Abstract

Many current wave-front-reconstruction systems use localized phase-slope measurements to estimate wave fronts distorted by atmospheric turbulence. Analytical expressions giving the performance of this class of adaptive-optics system are derived. Performance measures include the mean-square residual phase error across the aperture, the optical transfer function, the point-spread function, and the Strehl ratio. Numerical examples show that the mean-square residual error and the Strehl ratio are sensitive to variations of the photon noise in the wave-front sensor and to variations in the sensor spacing and the actuator spacing. The Strehl ratio degrades rapidly as the diameters of the individual slope sensors are made larger than the Fried seeing-cell diameter r0 and when the sensor signal levels fall below 100 counts per slope measurement. On the other hand, the resolution of the optical system is relatively unaffected by moderate changes in the photon noise or the densities of sensors and actuators. The diameter of the individual slope sensors can be as much as 1.5 times r0 without significant degradation in angular resolution. These performance measures are particularly important in the design of adaptive telescopes used for imaging in astronomy. For adaptive telescopes using laser guide stars, these measures can be used to determine the key design parameters for the laser.

© 1989 Optical Society of America

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References

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  1. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  2. E. P. Wallner, “Optimal wave-front correction using slope measurements,”J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  3. E. P. Wallner, “Comparison of wave front sensor configurations using optimal reconstruction and correction,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds., Proc. Soc. Photo-Opt. Instrum. Eng.351, 42–53 (1982).
    [CrossRef]
  4. 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]
  5. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,”J. Opt. Soc. Am. 67, 375–377 (1977).
    [CrossRef]
  6. R. J. Noll, “Phase estimates from slope-type wave-front sensors,”J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  7. J. Herrmann, “Least-squares wave front errors of minimum norm,”J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  8. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,”J. Opt. Soc. Am. 70, 998–1005 (1980).
    [CrossRef]
  9. K. Freischlad, C. Koliopoulos, “Wave front reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.551, 74–80 (1985).
    [CrossRef]
  10. J. Hermann, “Cross coupling and aliasing in modal wave-front estimation,”J. Opt. Soc. Am. 71, 989–992 (1981).
    [CrossRef]
  11. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,”J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  12. R. H. Hudgin, “Optimal wave-front estimation,”J. Opt. Soc. Am. 67, 378–382 (1977).
    [CrossRef]
  13. R. Hudgin, “Wave-front compensation error due to finite corrector-element size,”J. Opt. Soc. Am. 67, 393–395 (1977).
    [CrossRef]
  14. J. P. Gaffard, C. Boyer, “Adaptive optics for optimization of image resolution,” Appl. Opt. 26, 3772–3777 (1987).
    [CrossRef] [PubMed]
  15. D. P. Greenwood, “Mutual coherence function of a wave front corrected by zonal adaptive optics,”J. Opt. Soc. Am. 69, 549–554 (1979).
    [CrossRef]
  16. J. Y. Wang, “Optical resolution through a turbulent medium with adaptive phase compensations,”J. Opt. Soc. Am. 67, 383–390 (1977).
    [CrossRef]
  17. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,”J. Opt. Soc. Am. 68, 78–87 (1978).
    [CrossRef]
  18. L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
    [CrossRef]
  19. C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” submitted to Proc. IEEE.
  20. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 364.
  21. A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
    [CrossRef]
  22. E. S. Claflin, N. Baraket, “Configuring an electrostatic membrane mirror by least-squares fitting with analytically derived influence functions,” J. Opt. Soc. Am. A 3, 1833–1839 (1986).
    [CrossRef]
  23. P. M. Morse, H. Fesbback, Methods of Theoretical Physics, Part II (McGraw-Hill, New York, 1953), p. 1191.
  24. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,”J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  25. K. A. Winick, “Cramér–Rao lower bounds on the performance of charge-coupled-device optical position estimators,” J. Opt. Soc. Am. A 3, 1809–1815 (1986).
    [CrossRef]
  26. T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 160–171 (1989).
    [CrossRef]
  27. V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), p. 35.
  28. R. C. Smithson, Michal L. Peri, Robert S. Benson, “Quantitative simulation of image correction for astronomy with a segmented active mirror,” Appl. Opt. 27, 1615–1620 (1988).
    [CrossRef] [PubMed]
  29. R. C. Smithson, Michal L. Peri, “Partial correction of astronomical images with active mirrors,” J. Opt. Soc. Am. A 6, 92–97 (1989).
    [CrossRef]
  30. J. M. Beckers, F. J. Roddier, P. R. Eisenhardt, L. E. Goad, K.-L. Shu, “National Optical Astronomy observatories (NOAO) Infrared Adaptive Optics Program I: general description,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 290–297 (1986).
    [CrossRef]
  31. B. M. Welsh, C. S. Gardner, “Nonlinear resonant absorption effects on the design of resonance fluorescence lidars and laser guide stars,” Appl. Opt. 28, 4141–4153 (1989).
    [CrossRef] [PubMed]
  32. B. M. Welsh, C. S. Gardner, L. A. Thompson, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
    [CrossRef]
  33. D. L. Fried, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52–61 (1982).
    [CrossRef]

