Abstract

The use of limited degree-of-freedom adaptive optics in conjunction with statistical averaging and a linear image reconstruction algorithm is addressed. Image reconstruction is traded for full predetection compensation. It is shown through analytic calculations that the average optical transfer function (OTF) is significant for high spatial frequencies in the case of imaging through atmospheric turbulence with an adaptive optics system composed of a Hartmann-type wave-front sensor and a deformable mirror possessing far fewer actuators than one per atmospheric coherence diameter (r0). Statistical averaging is used to overcome the effects of measurement noise and randomness in individual realizations of the OTF. The imaging concept and signal-to-noise considerations are presented.

© 1991 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, Vol. 19, E. Wolf, ed. (North-Holland, Amsterdam, 1981).
    [CrossRef]
  2. G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox-Thompson and triple correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
    [CrossRef]
  3. J. C. Dainty, “Speckle imaging techniques,” in Digital Imaging Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 2–7 (1987).
  4. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  5. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  6. D. Korff, “Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence,” J. Opt. Soc. Am. 63, 971–980 (1973).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. E. P. Walner, “Comparison of wavefront sensor configurations using optimal reconstruction and correction,” in Wavefront Sensing, N. Bareket, C. Koliopoulos, eds., Proc. Soc. Photo-Opt. Instrum. Eng.351, 42–53 (1982).
  9. E. P. Walner, “Optimal wavefront correction using slope measurements,” J. Opt. Soc. Am 73, 1771–1776 (1983).
    [CrossRef]
  10. 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]
  11. P. Nisenson, R. Barakat, “Partial correction with adaptive optics,” J. Opt. Soc. Am. A 4, 2249–2253 (1987).
    [CrossRef]
  12. R. C. Smithson, M. L. Peri, “Partial correction of astronomical images with active mirrors,” J. Opt. Soc. Am. A 6, 92–97 (1989).
    [CrossRef]
  13. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
    [CrossRef]
  14. R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1987).
  15. J. P. Gaffard, C. Boyer, “Adaptive optics for optimization of image resolution,” Appl. Opt. 26, 3772–3777 (1987).
    [CrossRef] [PubMed]
  16. B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
    [CrossRef]
  17. C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
    [CrossRef]

1991 (1)

1990 (1)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

1989 (2)

1988 (1)

1987 (2)

1983 (1)

E. P. Walner, “Optimal wavefront correction using slope measurements,” J. Opt. Soc. Am 73, 1771–1776 (1983).
[CrossRef]

1978 (2)

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

J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
[CrossRef]

1973 (1)

Ayers, G. R.

Barakat, R.

Boyer, C.

Dainty, J. C.

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox-Thompson and triple correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
[CrossRef]

J. C. Dainty, “Speckle imaging techniques,” in Digital Imaging Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 2–7 (1987).

Gaffard, J. P.

Gardner, C. S.

Gonzalez, R. C.

R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1987).

Goodman, J. W.

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hardy, J. W.

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

Korff, D.

Markey, J. K.

Nisenson, P.

Northcott, M. J.

Peri, M. L.

Roddier, F.

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

Smithson, R. C.

Thompson, L. A.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Walner, E. P.

E. P. Walner, “Optimal wavefront correction using slope measurements,” J. Opt. Soc. Am 73, 1771–1776 (1983).
[CrossRef]

E. P. Walner, “Comparison of wavefront sensor configurations using optimal reconstruction and correction,” in Wavefront Sensing, N. Bareket, C. Koliopoulos, eds., Proc. Soc. Photo-Opt. Instrum. Eng.351, 42–53 (1982).

Wang, J. Y.

Welsh, B. M.

Wintz, P.

R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1987).

Appl. Opt. (1)

J. Opt. Soc. Am (1)

E. P. Walner, “Optimal wavefront correction using slope measurements,” J. Opt. Soc. Am 73, 1771–1776 (1983).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Proc. IEEE (2)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

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

Other (6)

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

E. P. Walner, “Comparison of wavefront sensor configurations using optimal reconstruction and correction,” in Wavefront Sensing, N. Bareket, C. Koliopoulos, eds., Proc. Soc. Photo-Opt. Instrum. Eng.351, 42–53 (1982).

