Abstract

Wave-front reconstruction with the use of the fast Fourier transform (FFT) and spatial filtering is shown to be computationally tractable and sufficiently accurate for use in large Shack–Hartmann-based adaptive optics systems (up to at least 10,000 actuators). This method is significantly faster than, and can have noise propagation comparable with that of, traditional vector–matrix-multiply reconstructors. The boundary problem that prevented the accurate reconstruction of phase in circular apertures by means of square-grid Fourier transforms (FTs) is identified and solved. The methods are adapted for use on the Fried geometry. Detailed performance analysis of mean squared error and noise propagation for FT methods is presented with the use of both theory and simulation.

© 2002 Optical Society of America

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References

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  1. K. Freischlad, C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 74–80 (1985).
    [CrossRef]
  2. K. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
    [CrossRef]
  3. K. Freischlad, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).
  4. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  5. D. L. 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]
  6. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  7. E. P. Wallner, “Optimal wave-front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  8. J. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, New York, 1998).
  9. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  10. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  11. E. M. Johansson, D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE2200, 372–383 (1994).
    [CrossRef]

1986 (1)

1983 (1)

1980 (1)

1978 (1)

1977 (2)

1976 (1)

Freischlad, K.

K. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
[CrossRef]

K. Freischlad, C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 74–80 (1985).
[CrossRef]

K. Freischlad, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).

Fried, D. L.

Gavel, D. T.

E. M. Johansson, D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE2200, 372–383 (1994).
[CrossRef]

Hardy, J.

J. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, New York, 1998).

Herrmann, J.

Hudgin, R. H.

Johansson, E. M.

E. M. Johansson, D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE2200, 372–383 (1994).
[CrossRef]

Koliopoulos, C. L.

K. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
[CrossRef]

K. Freischlad, C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 74–80 (1985).
[CrossRef]

Noll, R. J.

Wallner, E. P.

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (1)

Other (4)

K. Freischlad, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).

E. M. Johansson, D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE2200, 372–383 (1994).
[CrossRef]

J. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, New York, 1998).

K. Freischlad, C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 74–80 (1985).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Hudgin and Fried sensor geometries. The circles represent the Shack–Hartmann wave-front sensor locations. In the Hudgin geometry, the gradients are first differences. In the Fried geometry, the gradients are the average of the two nearest first differences to the subaperture.

Fig. 2
Fig. 2

Estimate error from reconstruction of just the gradients inside the aperture, with the rest set to zero. No noise was added so as to clearly isolate the effects of the boundary. The gradients were calculated directly from phase points by using Eqs. (1) and (2). Each curve in the plot is a slice across the aperture along a row of actuators. This simulation was done on a 6376-actuator system, which was 90 actuators in diameter on a 128×128 grid. The input phase aberration had an rms error of 1690 nm. The reconstruction had an rms error of 1002 nm for this trial. The error spans the aperture and is not easily removed. (Note how the error changes shape and sign from row 40 to row 88.) With the use of either of the boundary methods, the reconstruction error was essentially 0 nm.

Fig. 3
Fig. 3

The three types of gradients are shown in the Hudgin geometry at the edge of an aperture. Bold lines are the boundary gradients bx and by. These connect phase points across the aperture edge. Thin solid lines are the inside gradients ix and iy, which can be obtained from measurement. The dotted lines are the outside gradients. Note that a closed loop across the aperture edge does not sum to zero if the boundary gradients are set equal to zero.

Fig. 4
Fig. 4

Boundary method. Setting each closed loop across the aperture edge to zero results in an equation relating the unknown boundary gradients to the measured inside gradients and the zeroed outside gradients. In this example, the equations for the boundary gradients u1, u2, are as follows, starting from the upper left corner: -u1+u2=a, u2+u3=0, u3+u4=c+b, u4+u5=0, and u5-u6=d. The complete set of equations for the whole aperture forms a linear system, which is then solved for the estimate of the boundary gradients.

Fig. 5
Fig. 5

Extension method, shown for N=6. The values of the gradients closest to the aperture edge are repeated outside the aperture. For example, gradients a, c, and e are each topmost in their columns and are extended upward out of the aperture. The unmodified gradients, which are left as zeros, are not shown in this figure for clarity. The seam gradients are along the right and bottom edges. These gradients “connect” the spatially periodic copies of the wave-front phase, and they must be set so that every row of the x gradients and every column of the y gradients sum to zero. For example, the leftmost column in this case must satisfy the equation s1=-b-l-j. Examination of this figure shows how loop continuity is satisfied exactly by the extension method.

Fig. 6
Fig. 6

The coordinate transform from the Fried geometry produces two disjoint grids. One grid is connected by the dashed gradients, the other by the solid gradients. Combining these uncoupled grids introduces an unknown waffle error.

Fig. 7
Fig. 7

Waffle is a significant concern in Fried-geometry reconstructions in noisy conditions. For a representative sample case with noise, the errors of the boundary and extension reconstructions, as compared with a perfect reconstructor, are shown. At the top is the reduced error in the extension case after waffle removal. Global waffle is completely removed, while local waffle, which is due to noise, remains.