1989 (2)

1988 (1)

1987 (2)

J. P. Gaffard, C. Boyer, “Adaptive optics for optimization of image resolution,” Appl. Opt. 26, 3772–3777 (1987).
[CrossRef] [PubMed]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
[CrossRef]

1986 (2)

1983 (1)

1982 (1)

1981 (1)

1980 (2)

1979 (2)

1978 (3)

1977 (5)

1974 (1)

A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
[CrossRef]

1966 (1)

Baraket, N.

Beckers, J. M.

J. M. Beckers, F. J. Roddier, P. R. Eisenhardt, L. E. Goad, K.-L. Shu, “National Optical Astronomy observatories (NOAO) Infrared Adaptive Optics Program I: general description,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 290–297 (1986).
[CrossRef]

Benson, Robert S.

Boyer, C.

Claflin, E. S.

Cubalchini, R.

Eisenhardt, P. R.

J. M. Beckers, F. J. Roddier, P. R. Eisenhardt, L. E. Goad, K.-L. Shu, “National Optical Astronomy observatories (NOAO) Infrared Adaptive Optics Program I: general description,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 290–297 (1986).
[CrossRef]

Fesbback, H.

P. M. Morse, H. Fesbback, Methods of Theoretical Physics, Part II (McGraw-Hill, New York, 1953), p. 1191.

Freischlad, K.

K. Freischlad, C. Koliopoulos, “Wave front reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.551, 74–80 (1985).
[CrossRef]

Fried, D. L.

Gaffard, J. P.

Gardner, C. S.

B. M. Welsh, C. S. Gardner, “Nonlinear resonant absorption effects on the design of resonance fluorescence lidars and laser guide stars,” Appl. Opt. 28, 4141–4153 (1989).
[CrossRef] [PubMed]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
[CrossRef]

B. M. Welsh, C. S. Gardner, L. A. Thompson, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
[CrossRef]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” submitted to Proc. IEEE.

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 160–171 (1989).
[CrossRef]

Goad, L. E.

J. M. Beckers, F. J. Roddier, P. R. Eisenhardt, L. E. Goad, K.-L. Shu, “National Optical Astronomy observatories (NOAO) Infrared Adaptive Optics Program I: general description,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 290–297 (1986).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 364.

Greenwood, D. P.

Hardy, J. W.

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

Hermann, J.

Herrmann, J.

Hudgin, R.

Hudgin, R. H.

Kane, T. J.

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 160–171 (1989).
[CrossRef]

Koliopoulos, C.

K. Freischlad, C. Koliopoulos, “Wave front reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.551, 74–80 (1985).
[CrossRef]

Markey, J. K.

Morse, P. M.

P. M. Morse, H. Fesbback, Methods of Theoretical Physics, Part II (McGraw-Hill, New York, 1953), p. 1191.

Noll, R. J.

Peri, Michal L.

Roddier, F. J.

J. M. Beckers, F. J. Roddier, P. R. Eisenhardt, L. E. Goad, K.-L. Shu, “National Optical Astronomy observatories (NOAO) Infrared Adaptive Optics Program I: general description,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 290–297 (1986).
[CrossRef]

Shu, K.-L.

J. M. Beckers, F. J. Roddier, P. R. Eisenhardt, L. E. Goad, K.-L. Shu, “National Optical Astronomy observatories (NOAO) Infrared Adaptive Optics Program I: general description,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 290–297 (1986).
[CrossRef]

Smithson, R. C.

