Abstract

We introduce a multigrid preconditioned conjugate-gradient (MGCG) iterative scheme for computing open-loop wave-front reconstructors for extreme adaptive optics systems. We present numerical simulations for a 17-m class telescope with n=48756 sensor measurement grid points within the aperture, which indicate that our MGCG method has a rapid convergence rate for a wide range of subaperture average slope measurement signal-to-noise ratios. The total computational cost is of order n log n. Hence our scheme provides for fast wave-front simulation and control in large-scale adaptive optics systems.

© 2002 Optical Society of America

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References

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  1. M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  2. B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse-matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
    [CrossRef]
  3. F. U. Dowla, J. M. Brase, S. S. Olivier, C. A. Thompson, “Fast Fourier and wavelet transforms for wavefront reconstruction in adaptive optics,” in High-Resolution Wavefront Control: Methods, Devices, and Applications II, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds, Proc. SPIE4124, 118–127 (2000).
  4. L. A. Poyneer, D. T. Gavel, J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with the Fourier transform,” J. Opt. Soc. Am. A (to be published).
  5. D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  6. T. Fusco, J.-M. Conan, G. Rousset, L. M. Mugnier, V. Michau, “Optimal wave-front reconstruction strategies for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2527–2538 (2001).
    [CrossRef]
  7. Y. Saad, Iterative Methods for Sparse Linear Systems (PWS, Boston, Mass., 1996).
  8. U. Trottenberg, C. W. Oosterlee, A. Schüller, Multigrid (Academic, London, 2001).
  9. B. Parlett, The Symmetric Eigenvalue Problem (SIAM, Philadelphia, Pa., 1997).

2002 (1)

2001 (1)

1977 (1)

Brase, J. M.

L. A. Poyneer, D. T. Gavel, J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with the Fourier transform,” J. Opt. Soc. Am. A (to be published).

F. U. Dowla, J. M. Brase, S. S. Olivier, C. A. Thompson, “Fast Fourier and wavelet transforms for wavefront reconstruction in adaptive optics,” in High-Resolution Wavefront Control: Methods, Devices, and Applications II, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds, Proc. SPIE4124, 118–127 (2000).

Conan, J.-M.

Dowla, F. U.

F. U. Dowla, J. M. Brase, S. S. Olivier, C. A. Thompson, “Fast Fourier and wavelet transforms for wavefront reconstruction in adaptive optics,” in High-Resolution Wavefront Control: Methods, Devices, and Applications II, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds, Proc. SPIE4124, 118–127 (2000).

Ellerbroek, B. L.

Fried, D. L.

Fusco, T.

Gavel, D. T.

L. A. Poyneer, D. T. Gavel, J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with the Fourier transform,” J. Opt. Soc. Am. A (to be published).

Michau, V.

Mugnier, L. M.

Olivier, S. S.

F. U. Dowla, J. M. Brase, S. S. Olivier, C. A. Thompson, “Fast Fourier and wavelet transforms for wavefront reconstruction in adaptive optics,” in High-Resolution Wavefront Control: Methods, Devices, and Applications II, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds, Proc. SPIE4124, 118–127 (2000).

Oosterlee, C. W.

U. Trottenberg, C. W. Oosterlee, A. Schüller, Multigrid (Academic, London, 2001).

Parlett, B.

B. Parlett, The Symmetric Eigenvalue Problem (SIAM, Philadelphia, Pa., 1997).

Poyneer, L. A.

L. A. Poyneer, D. T. Gavel, J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with the Fourier transform,” J. Opt. Soc. Am. A (to be published).

Roggemann, M. C.

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

Rousset, G.

Saad, Y.

Y. Saad, Iterative Methods for Sparse Linear Systems (PWS, Boston, Mass., 1996).

Schüller, A.

U. Trottenberg, C. W. Oosterlee, A. Schüller, Multigrid (Academic, London, 2001).

Thompson, C. A.

F. U. Dowla, J. M. Brase, S. S. Olivier, C. A. Thompson, “Fast Fourier and wavelet transforms for wavefront reconstruction in adaptive optics,” in High-Resolution Wavefront Control: Methods, Devices, and Applications II, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds, Proc. SPIE4124, 118–127 (2000).

