Abstract

The theory of speckle in partially coherent, monochromatic light is extended to cover the entire range of object roughness. For “almost white-noise” objects it is shown how the speckle statistics are related to a Gaussian approximation of the object statistics. Within this approximation, general relations are derived for the spatial speckle correlation and the spatial Wiener spectrum. The theory is applied to calculate rms speckle contrast as function of object roughness, coherence of the illumination, and receiver plane defocusing.

© 1976 Optical Society of America

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  1. H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
    [Crossref]
  2. H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–38 (1974).
    [Crossref]
  3. H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
    [Crossref]
  4. J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
    [Crossref]
  5. J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of partially coherent light,” Nouv. Rev. Opt. 6, 189–195 (1975).
    [Crossref]
  6. H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16, 68–72 (1976).
    [Crossref]
  7. H. M. Pedersen, “The roughness dependence of partially developed, monochromatic speckle patterns,” Opt. Commun. 12, 156–159 (1974).
    [Crossref]
  8. J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
    [Crossref]
  9. H. M. Pedersen, “Object-roughness dependence of partially developed speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
    [Crossref]
  10. J. C. Dainty, “Coherent addition of a uniform beam to a speckle pattern,” J. Opt. Soc. Am. 62, 595–596 (1972).
    [Crossref]
  11. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, Vol. 9: Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 9–75.
    [Crossref]
  12. N. George, A. Jain, and R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
    [Crossref]
  13. W. T. Welford, “First order statistics of speckle produced by weak scattering media,” Opt. Quant. Elect. 7, 413–416 (1975).
    [Crossref]
  14. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon, New York, 1963), part I.
  15. J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
    [Crossref]
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  17. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, England, 1970).
  18. G. Parry, “Speckle patterns in partially coherent light,” in Topics in Applied Physics, Vol. 9: Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 77–121.
    [Crossref]
  19. T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optik 39, 258–267 (1974).
  20. J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill, New York, 1956), p. 83.
  21. H. H. Hopkins, “On the diffraction theory of optical imaging,” Proc. R. Soc. Ser. A, 217, 408–432 (1953) [reprinted in Selected Papers on Coherence and Fluctuations of Light, edited by L. Mandel and E. Wolf (Dover, New York, 1970), p. 141].
    [Crossref]
  22. W. Gautschi and W. F. Cahill, “Exponential Integral and related Functions,” in Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1965), pp. 227–251.
  23. E. Jakeman and P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen I. Theory,” J. Phys. A: Math. Gen. 8, 369–391 (1975); P. N. Pusey and E. Jakeman, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen II. Application to dynamic scattering in a liquid crystal,” J. Phys. A: Math. Gen. 8, 392–410 (1975).
    [Crossref]

1976 (2)

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16, 68–72 (1976).
[Crossref]

H. M. Pedersen, “Object-roughness dependence of partially developed speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
[Crossref]

1975 (7)

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
[Crossref]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[Crossref]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of partially coherent light,” Nouv. Rev. Opt. 6, 189–195 (1975).
[Crossref]

N. George, A. Jain, and R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

W. T. Welford, “First order statistics of speckle produced by weak scattering media,” Opt. Quant. Elect. 7, 413–416 (1975).
[Crossref]

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[Crossref]

E. Jakeman and P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen I. Theory,” J. Phys. A: Math. Gen. 8, 369–391 (1975); P. N. Pusey and E. Jakeman, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen II. Application to dynamic scattering in a liquid crystal,” J. Phys. A: Math. Gen. 8, 392–410 (1975).
[Crossref]

1974 (4)

T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optik 39, 258–267 (1974).

H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
[Crossref]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–38 (1974).
[Crossref]

H. M. Pedersen, “The roughness dependence of partially developed, monochromatic speckle patterns,” Opt. Commun. 12, 156–159 (1974).
[Crossref]

1972 (1)

1970 (1)

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[Crossref]

Asakura, T.

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16, 68–72 (1976).
[Crossref]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of partially coherent light,” Nouv. Rev. Opt. 6, 189–195 (1975).
[Crossref]

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
[Crossref]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[Crossref]

H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
[Crossref]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–38 (1974).
[Crossref]

Battin, R. H.

J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill, New York, 1956), p. 83.

Beckmann, P.

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, England, 1970).

Cahill, W. F.

W. Gautschi and W. F. Cahill, “Exponential Integral and related Functions,” in Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1965), pp. 227–251.

Dainty, J. C.

J. C. Dainty, “Coherent addition of a uniform beam to a speckle pattern,” J. Opt. Soc. Am. 62, 595–596 (1972).
[Crossref]

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[Crossref]

Fujii, H.

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16, 68–72 (1976).
[Crossref]

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
[Crossref]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–38 (1974).
[Crossref]

H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
[Crossref]

Gautschi, W.

