Abstract

Multigrid (MG) methods are presented for fast, efficient, flexible, and robust least-squares wavefront reconstruction in extremely high-resolution conventional adaptive optics, or ExAO. We demonstrate that MG can robustly handle a variety of sensor–actuator geometries, and it can accommodate deformable mirror influence function models that are more realistic than the common piecewise bilinear model. With MG one can also easily incorporate additional penalty, or regularization, terms to damp out the waffle mode in Fried geometry and to damp out instabilities due to actuators near the pupil boundary with poorly sensed influence. We present closed-loop simulation results that suggest that only one or two MG iterations per time step are needed to control an ExAO system.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Herrmann, "Least-squares wavefront errors of minimum norm," J. Opt. Soc. Am. 70, 28-35 (1980).
    [Crossref]
  2. M. P. Rimmer, "Method for evaluating lateral shearing interferograms," Appl. Opt. 13, 623-629 (1974).
    [Crossref] [PubMed]
  3. D. L. Fried, "Least-square fitting a wavefront distortion estimate to an array of phase-difference measurements," J. Opt. Soc. Am. 67, 370-375 (1977).
    [Crossref]
  4. H. Ren and R. Dekany, "Fast wavefront reconstruction by solving the Sylvester equation with the alternating direction implicit method," Opt. Express 12, 3279-3296 (2004), http://www.opticsexpress.org
    [Crossref] [PubMed]
  5. D. M. Young, Iterative Solutions of Large Linear Systems (Academic, 1972).
  6. U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid (Academic2001).
  7. W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd ed. (SIAM, 2000).
    [Crossref]
  8. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).
  9. L. Gilles, C. R. Vogel, and B. L. Ellerbroek, "Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction," J. Opt. Soc. Am. A 19, 1817-1822 (2002).
    [Crossref]
  10. 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]
  11. L. Gilles, "Order-N sparse minimum-variance open-loop reconstructor for extreme adaptive optics," Opt. Lett. 28, 1927-1929 (2003).
    [Crossref] [PubMed]
  12. L. Gilles, B. Ellerbroek, and C. R. Vogel, "Preconditioned conjugate gradient wave-front reconstructors for multi-conjugate adaptive optics," Appl. Opt. 42, 5233-5250 (2003).
    [Crossref] [PubMed]
  13. C. C. Douglas, G. Haase, and U. Langer, A Tutorial on Elliptic PDE Solvers and Their Parallelization (SIAM, 2003).
  14. K. Freischlad and C. L. Koliopoulos, "Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform," Proc. SPIE 551, 74-80 (1985).
  15. L. A. Poyneer, D. T. Gavel, and J. M. Base, "Fast wave-front reconstruction in large adaptive optics systems using the Fourier transform," J. Opt. Soc. Am. A 19, 2100-2111 (2002).
    [Crossref]
  16. D. L. Fried, "Least square fitting a wavefront distortion estimate to an array of phase difference measurements," J. Opt. Soc. Am. 67, 370-374 (1977).
    [Crossref]
  17. Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (SIAM, 2003).
    [Crossref]
  18. C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
    [Crossref]

2004 (1)

2003 (2)

2002 (3)

1985 (1)

K. Freischlad and C. L. Koliopoulos, "Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform," Proc. SPIE 551, 74-80 (1985).

1980 (1)

1977 (2)

1974 (1)

Base, J. M.

Briggs, W. L.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd ed. (SIAM, 2000).
[Crossref]

Dekany, R.

Douglas, C. C.

C. C. Douglas, G. Haase, and U. Langer, A Tutorial on Elliptic PDE Solvers and Their Parallelization (SIAM, 2003).

Ellerbroek, B.

Ellerbroek, B. L.

Freischlad, K.

K. Freischlad and C. L. Koliopoulos, "Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform," Proc. SPIE 551, 74-80 (1985).

Fried, D. L.

Gavel, D. T.

Gilles, L.

Haase, G.

C. C. Douglas, G. Haase, and U. Langer, A Tutorial on Elliptic PDE Solvers and Their Parallelization (SIAM, 2003).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).