Trottenberg, U.

U. Trottenberg, C. W. Oosterlee, A. Schüller, Multigrid (Academic, London, 2001).

Welsh, B.

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

J. Opt. Soc. Am. (1)

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

Other (6)

F. U. Dowla, J. M. Brase, S. S. Olivier, C. A. Thompson, “Fast Fourier and wavelet transforms for wavefront reconstruction in adaptive optics,” in High-Resolution Wavefront Control: Methods, Devices, and Applications II, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds, Proc. SPIE4124, 118–127 (2000).

L. A. Poyneer, D. T. Gavel, J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with the Fourier transform,” J. Opt. Soc. Am. A (to be published).

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

Y. Saad, Iterative Methods for Sparse Linear Systems (PWS, Boston, Mass., 1996).

U. Trottenberg, C. W. Oosterlee, A. Schüller, Multigrid (Academic, London, 2001).

B. Parlett, The Symmetric Eigenvalue Problem (SIAM, Philadelphia, Pa., 1997).

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Figures (2)

Fig. 1
Fig. 1

Fried geometry showing phase screen values ϕij and slope sensor location sij.

Fig. 2
Fig. 2

Top row: left, relative predictive error norm ϕ-ϕk/ϕ; right, relative system error norm ϕˆ-ϕk/ϕˆ versus CG iteration count for SNRs=1 and 10. Bottom row: left, true solution and right, predictive solution error ϕk-ϕ after k=2 CG iterations for SNR=10. r0=0.2 m, mesh size h=r0/3, square computational domain with 512×512 grid points (34 m×34 m).

Equations (27)

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s=Gϕ+η,
sxsy=GxGyϕ+ηxηy.
Cηη=ηηT=ηxηxTηxηyTηyηxTηyηyT=σ2I,
SNR=Gϕ21/2η21/2=Gϕ21/22nσ,
ϕˆ=Rˆs=[RˆxRˆy]sxsy=Rˆxsx+Rˆysy,
J(R)=defRs-ϕ2=Tr[Rs-ϕ][Rs-ϕ]T=Tr[RssTRT]-Tr[RsϕT]-Tr[ϕsTRT]+Tr[Cϕϕ],
δVJ(Rˆ)=limτ0 J(Rˆ+τV)-J(Rˆ)τ.
δVJ(Rˆ)=2Tr[{Rˆ(ssT)-ϕsT}VT].
Rˆ=ϕsTssT-1=CϕϕGT[GCϕϕGT+Cηη]-1.
Rˆ=[GTCηη-1G+Cϕϕ-1]-1GTCηη-1=[GTG+σ2Cϕϕ-1]-1GT.
C˜ϕϕ=F-1ΛF,
C=defC˜ϕϕ-1=F-1Λ-1F.
Λ(κ)=|δn(κ)|2=c2[|κ|2+1/L02]11/6.
Dϕ(P, Q)=[ϕ(P)-ϕ(Q)]2=6.88(r/r0)5/3,
[Gxϕ]ij=12[(ϕi+1,j+ϕi,j)-(ϕi+1,j+1+ϕi,j+1)],
[Gyϕ]ij=12[(ϕi+1,j+1+ϕi+1,j)-(ϕi,j+ϕi,j+1)].
Bϕ=GTs,B=L+σ2C.
h1>h2>>hl.
LhlL;Lhi-1Ihihi-1LhiIhi-1hi,i=l, l-1,, 2.
B=(σ2C+ωI)-(ωI-L),
uk+1=(σ2C+ωI)-1[f+(ωI-L)uk]=defS(u0, k)f.
W=i=1l[(4nR+2)FFT(Ni)+O(Ni)].
FFT(Ni)=i 4i+O(4i).
i=1lβ(i)1l+1β(x)dx.
W(4nR+2)1l+1x4xdx+Oi=1l 4i2(4nR+2)ln(4)N log2(N)+O(N).
ϕ=F -1{Λ1/2w},
ek=def ϕˆ-ϕkϕˆ

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