and D. T. Gavel, Opt. Lett. 28, 798 (2003).
    [CrossRef] [PubMed]
  5. D. G. MacMartin, J. Opt. Soc. Am. A 20, 1084 (2003).
    [CrossRef]
  6. B. L. Ellerbroek, “Comparison of least squares and minimal variance wavefront reconstruction for turbulence compensation in the presence of noise,” Rep. TR721R (Optical Sciences Company, Anaheim, Calif., 1986).
  7. B. L. Ellerbroek, J. Opt. Soc. Am. A 19, 1803 (2002).
    [CrossRef]
  8. U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid (Academic, San Diego, Calif., 2001).
  9. Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (PWS-Kent, Boston, Mass., 2003).
    [CrossRef]
  10. L. Gilles, B. L. Ellerbroek, and C. R. Vogel, Appl. Opt. 42, 5233 (2003).
    [CrossRef] [PubMed]

2003 (3)

2002 (3)

L. Gilles, C. R. Vogel, and B. L. Ellerbroek, J. Opt. Soc. Am. A 19, 1817 (2002).
[CrossRef]

L. A. Poyneer, D. T. Gavel, and J. M. Brase, J. Opt. Soc. Am. A 19, 2100 (2002).
[CrossRef]

B. L. Ellerbroek, J. Opt. Soc. Am. A 19, 1803 (2002).
[CrossRef]

Brase, J. M.

L. A. Poyneer, D. T. Gavel, and J. M. Brase, J. Opt. Soc. Am. A 19, 2100 (2002).
[CrossRef]

Ellerbroek, B. L.

L. Gilles, B. L. Ellerbroek, and C. R. Vogel, Appl. Opt. 42, 5233 (2003).
[CrossRef] [PubMed]

B. L. Ellerbroek, J. Opt. Soc. Am. A 19, 1803 (2002).
[CrossRef]

L. Gilles, C. R. Vogel, and B. L. Ellerbroek, J. Opt. Soc. Am. A 19, 1817 (2002).
[CrossRef]

B. L. Ellerbroek, “Comparison of least squares and minimal variance wavefront reconstruction for turbulence compensation in the presence of noise,” Rep. TR721R (Optical Sciences Company, Anaheim, Calif., 1986).

Gavel, D. T.

L. A. Poyneer, M. Troy, B. Macintosh, and D. T. Gavel, Opt. Lett. 28, 798 (2003).
[CrossRef] [PubMed]

L. A. Poyneer, D. T. Gavel, and J. M. Brase, J. Opt. Soc. Am. A 19, 2100 (2002).
[CrossRef]

Gilles, L.

L. Gilles, B. L. Ellerbroek, and C. R. Vogel, Appl. Opt. 42, 5233 (2003).
[CrossRef] [PubMed]

L. Gilles, C. R. Vogel, and B. L. Ellerbroek, J. Opt. Soc. Am. A 19, 1817 (2002).
[CrossRef]

Macintosh, B.

MacMartin, D. G.

Oosterlee, C. W.

U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid (Academic, San Diego, Calif., 2001).

Poyneer, L. A.

L. A. Poyneer, M. Troy, B. Macintosh, and D. T. Gavel, Opt. Lett. 28, 798 (2003).
[CrossRef] [PubMed]

L. A. Poyneer, D. T. Gavel, and J. M. Brase, J. Opt. Soc. Am. A 19, 2100 (2002).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Saad, Y.

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (PWS-Kent, Boston, Mass., 2003).
[CrossRef]

Schüller, A.

U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid (Academic, San Diego, Calif., 2001).

Trottenberg, U.

U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid (Academic, San Diego, Calif., 2001).

Troy, M.

Vogel, C. R.

L. Gilles, B. L. Ellerbroek, and C. R. Vogel, Appl. Opt. 42, 5233 (2003).
[CrossRef] [PubMed]

L. Gilles, C. R. Vogel, and B. L. Ellerbroek, J. Opt. Soc. Am. A 19, 1817 (2002).
[CrossRef]

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

L. Gilles, C. R. Vogel, and B. L. Ellerbroek, J. Opt. Soc. Am. A 19, 1817 (2002).
[CrossRef]

L. A. Poyneer, D. T. Gavel, and J. M. Brase, J. Opt. Soc. Am. A 19, 2100 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Other (4)

M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

B. L. Ellerbroek, “Comparison of least squares and minimal variance wavefront reconstruction for turbulence compensation in the presence of noise,” Rep. TR721R (Optical Sciences Company, Anaheim, Calif., 1986).

U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid (Academic, San Diego, Calif., 2001).

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (PWS-Kent, Boston, Mass., 2003).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Top left, computational cost of the sparse MGCG reconstructor. The phase grid mesh size is Δx=r0/3=0.16/3 m, and the Shack–Hartmann WFS subaperture size is d=Δx. Systems considered have N=332,652,1292,2572,5132 phase grid points, 80% of them inside the aperture domain. The corresponding telescope primary mirrors are of sizes D=1.7,3.4,6.8,13.6,27.3 m. Top right, rms reconstruction error in nm over 40 simulation runs. Two noise levels are considered: σpd=π/2 rad and σpd=π/8 rad rms phase difference. Bottom left, average SR over 40 simulation runs. Bottom right, cross section of the average point-spread function (PSF) core as a function of field of view (FoV) angle in units of λ/D (DL, diffraction limited).

Equations (4)

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s=Gϕ+η,
ϕˆ=Rˆs,
Rˆ=A-1GT,
A=GTG+σ2Cϕ-1,

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