Fig. 8
Fig. 8

Complete process of FT reconstruction. For the Hudgin-FT method, which is based on the Hudgin geometry, the gradient measurements are first extended. They are Fourier transformed, filtered, and inverse transformed before piston is removed. For the Fried-FT method, which is based on the Fried geometry, a few more steps are required. These are illustrated by the dashed arrows. The gradients are first converted to the two grids before they are extended and filtered. The two results are recombined, and then waffle and piston are removed.

Fig. 9
Fig. 9

Theoretical and simulation results of the noise propagation for the Hudgin-FT and Fried-FT methods. Aperture sizes vary on a 32×32 grid for the FFT. The solid curves are the noise propagators as determined theoretically. The data points are simulated noise propagator predictions. The simulation converges to the correct solution adequately enough to use it for large numbers of actuators, the theoretical calculations of which are computationally intractable.

Fig. 10
Fig. 10

Simulation results for noise propagation in comparison with the VMM and Freischlad’s square-aperture FT methods. In this DFT case, the reconstructions were done on the smallest size grid possible to correctly hold the aperture. (a) Hudgin geometry: The Hudgin-FT case lies between the square-aperture cases. (b) Fried geometry: The Fried-FT case is closer to the VMM case for smaller apertures but reaches square-aperture levels by 35,000 actuators.

Fig. 11
Fig. 11

Comparison of simulation results for noise propagation in the DFT and FFT cases. The DFT case is reconstructed on the smallest grid possible, while the FFT case uses power-of-2-sized grids. For both the Hudgin-FT and Fried-FT cases, there is a clear performance loss when the aperture diameter is small compared with the power-of-2-sized grid. Only the largest size apertures in a given power-of-2 grid approach ideal DFT results.

Fig. 12
Fig. 12

Theoretical results for the increase in noise propagation of a fixed 112-actuator system as the surrounding grid is increased in size. For both methods, increasing the grid size increases the total noise propagation in a regular manner.

Fig. 13
Fig. 13

Total mean squared error versus noise variance for two systems, a 448-actuator aperture on a 32×32 grid and an 11,304-actuator aperture on a 128×128 grid. Results for both the Hudgin-FT and Fried-FT methods are shown. The lines are the predicted performance, based on either theoretical or experimental noise propagation and an experimentally determined latent error. The data points are the results of simulation at various levels of noise. The effect of the noise propagator is clearly demonstrated in the differing slopes.

Equations (44)

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sx[m, n]=ϕ[m+1, n]-ϕ[m, n],
sy[m, n]=ϕ[m, n+1]-ϕ[m, n].
X[k, l]=F{x[m, n]}=1N2p=0N-1q=0N-1x[p, q]exp-j2πN(kp+lq).
Sx[k, l]=Φ[k, l]expj2πkN-1,
Sy[k, l]=Φ[k, l]expj2πlN-1.
Φˆ[k, l]=0,k, l=0exp-j2πkN-1Sx[k, l]+exp-j2πlN-1Sy[k,l]×4sin2πkN+sin2πlN-1,else.
sx[m, n]=ix[m, n]+bx[m, n],
sy[m, n]=iy[m, n]+by[m, n].
-u1+u2=a,u2+u3=0,
u3+u4=c+bu4+u5=0u5-u6=d.
Mu=c.
sx[m, n]=12(ϕ[m+1, n]-ϕ[m, n]+ϕ[m+1, n+1]-ϕ[m, n+1]),
sy[m, n]=12(ϕ[m, n+1]-ϕ[m, n]+ϕ[m+1, n+1]-ϕ[m+1, n]).
Φˆ[k, l]=0,k, l=0, k, l=N/2exp-j2πkN-1exp-j2πlN+1Sx[k, l]+exp-j2πlN-1exp-j2πkN+1Sy[k, l]×8sin2πkNcos2πlN+sin2πlNcos2πkN-1,else.
Sϕ(k)=0.023k-11/3r0-5/3,
σϕw24Sϕ(1/2d)Δk2.
σϕ2=1.03(D/r0)5/3
σϕw21.13(d/D)11/3σϕ2.
sa[m, n]=sx[m, n]+sy[m, n],
sb[m, n+1]=sx[m, n]-sy[m, n].
cv=m=0N-1n=0N-1ϕˆ[m, n]v[m, n]m=0N-1n=0N-1v[m, n]v[m, n].
ϕˆ-v[m, n]=ϕˆ[m, n]-cvv[m, n].
g=Hϕ+n.
ϕˆ=Mg.
=Mg-ϕ
b=E().
Λ=E[(-b)(-b)T].
mse=E(T)a.
b=(MH-I)ϕ,
Λ=MΛnMT.
msenp=mseσn2=Trace(MMT)a
b=(MH-I)mϕ,
Λ=(MH-I)Λϕ(HTMT-I)+MΛnMT.
mse=Trace(Λ)a.
mse
=Trace[(MH-I)Λϕ(HTMT-I)]+Trace(MΛnMT)a.
mse=mseϕ+msen(Λn).
mse=mseϕ+σn2 msenp.
msenp=0.46+0.087 ln a.
msenp=0.09753+1πln N.
msenp=0.17+0.13 ln a
msenp=0.6558+0.1603 ln a.
msenp=c+3πln(N-1),
msenp=0.1456 ln2 a-1.7922 ln a+7.6175

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