W. Gautschi and W. F. Cahill, “Exponential Integral and related Functions,” in Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1965), pp. 227–251.

George, N.

N. George, A. Jain, and R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

Goodman, J. W.

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[Crossref]

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

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, Vol. 9: Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 9–75.
[Crossref]

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical imaging,” Proc. R. Soc. Ser. A, 217, 408–432 (1953) [reprinted in Selected Papers on Coherence and Fluctuations of Light, edited by L. Mandel and E. Wolf (Dover, New York, 1970), p. 141].
[Crossref]

Jain, A.

N. George, A. Jain, and R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

Jakeman, E.

E. Jakeman and P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen I. Theory,” J. Phys. A: Math. Gen. 8, 369–391 (1975); P. N. Pusey and E. Jakeman, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen II. Application to dynamic scattering in a liquid crystal,” J. Phys. A: Math. Gen. 8, 392–410 (1975).
[Crossref]

Laning, J. H.

J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill, New York, 1956), p. 83.

McKechnie, T. S.

T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optik 39, 258–267 (1974).

Melville, R. D. S.

N. George, A. Jain, and R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

Ohtsubo, J.

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of partially coherent light,” Nouv. Rev. Opt. 6, 189–195 (1975).
[Crossref]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[Crossref]

Parry, G.

G. Parry, “Speckle patterns in partially coherent light,” in Topics in Applied Physics, Vol. 9: Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 77–121.
[Crossref]

Pedersen, H. M.

H. M. Pedersen, “Object-roughness dependence of partially developed speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
[Crossref]

H. M. Pedersen, “The roughness dependence of partially developed, monochromatic speckle patterns,” Opt. Commun. 12, 156–159 (1974).
[Crossref]

Pusey, P. N.

E. Jakeman and P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen I. Theory,” J. Phys. A: Math. Gen. 8, 369–391 (1975); P. N. Pusey and E. Jakeman, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen II. Application to dynamic scattering in a liquid crystal,” J. Phys. A: Math. Gen. 8, 392–410 (1975).
[Crossref]

Shindo, Y.

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16, 68–72 (1976).
[Crossref]

Spizzichino, A.

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

Welford, W. T.

W. T. Welford, “First order statistics of speckle produced by weak scattering media,” Opt. Quant. Elect. 7, 413–416 (1975).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, England, 1970).

Appl. Phys. (1)

N. George, A. Jain, and R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. A: Math. Gen. (1)

E. Jakeman and P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen I. Theory,” J. Phys. A: Math. Gen. 8, 369–391 (1975); P. N. Pusey and E. Jakeman, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen II. Application to dynamic scattering in a liquid crystal,” J. Phys. A: Math. Gen. 8, 392–410 (1975).
[Crossref]

Nouv. Rev. Opt. (2)

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
[Crossref]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of partially coherent light,” Nouv. Rev. Opt. 6, 189–195 (1975).
[Crossref]

Opt. Acta (1)

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[Crossref]

Opt. Commun. (7)

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16, 68–72 (1976).
[Crossref]

H. M. Pedersen, “The roughness dependence of partially developed, monochromatic speckle patterns,” Opt. Commun. 12, 156–159 (1974).
[Crossref]

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[Crossref]

H. M. Pedersen, “Object-roughness dependence of partially developed speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
[Crossref]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[Crossref]

H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
[Crossref]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–38 (1974).
[Crossref]

Opt. Quant. Elect. (1)

W. T. Welford, “First order statistics of speckle produced by weak scattering media,” Opt. Quant. Elect. 7, 413–416 (1975).
[Crossref]

Optik (1)

T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optik 39, 258–267 (1974).

Other (8)

J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill, New York, 1956), p. 83.

H. H. Hopkins, “On the diffraction theory of optical imaging,” Proc. R. Soc. Ser. A, 217, 408–432 (1953) [reprinted in Selected Papers on Coherence and Fluctuations of Light, edited by L. Mandel and E. Wolf (Dover, New York, 1970), p. 141].
[Crossref]

W. Gautschi and W. F. Cahill, “Exponential Integral and related Functions,” in Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1965), pp. 227–251.

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

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, Vol. 9: Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 9–75.
[Crossref]

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

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, England, 1970).

G. Parry, “Speckle patterns in partially coherent light,” in Topics in Applied Physics, Vol. 9: Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 77–121.
[Crossref]

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Figures (5)

FIG. 1
FIG. 1

Telecentric imaging geometry for the formation of image speckle patterns.

FIG. 2
FIG. 2

Speckle contrast in coherent light as function of object roughness, σϕ, number of scatterers in focal spot N, and the defocusing ζ.

