Abstract

A fresh look is taken at how light propagates through the atmosphere and how atmospheric turbulence affects images formed by large ground-based telescopes. Telescopes with fixed and adaptive optics are considered. The approach is based on a layered model of the atmosphere. It is shown that the atmosphere can be represented by an equivalent phase screen for the two quantities that determine most of the important image properties—the atmospheric modulation transfer function and the spectral correlation function. Techniques are described for measuring the parameters that define the equivalent phase screen. Expressions are given in terms of screen parameters for a number of image properties. Many of these properties are different from those in the conventional literature. Diffraction-limited cores in star images are discussed. An optimum wavelength at which resolution is maximized is also discussed. Resolution of the order of 0.05 arcsec is possible at this wavelength, but only if the telescope is near diffraction limited. The optimum wavelength can be used to produce maximum energy density at the focus of a ground-based laser beam directed at a target in space.

© 1991 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  2. R. E. Hufnagel, N. R. Stanley, “Modulation transfer function associated with image transmission through turbulent media,”J. Opt. Soc. Am. 54, 52–61 (1964).
    [CrossRef]
  3. A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).
  4. A. N. Kolmogorov, “Dissipation of energy in locally isotropic turbulence,” Dokl. Akad. Nauk SSSR 32, 16–18 (1941).
  5. 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]
  6. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  7. J. C. Dainty, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984).
  8. 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]
  9. J. W. Strohbehn, Laser Beam Propagation in the Atmosphere, Vol. 25 of Topics in Applied Physics (Springer-Verlag, Berlin, 1985).
  10. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.
  11. E. Wolf, Progress in Optics (North-Holland, Amsterdam, 1980), Vol. 19, p. 281.
  12. T. S. McKechnie, “Light propagation through the atmosphere and imaging in astronomy” (personal communication to J. C. Dainty et al., 1976).
  13. A. E. Gologly, “The optimization of the filter bandwidths and exposure times used in photographic recording,” M.S. thesis (Imperial College, London, 1976).
  14. T. S. McKechnie, “Cores in star images,”J. Opt. Soc. Am. 66, 635(A) (1976).
  15. E. Wolf, Progress in Optics (North-Holland, Amsterdam, 1985), p. 343.
  16. T. S. McKechnie, “Propagation of the spectral correlation function in a homogeneous medium,” J. Opt. Soc. Am. A 8, 339–345 (1991).
    [CrossRef]
  17. Also present on the 1976 trip to Royal Greenwich Observatory were J. C. Dainty, G. Parry, R. S. Scaddan, and J. G. Walker.
  18. R. F. Griffin, “On image structure, and the value and challenge of very large telescopes,” Observatory 93, 3–8 (1973).
  19. A. Papoulis, Probability, Random Variables and Stocastic Processes (McGraw-Hill, New York, 1965).
  20. At the second phase screen, the amplitude fluctuations are extremely weak, but obviously with further propagation stronger amplitude fluctuations develop.
  21. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).
  22. I. S. Reed, “On a moment theorem for complex Gaussian processes,”IRE Trans. Inf. Theory IT-8, 194–195 (1962).
    [CrossRef]
  23. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1970).
  24. J. W. Goodman, Stanford Electronics Laboratories Tech. Rep. TR2303-1 (SEL-63-140) (Stanford University, Stanford, Calif., 1963).
  25. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 82.
  26. R. A. Sprague, “Surface roughness measurement using white light speckle,” Appl. Opt. 11, 2811–2816 (1972).
    [CrossRef] [PubMed]
  27. M. Elbaum, M. Greenbaum, M. King, “Wavelength diversity technique for reduction of speckle,” Opt. Commun. 5, 171–174 (1972).
    [CrossRef]
  28. R. C. Smithson, M. L. Peri, R. S. Benson, “Quantitative simulation of image correction for Astronomy with a segmented active mirror,” Appl. Opt. 27, 1615–1620 (1988).
    [CrossRef] [PubMed]
  29. There may be practical limitations to the range of usable wavelengths set by the wavelength acceptance characteristics of the cavity mirror coatings.

