Abstract

We present two different implementations of the Fourier domain preconditioned conjugate gradient algorithm (FD-PCG) to efficiently solve the large structured linear systems that arise in optimal volume turbulence estimation, or tomography, for multi-conjugate adaptive optics (MCAO). We describe how to deal with several critical technical issues, including the cone coordinate transformation problem and sensor subaperture grid spacing. We also extend the FD-PCG approach to handle the deformable mirror fitting problem for MCAO.

© 2006 Optical Society of America

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References

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  1. J. M. Beckers, "Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics," in Proceedings of European Southern Observatory Conference and Workshop on Very Large Telescopes and Their Instrumentation, M. H. Ulrich, ed., Vol. 30 of ESO Conference andWorkshop Proceedings (European Southern Observatory, Garching, Germany, 1988), pp. 693-703.
  2. D. C. Johnston and B. M. Welsh, "Analysis of multi-conjugate adaptive optics," J. Opt. Soc. Am. A 11, 394-408 (1994).
    [CrossRef]
  3. T. Fusco, J. M. Conan, G. Rousset, L. M. Mugnier, and V. Michau, "Optimal wave-front reconstruction strategies for multi-conjugate adaptive optics," J. Opt. Soc. Am. A 18, 2527-2538 (2001).
    [CrossRef]
  4. R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, J-P. Veran, "Adaptive optics requirements definition for TMT," Advancements in Adaptive Optics, edited by D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, Proc. SPIE 5490, 879-890 (2004).
    [CrossRef]
  5. 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]
  6. G. Golub and C. VanLoan, Matrix Computations, 2nd Edition, Johns Hopkins University Press, 1989.
  7. 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]
  8. L. Gilles, B. L. Ellerbroek, and, C. R. Vogel, "Preconditioned conjugate gradient wave-front reconstructors for multi-conjugate adaptive optics," Appl. Opt. 42, 5233-5250 (2003).
    [CrossRef] [PubMed]
  9. B. L. Ellerbroek, L. Gilles, and C. R. Vogel, "Numerical simulations of multi-conjugate adaptive optics wavefront reconstruction on giant telescopes," Appl. Opt. 42, 4811-4818 (2003).
    [CrossRef] [PubMed]
  10. Q. Yang, C.R. Vogel, and B.L. Ellerbroek, "Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography," Appl. Opt. 45, 5281-5293 (2006).
    [CrossRef] [PubMed]
  11. J. W. Hardy, Adaptive Optics for Astronomical Telescopes, Oxford University Press, 1998.
  12. D. Fried, "Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements," J. Opt. Soc. Am. 67,370-375 (1977).
    [CrossRef]

2006 (1)

2003 (2)

2002 (2)

2001 (1)

1994 (1)

1977 (1)

Conan, J. M.

Ellerbroek, B. L.

Ellerbroek, B.L.

Fried, D.

Fusco, T.

Gilles, L.

Johnston, D. C.

Michau, V.

Mugnier, L. M.

Rousset, G.

Vogel, C. R.

Vogel, C.R.

Welsh, B. M.

Yang, Q.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Other (4)

J. M. Beckers, "Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics," in Proceedings of European Southern Observatory Conference and Workshop on Very Large Telescopes and Their Instrumentation, M. H. Ulrich, ed., Vol. 30 of ESO Conference andWorkshop Proceedings (European Southern Observatory, Garching, Germany, 1988), pp. 693-703.

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, J-P. Veran, "Adaptive optics requirements definition for TMT," Advancements in Adaptive Optics, edited by D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, Proc. SPIE 5490, 879-890 (2004).
[CrossRef]

G. Golub and C. VanLoan, Matrix Computations, 2nd Edition, Johns Hopkins University Press, 1989.

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

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Figures (3)

Fig. 1.
Fig. 1.

Illustration of cone-coordinate grid vs conventional equispaced grid. Cone-coordinate grid points in figure on the left have grid spacings that decrease with layer height. The conventional grid on the right has the same number of grid points at each layer height, but the grid spacing does not vary with layer height.

Fig. 2.
Fig. 2.

Sensor subaperture grid for a simulated 4-meter telescope. Green circles represent points in the computational grid. Blue stars represent vertices of 1/2 meter × 1/2 meter square high-order sensor subapertures. The red circle represents the outer edge of the clear aperture, or pupil.

Fig. 3.
Fig. 3.