Henson, V. E.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd ed. (SIAM, 2000).
[Crossref]

Herrmann, J.

Koliopoulos, C. L.

K. Freischlad and C. L. Koliopoulos, "Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform," Proc. SPIE 551, 74-80 (1985).

Langer, U.

C. C. Douglas, G. Haase, and U. Langer, A Tutorial on Elliptic PDE Solvers and Their Parallelization (SIAM, 2003).

McCormick, S. F.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd ed. (SIAM, 2000).
[Crossref]

Oosterlee, C. W.

U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid (Academic2001).

Poyneer, L. A.

Ren, H.

Rimmer, M. P.

Saad, Y.

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (SIAM, 2003).
[Crossref]

Schüller, A.

U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid (Academic2001).

Trottenberg, U.

U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid (Academic2001).

Vogel, C. R.

Young, D. M.

D. M. Young, Iterative Solutions of Large Linear Systems (Academic, 1972).

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

K. Freischlad and C. L. Koliopoulos, "Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform," Proc. SPIE 551, 74-80 (1985).

Other (7)

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (SIAM, 2003).
[Crossref]

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
[Crossref]

C. C. Douglas, G. Haase, and U. Langer, A Tutorial on Elliptic PDE Solvers and Their Parallelization (SIAM, 2003).

D. M. Young, Iterative Solutions of Large Linear Systems (Academic, 1972).

U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid (Academic2001).

W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd ed. (SIAM, 2000).
[Crossref]

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

DM actuator and WFS subaperture configuration for Fried geometry. WFS subapertures are indicated by the small squares. DM actuators are located at the corners of these squares. The solid circular curve indicates the boundary of the telescope pupil. Shaded squares within the pupil are the pupil-masked subapertures. Small open circles just outside the pupil indicate actuators whose influence extends inside the pupil.

Fig. 2
Fig. 2

(a) and (b) show one-dimensional splines that are used to construct 2D DM influence functions. (a) shows linear and cubic B-splines. (b) shows a clamped cubic B-spline. Circles (o) in (b) indicate influence function position at actuator node points. (c) shows a 2D influence function obtain by taking a tensor product of 1D linear splines; (d) shows a corresponding influence function obtained from a tensor product of cubic B-splines.

Fig. 3
Fig. 3

Grid representations of discrete Laplacian operators (L = PT P when the mask M = I) corresponding to (a) Hudgin geometry with bilinear influence functions and (b) Fried geometry with bilinear influence functions.

Fig. 4
Fig. 4

Solution errors for conventional least-squares wavefront reconstruction. Combinations of Hudgin or Fried geometry, bilinear or bicubic influence functions, and MG or MG-PCG iteration are shown: (a) Hudgin geometry, bilinear influence, and MG iteration; (b) Hudgin, bilinear, MG-PCG; (c) Fried, bilinear, MG; (d) Fried, bilinear, MG-PCG; (e) Fried, bicubic, MG; (f) Fried, bicubic, MG-PCG.

Fig. 5
Fig. 5

Phase errors for conventional least-squares wavefront reconstruction. Combinations of Hudgin or Fried geometry, bilinear or bicubic influence functions, and MG or MG-PCG iteration are shown: (a) Hudgin geometry, bilinear influence, and MG iteration; (b) Hudgin, bilinear, MG-PCG; (c) Fried, bilinear, MG; (d) Fried, bilinear, MG-PCG; (e) Fried, bicubic, MG; (f) Fried, bicubic, MG-PCG. Dashed horizontal lines indicate asymptotic values of phase error.

Fig. 6
Fig. 6

MG and MG-PCG performance for curvature penalized least-squares wavefront reconstruction with Fried geometry and bicubic influence functions. (a) shows the solution error for standalone MG iteration, (b) shows the solution error for PCG iteration with MG preconditioning, (c) shows the phase error for MG, and (d) shows the phase error for PCG with MG preconditioning. The dashed horizontal lines in (c) and (d) show the asymptotic value of the phase error.

