Abstract

The use of predetection compensation for the effects of atmospheric turbulence combined with postdetection image processing for imaging applications with large telescopes is addressed. Full and partial predetection compensation with adaptive optics is implemented by varying the number of actuators in the deformable mirror. The theoretical expression for the single-frame power spectrum signal-to-noise ratio (SNR) is reevaluated for the compensated case to include the statistics of the compensated optical transfer function. Critical to this analysis is the observation that the compensated optical transfer function does not behave as a circularly complex Gaussian random variable except at high spatial frequencies. Results from a parametric study of performance are presented to demonstrate improvements in power spectrum estimation for both point sources and an extended object and improvements in the Fourier phase spectrum estimation for an extended object. Full compensation is shown to provide a large improvement in the power spectrum SNR over the uncompensated case, while successively smaller amounts of predetection compensation provide smaller improvements, until a low degree of compensation gives results essentially identical to those of the uncompensated case. Three regions of performance were found with respect to the object Fourier phase spectrum estimate obtained from bispectrum postprocessing: (1) the fully compensated case in which bispectrum postprocessing provides no improvement in the phase estimate over that obtained from a fully compensated long-exposure image, (2) a partially compensated regime in which applying bispectrum postprocessing to the compensated images provides a phase spectrum estimation superior to that of the uncompensated bispectrum case, and (3) a poorly compensated regime in which the results are essentially indistinguishable from those of the uncompensated case. Accurate simulations were used to obtain some parameters for the power spectrum SNR analysis and to obtain the Fourier phase spectrum results.

© 1992 Optical Society of America

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References

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  1. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX.
    [CrossRef]
  2. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. Inst. Electr. Eng. 66, 651–697 (1978).
    [CrossRef]
  3. E. P. Walner, “Optimal wave-front correction using slope measurements,”J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  4. B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
    [CrossRef]
  5. P. Nisenson, R. Barakat, “Partial atmospheric correction with adaptive optics,” J. Opt. Soc. Am. A 4, 2249–2253 (1987).
    [CrossRef]
  6. M. C. Roggemann, “Limited degree-of-freedom adaptive optics and image reconstruction,” Appl. Opt. 30, 4227–4233 (1991).
    [CrossRef] [PubMed]
  7. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  8. A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  10. C. L. Matson, “Weighted-least-squares phase reconstruction from the bispectrum,” J. Opt. Soc. Am. A 8, 1905–1913 (1991).
    [CrossRef]
  11. M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Elect. Eng. (to be published).
  12. D. L. Hench, “Improved speckle imaging using limited degree-of-freedom adaptive optics,” TR-908 (Optical Sciences Company, Placentia, Calif., 1988).
  13. R. Holmes, S. M. Ebstein, “Partially compensated Knox–Thompson speckle imaging,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 478–491 (1990).
    [CrossRef]
  14. D. L. Fried, D. L. Hench, “Spectral bandwidth and adaptive optics considerations for white light speckle imagery,” TR-1033 (Optical Sciences Company, Placentia, Calif., 1990).
  15. J. C. Dainty, A. H. Greenaway, “Estimation of power spectra in speckle interferometry,”J. Opt. Soc. Am. 69, 786–790 (1979).
    [CrossRef]
  16. J. W. Goodman, J. F. Belsher, “Photon-limited images and their restoration,” Rome Air Development Center TR-76-50, interim report from contract F30602-75-C-0228 (Rome Air Development Center, Griffiss Air Force Base, N.Y., 1976).
  17. M. C. Roggemann, C. L. Matson, “Partially compensated speckle imaging: Fourier phase spectrum estimation,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 477–487 (1992).
    [CrossRef]
  18. B. L. Ellerbroek, “Comparison of least-squares and minimal variance reconstructors for turbulence compensation in the presence of noise: analysis and results,” (Optical Sciences Company, Placentia, Calif., 1986).
  19. G. Cochran, “Phase screen generation,” TR-663 (Optical Sciences Company, Placentia, Calif., 1985).