1991 (1)

1988 (1)

1976 (1)

T. S. McKechnie, “Cores in star images,”J. Opt. Soc. Am. 66, 635(A) (1976).

1973 (2)

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]

R. F. Griffin, “On image structure, and the value and challenge of very large telescopes,” Observatory 93, 3–8 (1973).

1972 (2)

R. A. Sprague, “Surface roughness measurement using white light speckle,” Appl. Opt. 11, 2811–2816 (1972).
[CrossRef] [PubMed]

M. Elbaum, M. Greenbaum, M. King, “Wavelength diversity technique for reduction of speckle,” Opt. Commun. 5, 171–174 (1972).
[CrossRef]

1970 (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

1966 (1)

1964 (1)

1962 (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,”IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

1941 (2)

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).

A. N. Kolmogorov, “Dissipation of energy in locally isotropic turbulence,” Dokl. Akad. Nauk SSSR 32, 16–18 (1941).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1970).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 82.

Benson, R. S.

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984).

T. S. McKechnie, “Light propagation through the atmosphere and imaging in astronomy” (personal communication to J. C. Dainty et al., 1976).

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Elbaum, M.

M. Elbaum, M. Greenbaum, M. King, “Wavelength diversity technique for reduction of speckle,” Opt. Commun. 5, 171–174 (1972).
[CrossRef]

Fried, D. L.

Gologly, A. E.

A. E. Gologly, “The optimization of the filter bandwidths and exposure times used in photographic recording,” M.S. thesis (Imperial College, London, 1976).

Goodman, J. W.

J. W. Goodman, Stanford Electronics Laboratories Tech. Rep. TR2303-1 (SEL-63-140) (Stanford University, Stanford, Calif., 1963).

Greenbaum, M.

M. Elbaum, M. Greenbaum, M. King, “Wavelength diversity technique for reduction of speckle,” Opt. Commun. 5, 171–174 (1972).
[CrossRef]

Griffin, R. F.

R. F. Griffin, “On image structure, and the value and challenge of very large telescopes,” Observatory 93, 3–8 (1973).

Hufnagel, R. E.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

King, M.

M. Elbaum, M. Greenbaum, M. King, “Wavelength diversity technique for reduction of speckle,” Opt. Commun. 5, 171–174 (1972).
[CrossRef]

Kolmogorov, A. N.

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).

A. N. Kolmogorov, “Dissipation of energy in locally isotropic turbulence,” Dokl. Akad. Nauk SSSR 32, 16–18 (1941).

Korff, D.

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

McKechnie, T. S.

T. S. McKechnie, “Propagation of the spectral correlation function in a homogeneous medium,” J. Opt. Soc. Am. A 8, 339–345 (1991).
[CrossRef]

T. S. McKechnie, “Cores in star images,”J. Opt. Soc. Am. 66, 635(A) (1976).

T. S. McKechnie, “Light propagation through the atmosphere and imaging in astronomy” (personal communication to J. C. Dainty et al., 1976).

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stocastic Processes (McGraw-Hill, New York, 1965).

Peri, M. L.

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,”IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Smithson, R. C.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 82.

Sprague, R. A.

Stanley, N. R.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1970).

Strohbehn, J. W.

J. W. Strohbehn, Laser Beam Propagation in the Atmosphere, Vol. 25 of Topics in Applied Physics (Springer-Verlag, Berlin, 1985).

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Wolf, E.

E. Wolf, Progress in Optics (North-Holland, Amsterdam, 1980), Vol. 19, p. 281.

E. Wolf, Progress in Optics (North-Holland, Amsterdam, 1985), p. 343.

Appl. Opt. (2)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Dokl. Akad. Nauk SSSR (2)

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).

A. N. Kolmogorov, “Dissipation of energy in locally isotropic turbulence,” Dokl. Akad. Nauk SSSR 32, 16–18 (1941).

IRE Trans. Inf. Theory (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,”IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Observatory (1)

R. F. Griffin, “On image structure, and the value and challenge of very large telescopes,” Observatory 93, 3–8 (1973).

Opt. Commun. (1)

M. Elbaum, M. Greenbaum, M. King, “Wavelength diversity technique for reduction of speckle,” Opt. Commun. 5, 171–174 (1972).
[CrossRef]

Other (16)

Also present on the 1976 trip to Royal Greenwich Observatory were J. C. Dainty, G. Parry, R. S. Scaddan, and J. G. Walker.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1970).