J. C. Dainty, “Speckle imaging techniques,” in Digital Imaging Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 2–7 (1987).

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

R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1987).

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Figures (9)

Fig. 1
Fig. 1

Diffraction-limited OTF, average short-exposure OTF, and average long-exposure OTF versus the normalized spatial frequency for the case of r0 = 10% telescope diameter.

Fig. 2
Fig. 2

Telescope pupil and subaperture and actuator geometry for the fully compensated system. Subapertures are represented by squares; actuators are represented by the symbol ●.

Fig. 3
Fig. 3

Diffraction-limited, average-compensated, average-short-exposure, and average-long-exposure OTF's versus the normalized spatial frequency for the fully compensated case.

Fig. 4
Fig. 4

Telescope pupil and subaperture and actuator geometry for Reduced Actuator Case I. Subapertures are represented by squares; actuators are represented by the symbol ●.

Fig. 5
Fig. 5

Diffraction-limited, average-compensated, average-short-exposure, and average-long-exposure OTF's versus normalized spatial frequency for Reduced Actuator Case I.

Fig. 6
Fig. 6

Telescope pupil and subaperture and actuator geometry for Reduced Actuator Case II. Subapertures are represented by squares; actuators are represented by the symbol ●.

Fig. 7
Fig. 7

Diffraction-limited, average-compensated, average-short-exposure, and average-long-exposure OTF's versus normalized spatial frequency for Reduced Actuator Case II.

Fig. 8
Fig. 8

SNR versus normalized spatial frequency for Reduced Actuator Case II, r0 = 10% of the telescope diameter, the average number of photoevents per integration time = 100, 1000, 10,000.

Fig. 9
Fig. 9

Block diagram of image measurement and reconstruction technique.

Equations (37)