1991

1989

1987

1983

1979

1978

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. Inst. Electr. Eng. 66, 651–697 (1978).
[CrossRef]

Barakat, R.

Belsher, J. F.

J. W. Goodman, J. F. Belsher, “Photon-limited images and their restoration,” Rome Air Development Center TR-76-50, interim report from contract F30602-75-C-0228 (Rome Air Development Center, Griffiss Air Force Base, N.Y., 1976).

Cochran, G.

G. Cochran, “Phase screen generation,” TR-663 (Optical Sciences Company, Placentia, Calif., 1985).

Dainty, J. C.

Ebstein, S. M.

R. Holmes, S. M. Ebstein, “Partially compensated Knox–Thompson speckle imaging,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 478–491 (1990).
[CrossRef]

Ellerbroek, B. L.

B. L. Ellerbroek, “Comparison of least-squares and minimal variance reconstructors for turbulence compensation in the presence of noise: analysis and results,” (Optical Sciences Company, Placentia, Calif., 1986).

Fried, D. L.

D. L. Fried, D. L. Hench, “Spectral bandwidth and adaptive optics considerations for white light speckle imagery,” TR-1033 (Optical Sciences Company, Placentia, Calif., 1990).

Gardner, C. S.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

J. W. Goodman, J. F. Belsher, “Photon-limited images and their restoration,” Rome Air Development Center TR-76-50, interim report from contract F30602-75-C-0228 (Rome Air Development Center, Griffiss Air Force Base, N.Y., 1976).

Greenaway, A. H.

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. Inst. Electr. Eng. 66, 651–697 (1978).
[CrossRef]

Hench, D. L.

D. L. Fried, D. L. Hench, “Spectral bandwidth and adaptive optics considerations for white light speckle imagery,” TR-1033 (Optical Sciences Company, Placentia, Calif., 1990).

D. L. Hench, “Improved speckle imaging using limited degree-of-freedom adaptive optics,” TR-908 (Optical Sciences Company, Placentia, Calif., 1988).

Holmes, R.

R. Holmes, S. M. Ebstein, “Partially compensated Knox–Thompson speckle imaging,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 478–491 (1990).
[CrossRef]

Lohmann, A. W.

Matson, C. L.

C. L. Matson, “Weighted-least-squares phase reconstruction from the bispectrum,” J. Opt. Soc. Am. A 8, 1905–1913 (1991).
[CrossRef]

M. C. Roggemann, C. L. Matson, “Partially compensated speckle imaging: Fourier phase spectrum estimation,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 477–487 (1992).
[CrossRef]

Nisenson, P.

Roddier, F.

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

Roggemann, M. C.

M. C. Roggemann, “Limited degree-of-freedom adaptive optics and image reconstruction,” Appl. Opt. 30, 4227–4233 (1991).
[CrossRef] [PubMed]

M. C. Roggemann, C. L. Matson, “Partially compensated speckle imaging: Fourier phase spectrum estimation,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 477–487 (1992).
[CrossRef]

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Elect. Eng. (to be published).

Walner, E. P.

Weigelt, G.

Welsh, B. M.

Wirnitzer, B.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. Inst. Electr. Eng.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. Inst. Electr. Eng. 66, 651–697 (1978).
[CrossRef]

Other

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Elect. Eng. (to be published).

D. L. Hench, “Improved speckle imaging using limited degree-of-freedom adaptive optics,” TR-908 (Optical Sciences Company, Placentia, Calif., 1988).

R. Holmes, S. M. Ebstein, “Partially compensated Knox–Thompson speckle imaging,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 478–491 (1990).
[CrossRef]

D. L. Fried, D. L. Hench, “Spectral bandwidth and adaptive optics considerations for white light speckle imagery,” TR-1033 (Optical Sciences Company, Placentia, Calif., 1990).

J. W. Goodman, J. F. Belsher, “Photon-limited images and their restoration,” Rome Air Development Center TR-76-50, interim report from contract F30602-75-C-0228 (Rome Air Development Center, Griffiss Air Force Base, N.Y., 1976).

