Abstract

A method is proposed for deriving the form of a turbulence-distorted wave front from Shack–Hartmann data by modeling the polychromatic imaging process in detail. In this way a higher order of reconstruction is possible than by methods that use only the barycenters (centers of gravity) of the individual spots. Simulations are presented showing the possible gains for a range of conditions of turbulence and photon flux. When photon noise is negligible the residual phase errors are approximately three times smaller than what is achievable by use of only the barycenters. The choice of functions for representing wave-front distortions is also considered with regard to the quality of the reconstruction obtained and to numerical efficiency.

© 1995 Optical Society of America

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References

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  2. J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
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  3. T. J. Schulz, “Estimation-theoretic approach to the deconvolution of atmospherically degraded images with wavefront sensor measurements,” in Digital Image Recovery and Synthesis II, Paul S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2029, 311–320 (1993).
    [CrossRef]
  4. R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
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  5. E. Thiébaut, J. M. Conan, “Implementation of additional image constraints for blind deconvolution using conjugate gradient minimization,” submitted to J. Opt. Soc. Am. A.
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    [CrossRef]
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    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 381.
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    [CrossRef] [PubMed]
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    [CrossRef]
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  15. R. Foy, “Speckle imaging review,” in Instrumentation for Ground Based Optical Astronomy, Present and Future, L. B. Robinson, ed. (Springer-Verlag, New York, 1988), pp. 345–359.
    [CrossRef]
  16. M. Tallon, A. Baranne, I. Belkine, R. Foy, M. Chatagnat, D. Dubet, B. Kohler, D. Lacroix, D. Robert, “SPID, a prototype for the very high angular resolution camera of the VLT,” Presented at the Workshop on Science with the VLT, Garching, Germany, June 28–July 1, 1994.
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    [CrossRef]

1994 (1)

1993 (1)

G. Roblin, D. Horville, “Etude de l’aberration propre à une trame de microlentilles,” J. Opt. (Paris) 24, 77–87 (1993).
[CrossRef]

1992 (2)

R. G. Lane, M. Tallon, “Wave-front reconstruction using a Shack–Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992).
[CrossRef] [PubMed]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

1990 (2)

1987 (1)

1976 (1)

1966 (1)

1965 (1)

Baranne, A.

M. Tallon, A. Baranne, I. Belkine, R. Foy, M. Chatagnat, D. Dubet, B. Kohler, D. Lacroix, D. Robert, “SPID, a prototype for the very high angular resolution camera of the VLT,” Presented at the Workshop on Science with the VLT, Garching, Germany, June 28–July 1, 1994.

Bates, R. H. T.

Belkine, I.

M. Tallon, A. Baranne, I. Belkine, R. Foy, M. Chatagnat, D. Dubet, B. Kohler, D. Lacroix, D. Robert, “SPID, a prototype for the very high angular resolution camera of the VLT,” Presented at the Workshop on Science with the VLT, Garching, Germany, June 28–July 1, 1994.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 381.

Chatagnat, M.

M. Tallon, A. Baranne, I. Belkine, R. Foy, M. Chatagnat, D. Dubet, B. Kohler, D. Lacroix, D. Robert, “SPID, a prototype for the very high angular resolution camera of the VLT,” Presented at the Workshop on Science with the VLT, Garching, Germany, June 28–July 1, 1994.

Conan, J. M.

E. Thiébaut, J. M. Conan, “Implementation of additional image constraints for blind deconvolution using conjugate gradient minimization,” submitted to J. Opt. Soc. Am. A.

Dainty, J. C.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Dubet, D.

M. Tallon, A. Baranne, I. Belkine, R. Foy, M. Chatagnat, D. Dubet, B. Kohler, D. Lacroix, D. Robert, “SPID, a prototype for the very high angular resolution camera of the VLT,” Presented at the Workshop on Science with the VLT, Garching, Germany, June 28–July 1, 1994.

Fontanella, J. C.

Foy, R.

M. Tallon, A. Baranne, I. Belkine, R. Foy, M. Chatagnat, D. Dubet, B. Kohler, D. Lacroix, D. Robert, “SPID, a prototype for the very high angular resolution camera of the VLT,” Presented at the Workshop on Science with the VLT, Garching, Germany, June 28–July 1, 1994.

R. Foy, “Speckle imaging review,” in Instrumentation for Ground Based Optical Astronomy, Present and Future, L. B. Robinson, ed. (Springer-Verlag, New York, 1988), pp. 345–359.
[CrossRef]

Fried, D. L.