J. W. Goodman, Stanford Electronics Laboratories Tech. Rep. TR2303-1 (SEL-63-140) (Stanford University, Stanford, Calif., 1963).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 82.

There may be practical limitations to the range of usable wavelengths set by the wavelength acceptance characteristics of the cavity mirror coatings.

A. Papoulis, Probability, Random Variables and Stocastic Processes (McGraw-Hill, New York, 1965).

At the second phase screen, the amplitude fluctuations are extremely weak, but obviously with further propagation stronger amplitude fluctuations develop.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

E. Wolf, Progress in Optics (North-Holland, Amsterdam, 1985), p. 343.

J. C. Dainty, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984).

J. W. Strohbehn, Laser Beam Propagation in the Atmosphere, Vol. 25 of Topics in Applied Physics (Springer-Verlag, Berlin, 1985).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

E. Wolf, Progress in Optics (North-Holland, Amsterdam, 1980), Vol. 19, p. 281.

T. S. McKechnie, “Light propagation through the atmosphere and imaging in astronomy” (personal communication to J. C. Dainty et al., 1976).

A. E. Gologly, “The optimization of the filter bandwidths and exposure times used in photographic recording,” M.S. thesis (Imperial College, London, 1976).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

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Figures (17)

Fig. 1
Fig. 1

Layered atmosphere. With an appropriate choice of layer thickness, the layers can be considered to act at statistically independent random phase screens.

Fig. 2
Fig. 2

Random phase screen atmospheric model. For the atmospheric MTF and the spectral correlation function, the atmosphere can be replaced by an equivalent phase screen.

Fig. 3
Fig. 3

Three mean levels associated with the equivalent wave. The two shown by dashed lines relate to the long- and short-exposure images.

Fig. 4
Fig. 4

Coordinate systems used in the analysis: (x, y) in the pupil plane and (X, Y) in the image plane.

Fig. 5
Fig. 5

Simulated star image in polychromatic light, using a phase screen to represent the atmosphere. Screen defined by σ =0.3 μm, w0 = 60 μm, and stopped down to D = 500 μm. The statistical character of the image (including the radial structure) is identical to that of an actual star image produced by a 1.25-m telescope in seeing defined by σ = 0.3 μm, w0 =0.15 m.

Fig. 6
Fig. 6

The same star image as in Fig. 5 but with a monochromatic filter. The radial structure is eliminated in favor of a granular structure.

Fig. 7
Fig. 7

Long-exposure intensity envelopes at various wavelengths for the 36-in. telescope at the RGO. Envelopes calculated using Eqs. (67) and (68) with σ1 = 0.3 μm, w0 = 0.15 m, D = 0.915 m, and λ0 = 0.55 μm.

Fig. 8
Fig. 8

Core detectability E is defined as the ratio a/b.

Fig. 9
Fig. 9

Core detectability [Eq. (74)] is critically dependent on the ratio σ/λ.

Fig. 10
Fig. 10

Cores as they might appear with the Mt. Palomar 200-in. telescope (assuming no aberrations). Envelopes calculated using Eqs. (67) and (68) with σ = 0.2 μm, w0 = 0.15 m, D = 5.08 m, and λ0 = 0.55 μm. (a) The core at a visible wavelength. (b) The core becomes more dominant at infrared wavelengths.

Fig. 11
Fig. 11

Effect of aberrations (defocus) on the core. Three quarters of a wave of defocus (0.75 W) eliminates the core even though the remaining portion of the seeing disk shows no significant change. Envelopes calculated using Eqs. (71) and (72) with σ = 0.2 μm, w0 = 0.15 m, D = 5.08 m, and λ = λ0 = 0.55 μm.