FD-PCG performance for the fitting step. The blue line shows the on-axis residual phase error, and the red line shows phase error at the edge of the 30-arcsecond corrected field of view of the telescope.

Equations (41)

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ϕ ( x , θ ) = [ ] x θ ,
[ ] x θ = = 1 n layer ψ ( x + z θ , z ) .
s i , j x = 1 h [ ( ϕ ( x i + h , y j ) ϕ ( x i , y j ) ) 2 + ( ϕ ( x i + h , y j + h ) ϕ ( x i , y j + h ) ) 2 ] + η i , j x ,
s i , j y = 1 h [ ( ϕ ( x i , y j + h ) ϕ ( x i , y j ) ) 2 + ( ϕ ( x i + h , y j + h ) ϕ ( x i + h , y j ) ) 2 ] + η i , j y ,
G = M Γ P .
s = + η .
ϕ DM = Pm .
m ( x , z k ) = [ H k a k ] ( x ) , k = 1 , , n DM .
ϕ DM = PH a ,
s = [ s h s t ] = [ G h ψ G t ψ ] + [ η h η t ] .
ψ est = arg min ψ { G h ψ s h C h 1 2 + G t ψ s t C t 1 2 + ψ C ψ 1 2 }
ψ est = ( G h T C h 1 G h + G t T C t 1 G t + C ψ 1 ) 1 ( G h T C h 1 s h + G t T C t 1 s t )
= [ C ψ G h T C ψ G t T ] [ G h C ψ G h T + C h G h C ψ G t T G t C ψ G h T G t C ψ G h T + C t ] 1 [ s h s t ]
C h 1 = N h 1 N h 1 T ( T T N h 1 T ) 1 T T N h 1
ψ est = ( A h A lr ) 1 ( b h + b t )
A h = G h T N h 1 G h + C ψ 1 ,
A lr = G h T N h 1 T ( T T N h 1 T ) 1 T T N h 1 G h G t T C t 1 G t ,
b h = G h T C h 1 s h , b t = G t T C t 1 s t .
A lr = U 1 U 1 T U 2 U 2 T ,
( A h A lr ) 1 = [ A h ( U 1 U 2 ) ( U 1 T U 2 T ) ] 1
= A h 1 + [ W 1 W 2 ] [ I W 1 T U 1 W 1 T U 2 W 2 T U 1 I + W 2 T U 2 ] 1 [ W 1 T W 2 T ] L
W 1 = A h 1 U 1 , W 2 = A h 1 U 2 .
ψ est = A h 1 b + Lb .
ψ ( c x + s , z ) = def ψ ˜ ( x ˜ + s ˜ ) .
x c x = def x ˜ .
ψ ˜ est = arg min ψ ˜ { G ˜ h ψ ˜ s h C h 1 2 + G t I c ψ ˜ s t C t 1 2 + ψ ˜ C ψ ˜ 1 2 }
ψ est = I c ψ ˜ est
[ A h ] ij = k = 1 n LGS P ki T S k P kj + δ ij B j , i , j = 1 , , n L ,
S k = Γ x T M N h 1 M Γ x + Γ y T M N h 1 M Γ y , k = 1 , , n LGS ,
P kj = F 1 P ̂ kj F ,
[ A h ] ij = F A ̂ ij F 1 with A ̂ ij = def k P ̂ ki * S ̂ k P ̂ kj + δ ij B ̂ j ,
S ̂ k = σ h 2 ( Γ ̂ x * M ̂ Γ ̂ x + Γ ̂ y * M ̂ Γ ̂ y ) , M ̂ = def FMF 1
σ h 2 ( Γ ̂ x 2 + Γ ̂ y 2 )
= def S ˜ k .
[ C ] ij = F 1 C ̂ ij F with C ̂ ij = k = 1 n LGS P ̂ ki * S ~ k P ̂ kj + δ ij B ̂ i ,
M S ( x ) = { 1 , if x is a subaperture vertex 0 , otherwise.
S ̃ k = σ h 2 ( Γ ̂ x * M ̂ S Γ ̂ x + Γ ̂ y * M ̂ S Γ ̂ y ) , M ̂ S = F 1 M S F ,
ϕ DM = MP DM H a ,
a pot = arg min a { MP DM H a ϕ est W 2 + a R 2 }
= A fit 1 ( H T P DM T MW ϕ est ) .
A fit = H T P DM T MWM P DM H + R .

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