Southwell, W. H.

Tatarski, V. I.

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), p. 35.

Thompson, L. A.

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
[CrossRef]

B. M. Welsh, C. S. Gardner, L. A. Thompson, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
[CrossRef]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” submitted to Proc. IEEE.

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 160–171 (1989).
[CrossRef]

Wallner, E. P.

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

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

Wang, J. Y.

Welsh, B. M.

B. M. Welsh, C. S. Gardner, “Nonlinear resonant absorption effects on the design of resonance fluorescence lidars and laser guide stars,” Appl. Opt. 28, 4141–4153 (1989).
[CrossRef] [PubMed]

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 160–171 (1989).
[CrossRef]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” submitted to Proc. IEEE.

B. M. Welsh, C. S. Gardner, L. A. Thompson, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
[CrossRef]

Winick, K. A.

Young, A. T.

A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
[CrossRef]

Appl. Opt. (3)

Astrophys. J. (1)

A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
[CrossRef]

J. Opt. Soc. Am. (15)

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,”J. Opt. Soc. Am. 56, 1372–1379 (1966).
[CrossRef]

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]

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,”J. Opt. Soc. Am. 67, 375–377 (1977).
[CrossRef]

R. H. Hudgin, “Optimal wave-front estimation,”J. Opt. Soc. Am. 67, 378–382 (1977).
[CrossRef]

J. Y. Wang, “Optical resolution through a turbulent medium with adaptive phase compensations,”J. Opt. Soc. Am. 67, 383–390 (1977).
[CrossRef]

R. Hudgin, “Wave-front compensation error due to finite corrector-element size,”J. Opt. Soc. Am. 67, 393–395 (1977).
[CrossRef]

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

J. Herrmann, “Least-squares wave front errors of minimum norm,”J. Opt. Soc. Am. 70, 28–35 (1980).
[CrossRef]

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,”J. Opt. Soc. Am. 70, 998–1005 (1980).
[CrossRef]

J. Hermann, “Cross coupling and aliasing in modal wave-front estimation,”J. Opt. Soc. Am. 71, 989–992 (1981).
[CrossRef]

D. L. Fried, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52–61 (1982).
[CrossRef]

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

D. P. Greenwood, “Mutual coherence function of a wave front corrected by zonal adaptive optics,”J. Opt. Soc. Am. 69, 549–554 (1979).
[CrossRef]

R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,”J. Opt. Soc. Am. 69, 972–977 (1979).
[CrossRef]

R. J. Noll, “Phase estimates from slope-type wave-front sensors,”J. Opt. Soc. Am. 68, 139–140 (1978).
[CrossRef]

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

Nature (1)

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
[CrossRef]

Proc. IEEE (1)

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

Other (9)

K. Freischlad, C. Koliopoulos, “Wave front reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.551, 74–80 (1985).
[CrossRef]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” submitted to Proc. IEEE.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 364.

P. M. Morse, H. Fesbback, Methods of Theoretical Physics, Part II (McGraw-Hill, New York, 1953), p. 1191.

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 160–171 (1989).
[CrossRef]

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), p. 35.

J. M. Beckers, F. J. Roddier, P. R. Eisenhardt, L. E. Goad, K.-L. Shu, “National Optical Astronomy observatories (NOAO) Infrared Adaptive Optics Program I: general description,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 290–297 (1986).
[CrossRef]

B. M. Welsh, C. S. Gardner, L. A. Thompson, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
[CrossRef]

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

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Figures (13)

Fig. 1
Fig. 1

Configuration of membrane mirror and actuator pad.

Fig. 2
Fig. 2

Membrane actuator response for actuators positioned along the x axis. The center of the membrane is at x = 0, and the edge is at x = 10.

Fig. 3
Fig. 3

Configuration of wave-front sensor.

Fig. 4
Fig. 4

Single Hartmann tilt sensor.

Fig. 5
Fig. 5

Average rms residual phase error versus subaperture photon count. The aperture size D ranges from 6L to 2L. The other parameters are constant: L = r0 and τ = 0.

Fig. 6
Fig. 6

Average rms residual phase error versus subaperture photon count. The subaperture size L ranges from 1.5r0 to 0.5r0. The other parameters are constant: D = 4L and τ = 0.