Fig. 12
Fig. 12

The optimum wavelength (λ = 2πσ) gives the highest central intensity. A range of wavelengths is near optimum (1.5 to 2.5 μm in case shown). Envelopes calculated using Eqs. (67) and (68) with σ = 0.3 μm, w0 = 0.15 m, D = 5.08 m, and λ0 = 0.55 μm.

Fig. 13
Fig. 13

Equipment used to measure the atmospheric seeing parameters σ, ρ(w), and T(w; λ).

Fig. 14
Fig. 14

Long-exposure intensity envelopes for Vega in visible light using the 36-in. telescope at the RGO. Over the visible spectrum, the intensity envelopes typically show little variation. Envelopes calculated using Eqs. (67) and (68) with σ1 = 0.3 μm, w0 = 0.15 m, D = 0.915 m, and λ0 = 0.55 μm. The atmospheric MTF’s shown in Fig. 15 were calculated from the λ = 0.55-μm envelope.

Fig. 15
Fig. 15

Long-exposure MTF’s and the waveheight autocorrelation function measured at the RGO.

Fig. 16
Fig. 16

A small reduction in σ (0.2 to 0.15 μm) causes a tenfold increase in the core strength at 0.55 μm (cf. Fig. 10). Envelopes calculated using Eqs. (67) and (68) with σ1 = 0.15 μm, w0 = 0.15 m, D = 5.08 m, and λ0 = 0.55 μm.

Fig. 17
Fig. 17

Laser beam focused on target in space. The coordinates (X, Y) and (X′, Y′) denote conjugate points.

Equations (101)