M. C. Roggemann, C. L. Matson, “Partially compensated speckle imaging: Fourier phase spectrum estimation,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 477–487 (1992).
[CrossRef]

B. L. Ellerbroek, “Comparison of least-squares and minimal variance reconstructors for turbulence compensation in the presence of noise: analysis and results,” (Optical Sciences Company, Placentia, Calif., 1986).

G. Cochran, “Phase screen generation,” TR-663 (Optical Sciences Company, Placentia, Calif., 1985).

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

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Figures (9)

Fig. 1
Fig. 1

Examples of fully and partially compensated pupil geometries. The squares represent the locations of Hartmann WFS subapertures, and the dots represent the locations of actuators: (a) fully compensated, with actuators separated by 0.1 × the telescope diameter; (b) partially compensated, with actuators separated by 0.2 × the telescope diameter.

Fig. 2
Fig. 2

Statistics of the compensated OTF for the pupil geometries shown in Fig. 1. (a) Curve 1 is for the pupil geometry in Fig. 1(a); curve 2 is for the pupil geometry in Fig. 1(b); the diffraction-limited OTF is provided for comparison. (b) Curve 1 is the variance of Re(H) for the pupil geometry in Fig. 1(a); curve 2 is the variance of Im(H) for the pupil geometry in Fig. 1(a); curve 3 is the variance of Re(H) for the pupil geometry in Fig. 1(b); curve 4 is the variance of Im(H) for the pupil geometry in Fig. 1(b). All the plots are ux-axis slices.

Fig. 3
Fig. 3

Comparison of theory and simulation for the power spectrum SNR for (a) the pupil geometry in Fig. 1(a) and (b) the pupil geometry in Fig. 1(b). The solid curves show the simulation result, and the dashed curves show the theoretical calculation obtained by using the statistical model for the modulus of the compensated OTF. All the plots are ux-axis slice.

Fig. 4
Fig. 4

Theoretical and simulation long-exposure OTF’s for a 1.6-m-diameter unobscured telescope with r0 = 10 cm. The solid curve is the theoretical prediction, and the dashed curve is the simulation output.

Fig. 5
Fig. 5

Compensated expected single-frame power spectrum SNR for r0 = 0.1 m for the actuator grid spacings of 1, 10 cm; 2, 20 cm; 3, 30 cm; and 4, 50 cm. The object is a point source that provides 10,000 photoevents per frame, on average.

Fig. 6
Fig. 6

Compensated expected single-frame power spetrum SNR for r0 = 0.2 m for the actuator grid spacings of 1, 10 cm; 2 20 cm; 3, 30 cm; and 4, 50 cm. The object is a point source that provides 10,000 photoevents per frame, on average.

Fig. 7
Fig. 7

Object used to examine the phase spectrum estimation with bispectrum postprocessing of compensated images.

Fig. 8
Fig. 8

Radially averaged mean-square phase spectrum estimation errors for the object in Fig. 7 as a function of the spatial frequency magnitude for the following actuator grid spacing cases: (a) 10 cm, (b) 20 cm, (c) 30 cm, and (d) 50 cm. The curves are labeled as follows: 1, uncompensated long-exposure case; 2, uncompensated bispectrum result; 3, compensated long-exposure case; and 4, compensated bispectrum result.

Fig. 9
Fig. 9

Radially averaged power spectrum SNR for 500-frame ensemble of the object in Fig. 6. From top to bottom the curves represent actuator spacing grids of 10, 20, 30, and 50 cm, with the bottom curve showing the case of uncompensated power spectrum estimation.