Glindemann, A.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Horville, D.

G. Roblin, D. Horville, “Etude de l’aberration propre à une trame de microlentilles,” J. Opt. (Paris) 24, 77–87 (1993).
[CrossRef]

Kohler, B.

M. Tallon, A. Baranne, I. Belkine, R. Foy, M. Chatagnat, D. Dubet, B. Kohler, D. Lacroix, D. Robert, “SPID, a prototype for the very high angular resolution camera of the VLT,” Presented at the Workshop on Science with the VLT, Garching, Germany, June 28–July 1, 1994.

Lacroix, D.

M. Tallon, A. Baranne, I. Belkine, R. Foy, M. Chatagnat, D. Dubet, B. Kohler, D. Lacroix, D. Robert, “SPID, a prototype for the very high angular resolution camera of the VLT,” Presented at the Workshop on Science with the VLT, Garching, Germany, June 28–July 1, 1994.

Lane, R. G.

Noll, R. J.

Parenti, R. J.

Primot, J.

Robert, D.

M. Tallon, A. Baranne, I. Belkine, R. Foy, M. Chatagnat, D. Dubet, B. Kohler, D. Lacroix, D. Robert, “SPID, a prototype for the very high angular resolution camera of the VLT,” Presented at the Workshop on Science with the VLT, Garching, Germany, June 28–July 1, 1994.

Roblin, G.

G. Roblin, D. Horville, “Etude de l’aberration propre à une trame de microlentilles,” J. Opt. (Paris) 24, 77–87 (1993).
[CrossRef]

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Vol. 19 of Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376.
[CrossRef]

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Rousset, G.

Saisela, R. J.

Schulz, T. J.

T. J. Schulz, “Estimation-theoretic approach to the deconvolution of atmospherically degraded images with wavefront sensor measurements,” in Digital Image Recovery and Synthesis II, Paul S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2029, 311–320 (1993).
[CrossRef]

Tallon, M.

R. G. Lane, M. Tallon, “Wave-front reconstruction using a Shack–Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992).
[CrossRef] [PubMed]

M. Tallon, A. Baranne, I. Belkine, R. Foy, M. Chatagnat, D. Dubet, B. Kohler, D. Lacroix, D. Robert, “SPID, a prototype for the very high angular resolution camera of the VLT,” Presented at the Workshop on Science with the VLT, Garching, Germany, June 28–July 1, 1994.

Thiébaut, E.

E. Thiébaut, J. M. Conan, “Implementation of additional image constraints for blind deconvolution using conjugate gradient minimization,” submitted to J. Opt. Soc. Am. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 381.

Appl. Opt. (1)

J. Opt. (Paris) (1)

G. Roblin, D. Horville, “Etude de l’aberration propre à une trame de microlentilles,” J. Opt. (Paris) 24, 77–87 (1993).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Eng. (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Waves Random Media (1)

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Other (7)

E. Thiébaut, J. M. Conan, “Implementation of additional image constraints for blind deconvolution using conjugate gradient minimization,” submitted to J. Opt. Soc. Am. A.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Vol. 19 of Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376.
[CrossRef]

T. J. Schulz, “Estimation-theoretic approach to the deconvolution of atmospherically degraded images with wavefront sensor measurements,” in Digital Image Recovery and Synthesis II, Paul S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2029, 311–320 (1993).
[CrossRef]

D. L. Fried, ed., Feature issue on adaptive optics, J. Opt. Soc. Am.67, 269–422 (1977).

R. Foy, “Speckle imaging review,” in Instrumentation for Ground Based Optical Astronomy, Present and Future, L. B. Robinson, ed. (Springer-Verlag, New York, 1988), pp. 345–359.
[CrossRef]

M. Tallon, A. Baranne, I. Belkine, R. Foy, M. Chatagnat, D. Dubet, B. Kohler, D. Lacroix, D. Robert, “SPID, a prototype for the very high angular resolution camera of the VLT,” Presented at the Workshop on Science with the VLT, Garching, Germany, June 28–July 1, 1994.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 381.

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Figures (5)

Fig. 1
Fig. 1

Restricted influence of individual microlenses on the image. The upper plane represents a 3 × 3 array of microlenses and the lower one the resulting image. The intensity over each shaded region in the image is calculated exclusively from the phase over the similarly shaded region of the pupil. Thus each point in the image depends on the four microlenses above it, and each microlens influences a square region of the image of four times its own area.