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σ i 2 = h i ( x ) 2
ρ i ( w ) = h i ( x + w ) h i ( x ) / h i ( x ) 2 ,
exp [ 2 π i h 1 ( x ) / λ 1 ] ,             exp [ 2 π i h 1 ( x ) / λ 2 ] .
S 1 ( w ; λ 1 , λ 2 ) = exp [ 2 π i h 1 ( x + w ) / λ 1 ] exp [ - 2 π i h 1 ( x ) / λ 2 ] ,
B 1 ( x ; λ 1 ) exp [ 2 π i h 2 ( x ) / λ 1 ] ,             B 1 ( x ; λ 2 ) exp [ 2 π i h 2 ( x ) / λ 2 ] .
S 2 ( w ; λ 1 , λ 2 ) = B 1 ( x + w ; λ 1 ) exp [ 2 π i h 2 ( x + w ) / λ 1 ] × B 1 * ( x ; λ 2 ) exp [ - 2 π i h 2 ( x ) / λ 2 ] .
S 2 ( w ; λ 1 , λ 2 ) = B 1 * ( x + w ; λ 1 ) B 1 ( x ; λ 2 ) × exp [ 2 π i h 2 ( x + w ) / λ 1 ] exp [ - 2 π i h 2 ( x ) / λ 2 ] .
B 1 ( x + w ; λ 1 ) B 1 * ( x ; λ 2 ) = exp [ 2 π i h 1 ( x + w ) / λ 1 ] × exp [ - 2 π i h 1 ( x ) / λ 2 ] .
S 2 ( w ; λ 1 , λ 2 ) = exp [ 2 π i h 1 ( x + w ) / λ 1 ] exp [ - 2 π i h 1 ( x ) / λ 2 ] × exp [ 2 π i h 2 ( x + w ) / λ 1 ] exp [ - 2 π i h 2 ( x ) / λ 2 ] .
S ( w ; λ 1 , λ 2 ) = exp [ 2 π i h 1 ( x + w ) / λ 1 ] exp [ - 2 π i h 1 ( x ) / λ 2 ] × exp [ 2 π i h 2 ( x + w ) / λ 1 ] exp [ - 2 π i h 2 ( x ) / λ 2 ] × × exp [ 2 π i h n ( x + w ) / λ 1 ] exp [ - 2 π i h n ( x ) / λ 2 ] .
H ( x ) = h 1 ( x ) + h 2 ( x ) + + h n ( x ) ,
S ( w ; λ 1 , λ 2 ) = exp [ 2 π i ( H ( x + w ) / λ 1 - H ( x ) / λ 2 ) ] .
σ 2 = H ( x ) 2
ρ ( w ) = H ( x + w ) H ( w ) / H ( x ) 2 ,
S ( w ; λ 1 , λ 2 ) = exp { - 2 π 2 σ 2 [ 1 λ 1 2 + 1 λ 2 2 - 2 ρ ( w ) λ 1 λ 2 ] } .
T ( w ; λ ) = exp { - 4 π 2 σ 2 λ 2 [ 1 - ρ ( w ) ] } .
σ 2 = i = 1 n σ i 2
ρ ( w ) = 1 σ 2 i = 1 n σ i 2 ρ i ( w ) .
h i ( x ) = z i - 1 z i [ n ( x , z ) - n ( x , z ) ] n ( x , z ) d z ,
N ( x , z ) = n ( x , z ) - n ( x , z ) n ( x , z ) .
N ( x , z ) = 0.
σ i 2 = z i - 1 z i N ( x , z ) N ( x , z ) d z d z ,
ρ i ( w ) = 1 σ i 2 z i - 1 z i N ( x + w , z ) N ( x , z ) d z d z .
N ( x + w , z ) N ( x , z ) = N ( x + w , z ) N ( x , z ) = 0 ,
0 z N ( x + w , z ) N ( x , z ) d z d z = 0 z 1 N ( x + w , z ) N ( x , z ) d z d z + z 1 z 2 + z 2 z 3 + + z n - 1 z n N ( x + w , z ) N ( x , z ) d z d z .
σ 2 = 0 z N ( x , z ) N ( x , z ) d z d z ,
ρ ( w ) = 1 σ 2 0 z N ( x + w , z ) N ( x , z ) d z d z .
T ( w ; λ ) = exp { - 4 π 2 λ 2 0 z [ N ( x , z ) N ( x , z ) - N ( x + w , z ) N ( x , z ) ] d z d z } .