Fig. 7
Fig. 7

Average rms residual phase error versus subaperture photon count. The dimensionless quantity /r0 ranges from 0 to 2.5. The other parameters are constant: D = 4L and L = r0.

Fig. 8
Fig. 8

OTF for the subaperture photon count ranging from 10 to 500. The other parameters are constant: D = 4L, L = r0, and τ = 0.

Fig. 9
Fig. 9

PSF for the subaperture photon count ranging from 10 to 500. The other parameters are constant: D = 4L, L = r0, and τ = 0.

Fig. 10
Fig. 10

Strehl ratio versus subaperture photon count. The other parameters are constant: D = 4L, L = r0, and τ = 0.

Fig. 11
Fig. 11

OTF for the subaperture size L ranging from 1.5r0 to 0.5r0. The other parameters are constant: D = 4L, τ = 0, and N = 500.

Fig. 12
Fig. 12

PSF for the subaperture size L ranging from 1.5r0 to 0.5r0. The other parameters are constant: D = 4L, τ = 0, and N = 500.

Fig. 13
Fig. 13

Strehl ratio versus subaperture size. The other parameters are constant: D = 4L, τ = 0, and N = 500.

Equations (43)

Equations on this page are rendered with MathJax. Learn more.

d 2 x W A ( x ) = 1 ,
ϕ ( x , t ) = ψ ( x , t ) - d 2 x W A ( x ) ψ ( x , t ) .
s n ( t ) = d 2 x W n ( x ) [ ϕ ( x , t ) · d ^ n ] + α n ( t ) ,
s n ( t ) = - d 2 x [ W n ( x ) · d ^ n ] ϕ ( x , t ) + α n ( t ) .
c j ( t ) = n M j n s n ( t ) ,
ϕ ˜ ( x , t ) = j - t c j ( ξ ) r j ( x , t - ξ ) d ξ ,
ϕ ˜ ( x , t ) - j c j ( t ) r j ( x ) .
( x , t , τ ) = ϕ ˜ ( x , t - τ ) - ϕ ( x , t ) = j r j ( x ) n M j n s n ( t - τ ) - ϕ ( x , t ) ,
2 ( x , t , τ ) = j j n n r j ( x ) r j ( x ) M j n M j n × s n ( t - τ ) s n ( t - τ ) - 2 j n r j ( x ) M j n × s n ( t - τ ) ϕ ( x , t ) + ϕ 2 ( x , t ) ,
( τ ) 2 = d 2 x W A ( x ) 2 ( x , τ ) .
( τ ) 2 = j j n n M j n M j n S n n R j j - 2 j n M j n A j n ( τ ) + 0 2 ,
S n n = s n ( t ) s n ( t ) = d 2 x d 2 x W n s ( x ) W n s ( x ) ϕ ( x , t ) ϕ ( x , t ) + α n ( t ) α n ( t ) ,
R j j = d 2 x W A ( x ) r j ( x ) r j ( x ) ,
A j n ( τ ) = d 2 x W A ( x ) r j ( x ) s n ( t - τ ) ϕ ( x , t ) = - d 2 x d 2 x W A ( x ) r j ( x ) W n s ( x ) × ϕ ( x , t ) ϕ ( x , t - τ ) ,
0 2 = d 2 x W A ( x ) ϕ 2 ( x , t ) .
M j n * ( τ ) = R j j - 1 A j n ( τ ) S n n - 1 ,
( τ ) 2 min = 0 2 - R j j - 1 A j n ( τ ) S n n - 1 A j n ( τ ) .
H ( ρ ) = d 2 x W A ( x ) E ( x ) W A * ( x - ρ ) E * ( x - ρ ) d 2 x W A ( x ) E ( x ) 2 ,
H ( ρ ) = d 2 x W A ( x ) E ( x ) W A * ( x - ρ ) E * ( x - ρ ) exp { j [ ξ ( x ) - ξ ( x - ρ ) ] } d 2 x W A ( x ) E ( x ) 2 ,