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i ( x ) = h ( x ) * o ( x ) ,
I ( u ) = H ( u ) I ( u ) ,
H LE ( u ) = exp [ 3.44 ( λ m f u / r 0 ) 5 / 3 ] H 0 ( u ) ,
H SE ( u ) = exp { 3.44 ( λ m f u / r 0 ) 5 / 3 [ 1 ( λ m f u / D 0 ) 1 / 3 ] } H 0 ( u ) ,
W A ( x ) d 2 x = 1 .
ϕ ( x ) = ψ ( x ) W A ( x ) ψ ( x ) d 2 x .
D ( x , x ) = 6.8839 ( | x x | / r 0 ) 5 / 3 .
ϕ ~ ( x ) = j c j r j ( x ) ,
( x ) = ϕ ~ ( x ) ϕ ( x ) .
c j = j M j n s n ,
M j n = R j j 1 A j n S n n 1 .
R j j = r j ( x ) r j ( x ) d 2 x ,
S n n = ( 1 / 2 ) W n s ( x ) W n s ( x ) D ( x , x ) d 2 x d 2 x + α n α n ,
A j n = W A ( x ) r j ( x ) W n s ( x ) [ ( 1 / 2 ) D ( x , x ) + g ( x ) ] d 2 x d 2 x ,
g ( x ) = D ( x , x ) W A ( x ) d 2 x .
r j ( x ) = exp ( | x x j | 2 / L 2 ) W A ( x ) exp ( | x x j | 2 / L 2 ) d 2 x ,
H ( ρ ) = N 1 W A ( x ) W A * ( x ρ ) exp { j [ ( x ) ( x ρ ) ] } d 2 x = N 1 exp [ D ( x , x ρ ) / 2 ]
× W A ( x ) W A * ( x ρ ) exp { ( 1 2 ) j i [ r j ( x ) r j ( x ρ ) ] × [ r i ( x ) r i ( x ρ ) ] C j i + j [ r j ( x ) r j ( x ρ ) ] × c j [ ϕ ( x ) ϕ ( x ρ ) ] } d 2 x ,
C j i = c j c i = n m M j n M i m S n m ,
c j [ ϕ ( x ) ϕ ( x ρ ) ] = ( 1 2 ) n M j n W n s ( x ) × [ D ( x , x ) D ( x ρ , x ) ] d 2 x ,
N = | W A ( x ) | 2 d 2 x ,
u = ρ / λ f .
var { H ( ρ ) } = | H ( ρ ) | 2 | H ( ρ ) | 2 .
| H ( ρ ) | 2 = N 2 W A ( x 1 ) W A * ( x 1 ρ ) exp { j [ ( x 1 ) ( x 1 ρ ) ] } d 2 x 1 × W A * ( x 2 ) W A ( x 2 ρ ) exp { j [ ( x 2 ) ( x 2 ρ ) ] } d 2 x 2
= N 2 W A ( x 1 ) W A * ( x 1 ρ ) W A * ( x 2 ) W A ( x 2 ρ ) exp ( j { [ ( x 1 ) ( x 1 ρ ) ] [ ( x 2 ) ( x 2 ρ ) ] } ) d 2 x 1 d 2 x 2 .
| H ( ρ ) | 2 = N 2 W A ( x 1 ) W A * ( x 1 ρ ) W A * ( x 2 ) W A ( x 2 ρ ) × exp ( ( ½ ) { [ ( x 1 ) ( x 1 ρ ) [ ( x 2 ) ( x 2 ρ ) ] } 2 ) d 2 x 1 d 2 x 2 .
( 1 / 2 ) { [ ( x 1 ) ( x 1 ρ ) ] [ ( x 2 ) ( x 2 ρ ) ] } 2 = ( 1 / 2 ) [ ( x 1 ) ( x 1 ρ ) ] 2 + ( 1 / 2 ) [ ( x 2 ) ( x 2 ρ ) ] 2 + [ ( x 1 ) ( x 1 ρ ) ] [ ( x 2 ) ( x 2 ρ ) ] .
( 1 / 2 ) [ ( x ) ( x ρ ) ] 2 = ( 1 / 2 ) j i [ r j ( x ) r j ( x ρ ) ] [ r i ( x ) r i ( x ρ ) ] C j i + j [ r j ( x ) r j ( x ) ρ ) ] c j [ ϕ ( x ) ϕ ( x ρ ) ] + ( 1 / 2 ) D ( x , x ρ ) .
( x 1 ) ( x 2 ) = j i c j c i r j ( x 1 ) r i ( x 2 ) j c j r j ( x 1 ) ϕ ( x 2 ) j c j r j ( x 2 ) ϕ ( x 1 ) + ϕ ( x 1 ) ϕ ( x 2 ) ,
ϕ ( x 1 ) ϕ ( x 2 ) = ( ½ ) D ( x 1 , x 2 ) + g ( x 1 ) + g ( x 2 ) a ,
a = W A ( x ) g ( x ) d 2 x .
[ ( x 1 ) ( x 1 ρ ) ] [ ( x 2 ) ( x 2 ρ ) ] = j i c j i [ r j ( x 1 ) r j ( x 1 ρ ) ] [ r i ( x 2 ) r i ( x 2 ρ ) ] j [ r j ( x 1 ) r j ( x 1 ρ ) ] c j [ ϕ ( x 2 ) ϕ ( x 2 ρ ) ] j [ r j ( x 2 ) r j ( x 2 ρ ) ] c j [ ϕ ( x 1 ) ϕ ( x 1 ρ ) ] D ( x 1 , x 2 ) + D ( x j , x 2 ρ ) .
SNR ( u ) = | I ( u ) | / [ var { I ( u ) } ] 1 / 2 ,
SNR ( u ) = K 1 / 2 | H ( u ) O n ( u ) | ,
SNR ( u ) = K | H ( u ) O n ( u ) | / [ K + K 2 | O n ( u ) | 2 var { H ( u ) } ] 1 / 2 ,
SNR E ( u ) = ( M 1 / 2 ) SNR 1 ( u ) .
O ~ ( u ) = I j ( u ) / P m ( u ) .

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