S ( w ; λ 1 , λ 2 ) = exp { - 2 π 2 [ ( 1 λ 1 2 + 1 λ 2 2 ) 0 z N ( x , z ) N ( x , z ) d z d z - 2 λ 1 λ 2 0 z N ( x + w , z ) N ( x , z ) d z d z ] } .
H ( x ) = n - 1 n 0 - 1 H 0 ( x ) ,
σ = n - 1 n 0 - 1 σ 0
ρ ( w ) = ρ 0 ( w ) ,
S ( w ; λ 1 , λ 2 ) = exp { - 2 π 2 σ 0 2 [ ( n 1 - 1 ) 2 λ 1 2 ( n 0 - 1 ) 2 + ( n 2 - 1 ) 2 λ 2 2 ( n 0 - 1 ) 2 - 2 ρ 0 ( w ) ( n 1 - 1 ) ( n 2 - 1 ) λ 1 λ 2 ( n 0 - 1 ) 2 ] } ,
T ( w ; λ ) = exp { - 4 π 2 σ 0 2 ( n - 1 ) 2 λ 2 ( n 0 - 1 ) 2 [ 1 - ρ 0 ( w ) ] } ,
H l ( x ) = H ( x ) - H ¯ l ( x ) .
σ l 2 = H l 2 ( x ) ,
ρ l ( w ) = H l ( x + w ) H l ( x ) H l 2 ( x ) .
S l ( w ; λ 1 , λ 2 ) = exp { - 2 π 2 σ l 2 [ 1 λ l 2 + 1 λ 2 2 - 2 ρ l ( w ) λ 1 λ 2 ] } ,
T l ( w ; λ ) = exp { - 4 π 2 σ l 2 λ 2 [ 1 - ρ l ( w ) ] } .
H s ( x ) = H ( x ) - H ¯ s ( x ) .
σ s 2 = H s 2 ( x ) ,
ρ s ( w ) = H s ( x + w ) H s ( x ) H s 2 ( x ) .
S s ( w ; λ 1 , λ 2 ) = exp { - 2 π 2 σ s 2 [ 1 λ 1 2 + 1 λ 2 2 - 2 ρ s ( w ) λ 1 λ 2 ] } ,
T s ( w ; λ ) = exp { - 4 π 2 σ s 2 λ 2 [ 1 - ρ s ( w ) ] } .
R 2 = X 2 + Y 2 .
α = R F ( rad ) = 4.85 × 10 - 6 R F ( arcsec ) .
A ( X , Y ; λ ) 1 λ - B ( x , y ; λ ) K ( x , y ; λ ) × exp [ - 2 π i ( x X + y Y ) F λ ] d x d y ,
K ( x , y ; λ ) = 1 for x 2 + y 2 < 0.25 D 2 = 0 otherwise .
A ( X , Y ; λ 1 ) A * ( X , Y ; λ 2 ) 1 λ 1 λ 2 × - B ( x , y , λ 1 ) B * ( x , y ; λ 2 ) K ( x y ; λ 1 ) K * ( x , y ; λ 2 ) × exp [ - 2 π i F ( x X + y Y λ 1 - x X + y Y λ 2 ) ] d x d x d y d y .
S s ( x - x , y - y ; λ 1 , λ 2 ) = B ( x , y ; λ 1 ) B * ( x , y ; λ 2 ) [ B ( x , y ; λ 1 ) B * ( x , y ; λ 1 ) B ( x , y ; λ 2 ) B * ( x , y ; λ 2 ) ] 0.5 .
C ( X , Y , X , Y ; λ 1 , λ 2 ) = A ( X , Y ; λ 1 ) A * ( X , Y ; λ 2 ) [ I ( X , Y ; λ 1 ) I ( X , Y ; λ 2 ) ] 0.5 ,
I ( X , Y ; λ ) = A ( X , Y ; λ ) A * ( X , Y ; λ ) .
I ( X , Y ) = 0 I ( X , Y ; λ ) G ( λ ) d λ ,
[ I ( X , Y ) - I ( X , Y ) ] [ I ( X , Y ) - I ( X , Y ) ] = 0 G ( λ 1 ) G ( λ 2 ) [ I ( X , Y ; λ 1 ) - I ( X , Y ; λ 1 ) ] × [ I ( X , Y ; λ 2 ) - I ( X , Y ; λ 2 ) ] d λ 1 d λ 2 .
I ( X , Y ; λ 1 ) I ( X , Y ; λ 2 ) - I ( X , Y ; λ 1 ) × I ( X , Y ; λ 2 ) = A ( X , Y ; λ 1 ) A * ( X , Y ; λ 2 ) 2 .