Fig. 2
Fig. 2

Comparison of the eigenvalues of Zernike bases as a function of the number of polynomials included. The solid curve shows the eigenvalues of the covariance matrix for the first 3000 Zernike polynomials. For bases with the number of elements as on the abscissa, the dotted curve shows the eigenvalue of the covariance matrix at the fifteenth percentile and the short-dashed curve at the thirtieth. The largest eigenvalue for the covariance matrix of the polynomials excluded from the basis is shown by the long-dashed curve. Thus 15% of the eigenvalues would lie below the dotted curve. For bases of more than a couple of hundred elements these correspond to wave-front variations that are more than an order of magnitude less energetic than the most important mode of the functions excluded, as shown by the long-dashed curve.

Fig. 3
Fig. 3

Deviations from a 5/3 slope for the structure functions of wave fronts generated from bases of various sizes as marked. The wave-front diameter is D, and r is the scale of interest as shown in the expression on the abscissa. Each curve represents a mean of 1000 wave fronts. Wave fronts following Kolmogorov statistics should have power spectra of index 5/3 and so would reduce to horizontal lines on this graph. A plane surface would give an index of 2, so the deviations at small r toward a slope of 1/3 in the figure indicate that the surfaces in question are essentially flat (but not horizontal) on these scales.

Fig. 4
Fig. 4

Comparison of the present global method and the optimal reconstruction by centers of gravity for a typical simulated wave front (top left). To its right is the Shack–Hartmann image as described in Section 2. The reconstruction from the mean slopes, but without passing though this image and therefore without photon noise, is shown at the lower left. This reconstruction used the first 55 Karhunen–Loève functions, which a posteori gives the best reconstruction for this surface, having σp = 0.59 (see text) for D/r0 = 8. At the lower right is the global reconstruction with 900 Karhunen–Loève functions derived from the image shown. It has σp = 0.19.

Fig. 5
Fig. 5

Comparison of the residual errors for different types of fit, noise level, and basis function. The vertical axis shows the standard deviation of the difference between the original and the reconstructed phase. All the wave fronts have D/r0 = 8 with subpupils of width r0. For other values of D/r0 but the same microlens array, the rms errors scale as (D/r0)5/6. The two dashed curves show the optimal reconstructions by Karhunen–Loève functions for wave fronts generated from 1000 (lower curve) and 3000 (upper curve) functions. The dotted curve shows the optimal Zernike reconstruction for truly Kolmogorov wave fronts. The reconstruction by mean slopes is shown by the dotted–dashed curve, and reconstructions by the present global approach for various numbers of photons per spot are shown by the solid curves.

Equations (18)

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u ( x ) = exp ( 2 π i D / λ ) D ( 1 + cos χ ) d y ,
χ = tan - 1 ( x - y / z ) , D = [ ( z + p ( y ) λ 0 / 2 π ] 2 + ( x - y ) 2 ] 1 / 2 ,
h ( x ) = f ( λ ) u ( x ) u * ( x ) d λ ,
P ( p g ) = P ( g p ) P ( p ) P ( g ) ,
= g + p ,
g = - log P ( g p ) ,
p = - log P ( p ) .
i i ( g i - h i ) 2 h i ,
u ( x ) = exp { 2 π i [ z 2 + ( x - y ) 2 ] 1 / 2 λ } exp [ i p ( y ) λ 0 λ ] d y .
u i j = k l [ Q i j k l exp ( i p k l ) ] λ 0 / λ ,
Q i j k l = exp { 2 π i λ 0 [ z 2 + ( w p i - w i k ) 2 + ( w p j - w i l ) 2 ] 1 / 2 }
u i = j [ R i j exp ( i p j ) ] λ 0 / λ ,
q i = exp ( i p i λ 0 λ ) ,
h i p j = 2 λ 0 λ [ Re u i ( - Re R i j Im q j - Im R i j Re q j ) + Im u i ( Re R i j Re q j - Im R i j Im q j ) ] = 2 Im ( u i R i j * q j * ) .
g p j = i 4 λ 0 λ ( h i - g i ) h i Im ( u i R i j * q j * ) ,
h i = u i u i * ,             u i = k R i k q k .
{ K n i o K n i e } = α n i { sin n π ϕ cos n π ϕ } f n i ( r ) ,
σ 2 ( n ) = n + 1 cov ( f i , f i ) ,

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