H ( ρ ) = d 2 x W A ( x ) W A * ( x - ρ ) exp { j [ ( x , t , τ ) - ( x - ρ , t , τ ) ] } d 2 x W A ( x ) 2 .
H ( ρ ) = d 2 x W A ( x ) W A * ( x - ρ ) exp { j [ ( x , t , τ ) - ( x - ρ , t , τ ) ] } d 2 x W A ( x ) 2 .
H ( ρ ) = d 2 x W A ( x ) W A * ( x - ρ ) exp [ ( - 1 / 2 ) [ ( x , t , τ ) - ( x - ρ , t , τ ) ] 2 ] d 2 x W A ( x ) 2 .
H ( ρ ) = exp { - [ ϕ ( x , t ) - ϕ ( x - ρ , t ) ] 2 2 } d 2 x W A ( x ) 2 d 2 x W A ( x ) W A * ( x - ρ ) × ( exp { ( - 1 / 2 ) j i [ r j ( x ) - r i ( x - ρ ) ] C j i - r j ( x - ρ ) ] [ r i ( x ) - r i ( x - ρ ) ] C j i + j [ r j ( x ) - r j ( x - ρ ) ] c j ( t - τ ) [ ϕ ( x , t ) - ϕ ( x - ρ , t ) ] } ) ,
C j i = c j ( t ) c i ( t ) = n m M j n M i m S n m .
D ( x , x , τ ) = [ ψ ( x , t ) - ψ ( x , t - τ ) ] 2 .
S n n = ( - 1 / 2 ) d 2 x d 2 x W n s ( x ) W n s ( x ) D ( x , x , 0 ) + α n ( t ) α n ( t ) ,
A j n ( τ ) = - d 2 x d 2 x W A ( x ) r j ( x ) W n s ( x ) × [ ( - 1 / 2 ) D ( x , x , τ ) + g ( x , τ ) ] ,
g ( x , τ ) = ( 1 / 2 ) d 2 x W A ( x ) D ( x , x , τ ) .
H ( ρ ) = exp [ - D ( x , x - ρ , 0 ) 2 ] d 2 x W A ( x ) 2 d 2 x 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 i j + j [ r j ( x ) - r j ( x - ρ ) ] × c j ( t - τ ) [ ϕ ( x , t ) - ϕ ( x - ρ , t ) ] } ) ,
c j ( t - τ ) [ ϕ ( x , t ) - ϕ ( x - ρ , t ) ] = ( 1 / 2 ) n M j n d 2 x W n s ( x ) [ D ( x , x , τ ) - D ( x - ρ , x , τ ) ] .
α n ( t ) α n ( τ ) = σ α n 2 k n n ( t - τ ) ,
δ n n = { 1 n th and n th subapertures coincide 0 otherwise .
r j ( x , y ) exp [ - ( x - x j ) 2 - ( y - y j ) 2 L a 2 ] ,
2 r j ( x , y ) = - P j ( x , y ) / T ,
r j ( α ρ , ϕ ) = α 2 2 π T 0 2 π d ϕ ( 0 ρ ρ d ρ { ln ( 1 / ρ ) - n = 1 1 n [ ( ρ ρ ) n ] - ( ρ / ρ ) n ] cos [ n ( ϕ - ϕ ) ] } P j ( ρ , ϕ ) + ρ 1 ρ d ρ { ln ( 1 / ρ ) - n = 1 1 n [ ( ρ ρ ) n - ( ρ / ρ ) n ] × cos [ n ( ϕ - ϕ ) ] } P j ( ρ , ϕ ) ) ,
σ α = 2 π Δ x λ f L ,
Δ x = η σ I N 1 / 2 ,
σ I = { 1.22 λ f L 2 2 r 0 L > r 0 1.05 λ f L 2 2 L L r 0 .
σ α = { 0.86 π η N 1 / 2 r 0 L > r 0 0.74 π η N 1 / 2 L L r 0 .
D ( x , x , τ ) = D ( x , x + v τ ) = D ( x - x - v τ ) ,
D ( x , x , τ ) = 6.8839 ( x - x - v τ r 0 ) 5 / 3 .
s ( u / λ f D , v / λ f D ) = F 2 - 1 [ H ( x , y ) ] ( λ f D ) 2 ,
s ( u / λ f D ) = d v s ( u / λ f D , v / λ f D ) = d x H ( x , 0 ) λ f D exp [ j 2 π x ( u λ f D ) ] = F 1 - 1 [ H ( x , 0 ) ] λ f D ,

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