I ( X , Y ; ) I ( X , Y ) I ( X , Y ) ( I ( X , Y ) = - 1.0 = 0 G ( λ 1 ) G ( λ 2 ) A ( X , Y ; λ 1 ) A * ( X , Y ; λ 2 ) 2 d λ 1 d λ 2 [ 0 G ( λ ) I ( X , Y ; λ ) d λ ] 0.5 [ 0 G ( λ ) I ( X , Y ; λ ) d λ ] 0.5 .
[ I ( X , Y ) - I ( X , Y ) ] 2 = 0 G ( λ 1 ) G ( λ 2 ) × A ( X , Y ; λ 1 ) A * ( X , Y ; λ 2 ) 2 d λ 1 d λ 2 ,
M ( R ) = 0 G ( λ ) I ( R ; λ ) d λ 0 G ( λ 1 ) G ( λ 2 ) A ( R ; λ 1 ) A * ( R ; λ 2 ) 2 d λ 1 d λ 2 .
A ( R ; λ 1 ) A * ( R ; λ 2 ) π D 2 2 λ 1 λ 2 J 1 [ π D R F ( 1 λ 1 - 1 λ 2 ) ] π D R F ( 1 λ 1 - 1 λ 2 ) × 0 2 π S s ( w ; λ 1 , λ 2 ) J 0 [ π w R F ( 1 λ 1 + 1 λ 2 ) ] w d w ,
C ( R ; λ 1 λ 2 ) = A ( R ; λ 1 ) A * ( R ; λ 2 ) [ I ( R ; λ 1 ) I ( R ; λ 2 ) ] 0.5 .
C ( R ; λ 1 , λ 2 ) = 2 J 1 [ π D R F ( 1 λ 1 - 1 λ 2 ) ] π D R F ( 1 λ 1 - 1 λ 2 ) 0 2 π S s ( w ; λ 1 , λ 2 ) J 0 [ π w R F ( 1 λ 1 + 1 λ 2 ) ] w d w [ 0 2 π T s ( w ; λ 1 ) J 0 ( 2 π w R F λ 1 ) w d w 0 2 π T s ( w ; λ 2 ) J 0 ( 2 π w R F λ 2 ) w d w ] 0.5 .
C ( R ; λ 1 , λ 2 ) = 2 J 1 [ π D R F ( 1 λ 1 - 1 λ 2 ) ] π D R F ( 1 λ 1 - 1 λ 2 ) × exp [ - 2 π 2 σ s 2 ( 1 λ 1 - 1 λ 2 ) 2 ] .
PDF ( I ) = g M I M - 1 exp ( - g I ) Γ ( M ) for I > 0 = 0 otherwise ,
g = M I .
I ( X , Y ; λ ) 1 λ 2 - B ( x + u , y + v ; λ ) B * ( x , y ; λ ) × [ - K ( x + u , y + v ; λ ) K * ( x , y ; λ ) d x d y ] × exp [ - 2 π i ( u X + v Y ) F λ ] d u d v ,
I ( X , Y ; λ ) 1 λ 2 - T ( u , v ; λ ) T t ( u , v ; λ ) × exp [ - 2 π i ( u X + v Y ) F λ ] d u d v ,
T t ( w ; λ ) = 2 π cos - 1 ( w D ) - 2 w π D ( 1 - w 2 D 2 ) 0.5 ,
w 2 = u 2 + v 2 .
I ( R ; λ ) = 4 λ 0 2 π D 2 λ 2 0 2 π exp { - 4 π 2 σ 2 λ 2 [ 1 - ρ ( w ) ] } × [ 2 π cos - 1 ( w D ) - 2 w π D ( 1 - w 2 D 2 ) 0.5 ] × J 0 ( 2 π w R F λ ) w d w .
ρ 1 ( w ) = exp ( - w 2 w 0 2 ) ,
T c = exp ( - 4 π 2 σ 2 λ 2 ) .
T v ( u , v ; λ ) = T ( u , v ; λ ) - T c .
1 λ 2 exp ( - 4 π 2 σ 2 λ 2 ) - T t ( u , v ; λ ) exp [ - 2 π i ( u x + v y ) F λ ] d u d v .
exp ( - 4 π 2 σ 2 / λ 2 )
I ( R ; λ ) = λ 0 2 λ 2 exp ( - 4 π 2 σ 2 λ 2 ) × [ P ( R ; λ ) + 4 ω 0 2 D 2 n = 1 ( 2 π σ / λ ) 2 n n n ! exp ( - π 2 w 0 2 R 2 n F 2 λ 2 ) ] ,
P ( R ; λ ) = 64 D 4 | 0 D / 2 r exp ( - i π r 2 Z λ F 2 ) J 0 ( 2 π r R λ F ) d r | 2 ,
P ( R ; λ ) = 4 { J 1 [ π D R / F ( λ ) ] [ π D R / F ( λ ) ] } 2 .