Equations (36)

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H ( u ) = r ( u ) + j i ( u ) ,
H ( u ) = N - 1 P ( x ) P ( x - u λ f ) d 2 x ,
N = P ( x ) 2 d 2 x ,
P ( x ) = P ( x ) exp [ j ( x ) ] ,
r ( u ) = s ( u ) + n r ( u ) ,
i ( u ) = n i ( u ) ,
s ( u ) = r ( u ) ,
H ( u ) = s ( u ) .
d ( x ) = k = 1 K δ ( x - x k ) ,
D ( u ) = d ( x ) exp ( - j 2 π u x ) d 2 x
= k = 1 K exp ( - j 2 π u x k ) .
ϕ i ( u ) = D ( u ) 2 .
Q ( u ) = D ( u ) 2 - K .
SNR Q ( u ) = Q ( u ) { var [ Q ( u ) ] } 1 / 2 ,
Q ( u ) = ( K ¯ ) 2 O ^ ( u ) 2 H ( u ) 2 ,
var [ Q ( u ) ] = ( K ¯ ) 2 + ( K ¯ ) 2 O ^ ( 2 u ) 2 H ( 2 u ) 2 + 2 ( K ¯ ) 3 O ^ ( u ) 2 H ( u ) 2 + ( K ¯ ) 4 O ^ ( u ) 4 × { H ( u ) 4 - [ H ( u ) 2 ] 2 } .
H ( u ) 4 = 3 σ r 4 + 6 σ r 2 s 2 + s 4 + 2 ( σ r 2 + s 2 ) σ i 2 + 3 σ i 4 .
D ( x 1 , x 2 ) = 6.88 ( x 1 - x 2 r 0 ) 5 / 3 ,
H LE ( u ) = H 0 ( u ) exp [ - 0.5 D ( u λ f ) ] ,
r j ( x ) = exp [ - ( x - x j ) 2 L 2 ] ,
SNR n = n SNR 1 ,
Q ( u ) = ( K ¯ ) 2 ϕ ^ i ( u ) ,
ϕ ^ i ( u ) = O ^ ( u ) 2 H ( u ) 2 ,
Q ( u ) = ( K ¯ ) 2 O ^ ( u ) 2 H ( u ) 2 ,
var [ Q ( u ) ] = D ( u ) 4 - 2 K D ( u ) 2 + K 2 - ( K ¯ ) 4 ϕ ^ i 2 ( u ) .
D ( u ) 4 = K ¯ + 2 ( K ¯ ) 2 + 4 ( 1 + K ¯ ) ( K ¯ ) 2 ϕ ^ i ( u ) + ( K ¯ ) 2 ϕ ^ i ( 2 u ) + Λ ( u ) 4 ,
Λ ( u ) 4 uncomp = 2 ( K ¯ ) 4 O ^ ( u ) 4 [ H ( u ) 2 ] 2 .
Λ ( u ) 4 comp = ( K ¯ ) 4 O ^ ( u ) 4 [ H ( u ) 4 ] .
K D ( u ) 2 = K ¯ + ( K ¯ ) 2 + ( K ¯ ) 2 ( K ¯ + 2 ) ϕ ^ i ( u ) ,
K 2 = ( K ¯ ) 2 + K ¯ ,
var [ Q ( u ) ] = ( K ¯ ) 2 + ( K ¯ ) 2 O ^ ( 2 u ) 2 H ( 2 u ) 2 + 2 ( K ¯ ) 3 O ^ ( u ) 2 H ( u ) 2 + ( K ¯ ) 4 O ^ ( u ) 4 × { H ( u ) 4 - [ H ( u ) 2 ] 2 } .
H ( u ) 4 = { [ s ( u ) + n r ( u ) ] 2 + [ n i ( u ) ] 2 } 2 .
H ( u ) 4 = n i 4 + 2 n i 2 n r 2 + n r 4 + 4 n i 2 n r s + 4 n r 3 s + 2 n i 2 s 2 + 6 n r 2 s 2 + 4 n r s 3 + s 4 ,
n r 4 = 3 σ r 4 ,
n i 4 = 3 σ i 4 .
H ( u ) 4 = 3 σ r 4 + 6 σ r 2 s 2 + s 4 + 2 ( σ r 2 + s 2 ) σ i 2 + 3 σ i 4 ,

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