Fig. 7
Fig. 7

Pupil-masked phase errors obtained (a) without curvature regularization and (b) with curvature regularization.

Tables (3)

Tables Icon

Table 1 Closed-Loop rms Phase Errors for Regularized Least-Squares Wavefront Reconstruction a

Tables Icon

Table 2 Closed-Loop rms Phase Errors for Regularized Least-Squares Wavefront Reconstruction a

Tables Icon

Table 3 Closed-Loop rms Phase Errors for Bilinear and Bicubic DMs with Varying SNRs a

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

s = G ϕ + η ,
G = [ G x , G y ] = [ M Γ x , M Γ y ] ,
[ Γ x ϕ ] i = Ω i ϕ x d x ,
ϕ DM = H a = j a j h j ( x ) .
h j ( x , y ) = b ( x x j 1 Δ ) b ( y y j 2 Δ ) .
b ( z ) = { z + 1 , 1 z 0 , 1 z , 0 z 1 , 0 ,       otherwise,
b ( z ) = 1 4 × { 1 + 3 ( 1 z ) + 3 ( 1 z ) 2 3 ( 1 z ) 3 , 0 z 1 , ( 2 z ) 3 , 1 z 2 , 0 , z 2 ,
[ P x ] i j = [ G x H ] i j = Ω i h j x d x ,
[ Γ x h ] i = 1 2 [ h ( x i 1 + 1 , y i 2 ) h ( x i 1 , y i 2 ) + h ( x i 1 + 1 , y i 2 + 1 ) h ( x i 1 , y i 2 + 1 ) ] ,
[ Γ x h ] i = 1 2 [ h ( x i 1 , y i 2 + 1 ) h ( x i 1 , y i 2 ) + h ( x i 1 + 1 , y i 2 + 1 ) h ( x i 1 + 1 , y i 2 ) ] .
ϕ = 2 ϕ x 2 + 2 ϕ y 2 .
( ϕ DM ) = [ ϕ DM ( x ) ] 2 d x
= j k a j a k h j ( x ) h k ( x ) d x [ R ] j k
= a T R a .
R ( ϕ DM ) = L Hudgin a 2 = a T L Hudgin L T Hudgin a R Hudgin
ϕ LS = H a LS ,
J LS ( a ) = Pa - s C η - 1 2 = ( P a s ) T C η       1 ( P a s ) .
a LS = a null + P η       s ,
P η     = lim α 0 + ( P T C η     1 P + α I ) 1 P T C η     1
ϕ LSreg = H a LSreg ,
a LSreg = arg min a { Pa - s C η - 1 2 + β a T Ra + α a 2 }
    = ( P T C η - 1 P + β R + α I ) 1 P T C η       1 s ,
A = P T C η       1 P + β R + α I ,
A h = L h + D h + U h ,
x k + 1 = x k + ω D h - 1 ( b h A h x k ) .
cost V = 4 / 3 ( ν pre + ν post + 1 ) r N h 1 4 / 3 ( ν pre + ν post + 1 ) n z ( A ) .
M 1 y = V   cycle ( A , y , 0 ) .
cost MG-PCG = n z ( A ) + cost V .
E phase ( k ) = M ( ϕ H a k ) , k = 0 , 1 , ,
E soln ( k ) = M H ( a * a k ) M H a * , k = 0 , 1 , .
E phase ( ) = M ( ϕ H a ) M ϕ ,
total cost of MG-PCG   n PCG × nz ( A ) × [ 1 + 4 / 3 ( ν pre + ν post + 1 ) ] ,
a ( t ) = a ( t 1 ) + g A 1 P s CL ( t 1 ) .
s CL ( t ) = G [ ϕ atmos ( t ) ϕ DM ( t ) ] + η ( t ) .
ϕ atmos ( x , t ) = ϕ 0 ( x t v ) .
M ( ϕ atmos ( t ) ϕ DM ( t ) ) M ϕ atmos ( t ) ) ,

Metrics