E = D 2 P ( 0 ; λ ) 4 w 0 2 n = 1 ( 2 π σ / λ ) 2 n n n ! .
a 1 λ 2 exp ( - 4 π 2 σ 2 λ 2 ) P ( 0 ; λ ) .
d a d λ = 0 ,
λ m = 2 π σ .
a m 1 λ m 2 1 e P ( 0 ; λ ) .
C ( R ; λ 1 , λ 2 ) = { 2 J 1 [ π D R F ( 1 λ 1 - 1 λ 2 ) ] π D R F ( 1 λ 1 - 1 λ 2 ) - ( D 2 D 2 ) × 2 J 1 [ π D R F ( 1 λ 1 - 1 λ 2 ) ] π D R F ( 1 λ 1 - 1 λ 2 ) } ( D 2 D 2 - D 2 ) × exp [ - 2 π 2 σ 2 ( 1 λ 1 - 1 λ 2 ) 2 ] .
C ( 0 ; λ 1 , λ 2 ) = exp [ - 2 π 2 σ 2 ( 1 λ 1 - 1 λ 2 ) 2 ] .
I ( R ; λ 1 ) I ( R ; λ 2 ) I ( R ; λ 1 ) I ( R ; λ 2 ) - 1 = C ( R ; λ 1 , λ 2 ) 2 .
σ 2 = - ln [ I ( 0 ; λ 1 ) I ( 0 ; λ 2 ) I ( 0 ; λ 1 ) I ( 0 ; λ 2 ) - 1 ] { λ 1 2 λ 2 2 [ 4 π 2 ( λ 1 - λ 2 ) 2 ] } ,
σ a 2 = σ 2 + σ w 2 .
σ a - σ σ = σ w 2 2 σ 2 .
T ( u , v ; λ ) λ 2 T t ( u , v ; λ ) - I ( X , Y ; λ ) × exp [ 2 π i ( u X + v Y ) F λ ] d X d Y .
T ( w ; λ ) 2 π λ 2 T t ( w ; λ ) 0 I ( R ; λ ) J 0 ( 2 π i R w F λ ) R d R .
ρ ( w ) = 1 + ( λ 2 4 π 2 σ 2 ) ln [ T ( w ; λ ) ] .
a 1 1 λ 1 2 exp ( - 4 π 2 σ 2 λ 1 2 ) P ( 0 ; λ 1 ) G ( λ 1 ) ,
a 2 1 λ 2 2 exp ( - 4 π 2 σ 2 λ 2 2 ) P ( 0 ; λ 2 ) G ( λ 2 ) ,
σ 2 = λ 1 2 λ 2 2 4 π 2 ( λ 1 2 - λ 2 2 ) 2 ln [ a 1 λ 1 2 P ( 0 ; λ 2 ) G ( λ 2 ) a 2 λ 2 2 P ( 0 ; λ 1 ) G ( λ 1 ) ] .
rect ( x D ) = 1.0 for x < D 2 = 0     otherwise ,            
A ( X ; λ 1 ) A * ( X ; λ 2 ) 1 ( λ 1 λ 2 ) 0.5 - S ( u ; λ 1 , λ 2 ) rect ( x + u D ) × rect ( x D ) exp [ - 2 π i x X F ( 1 λ 1 - 1 λ 2 ) ] exp ( - 2 π i u X F λ 1 ) d x d u ,
S c = exp [ - 2 π 2 σ 2 ( 1 λ 1 2 + 1 λ 2 2 ) ] ,
S v ( u ; λ 1 λ 2 ) = S ( u ; λ 1 , λ 2 ) - S c .
A ( X ; λ 1 ) A * ( X ; λ 2 ) 1 ( λ 1 λ 2 ) 0.5 { S c D 2 sinc ( D X F λ 1 ) sinc ( D X F λ 2 ) + - D sinc [ D X F ( 1 λ 1 - 1 λ 2 ) ] S v ( u ; λ 1 , λ 2 ) × exp [ - i π u X F ( 1 λ 1 + 1 λ 2 ) ] d u } ,
C ( X ; λ 1 , λ 2 ) = A ( X ; λ 1 ) A * ( X ; λ 2 ) [ A ( X ; λ 1 ) A * ( X ; λ 1 ) A ( X ; λ 2 ) A * ( X ; λ 2 ) ] 0.5 .
A ( R ; λ 1 ) A * ( R ; λ 2 ) 1 λ 1 λ 2 { S c π 2 D 4 4 J 1 ( π R D F λ 1 ) J 1 ( π R D F λ 2 ) ( π R D F λ 1 ) ( π R D F λ 2 ) + π D 2 2 J 1 [ π R D F ( 1 λ 1 - 1 λ 2 ) ] [ π R D F ( 1 λ 1 - 1 λ 2 ) ] 0 2 S v ( w ; λ 1 λ 2 ) × J 0 [ π w R F ( 1 λ 1 + 1 λ 2 ) ] w d w } ,

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