FIG. 3
FIG. 3

Speckle contrast as function of defocusing ζ, for different coherence conditions β and (a) σϕ = 2, (b) σϕ = 1, (c) σϕ = 0.5. The curves are for N = 10.

FIG. 4
FIG. 4

Speckle contrast as function of object roughness σϕ under varying coherence conditions β, (a) in focus: ζ = 0, (b) strongly defocused: ζ = 10 (Neff = total number of scatterers contributing to a point in the defocused pattern, βeff = total number of incoherent contributions to point in defocused pattern).

FIG. 5
FIG. 5

Speckle contrast as function of object roughness σϕ in partially coherent illumination β = 0.5 shown for different defocusing ζ. The curves are shown in a log-log scale to visualize the low-roughness dependence.

Tables (1)

Tables Icon

TABLE I Parameters determining the speckle contrast after Eqs. (33) and (34) for three Gaussian phase objects with different phase correlation functions cϕ(ξ).

Equations (35)

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h ( x , z ) = F - 1 { H ( f ) exp [ 2 π i z ( 1 / λ 2 - f 2 ) 1 / 2 ] } e i k z F - 1 { [ H ( f ) exp ( - π i λ z f 2 ) ] } ,
V ( x 1 , x 2 ) = γ ( x 1 - x 2 ) h ( x 1 ) h * ( x 2 ) = V * ( x 2 , x 1 ) .
I ( x ) = t ( x 1 ) t * ( x 2 ) V ( x - x 1 , x - x 2 ) d 2 x 1 d 2 x 2 .
I ( x ) = t ( x 1 ) t * ( x 2 ) V ( x - x 1 , x - x 2 ) d 2 x 1 d 2 x 2 ,
R ( x 1 , x 2 , x 3 , x 4 ) = t ( x 1 ) t * ( x 2 ) t ( x 3 ) t * ( x 4 ) - t ( x 1 ) t * ( x 2 ) t ( x 3 ) t * ( x 4 ) ,
C I ( x 1 , x 2 ) = Δ I ( x 1 ) Δ I ( x 2 ) = I ( x 1 ) I ( x 2 ) - I ( x 1 ) I ( x 2 ) = R ( x 1 - x 1 , x 1 - x 2 , x 2 - x 3 , x 2 - x 4 ) × V ( x 1 , x 2 ) V ( x 3 , x 4 ) d 2 x 1 d 2 x 2 d 2 x 3 d 2 x 4 ,
R = t 1 t 2 * t 3 t 4 * - t 1 t 2 * t 3 t 4 * t 1 t 2 * t 3 t 4 * - t 1 t 2 * t 3 t 4 * + t 1 t 3 t 2 * t 4 * - t 1 t 2 * t 3 t 4 * + t 1 t 4 * t 2 * t 3 - t 1 t 2 * t 3 t 4 * ,
R t 1 t 3 t 2 * t 4 * + t 1 t 4 * t 2 * t 3 - 2 t 1 t 2 * t 3 t 4 * ,
X 1 X 2 X 3 X 4 = X 1 X 2 X 3 X 4 + X 1 X 3 X 2 X 4 + X 1 X 4 X 2 X 3 - 2 X 1 X 2 X 3 X 4 .
C t = Δ t 1 Δ t 2 * = t 1 t 2 * - t 1 t 2 * , C ˜ t = Δ t 1 Δ t 2 = t 1 t 2 - t 1 t 2 ,
R ( x 1 , x 2 , x 3 , x 4 ) C ˜ t ( x 3 - x 1 ) C ˜ t * ( x 4 - x 2 ) + C ˜ t ( x 3 - x 1 ) t 0 * 2 + t 0 2 C ˜ t * ( x 4 - x 2 ) + C t ( x 4 - x 1 ) C t * ( x 3 - x 2 ) + t 0 2 [ C t ( x 4 - x 1 ) + C t * ( x 3 - x 2 ) ] .
t 0 = exp ( i ϕ ) = exp ( - 1 2 σ ϕ 2 ) ,
C t ( ξ ) = exp { i [ ϕ ( x ) - ϕ ( x + ξ ) ] } - t 0 2 = exp ( - σ ϕ 2 ) { exp [ σ ϕ 2 c ϕ ( ξ ) ] - 1 } ,
C ˜ t ( ξ ) = exp { i [ ϕ ( x ) + ϕ ( x + ξ ) ] } - t 0 2 = exp ( - σ ϕ 2 ) { exp [ - σ ϕ 2 c ϕ ( ξ ) ] - 1 } .
R ( x 1 , x 2 , x 3 , x 4 ) exp ( - 2 σ ϕ 2 ) × [ ( exp { - σ ϕ 2 [ c ϕ ( x 3 - x 1 ) + c ϕ ( x 4 - x 2 ) ] } - 1 ) + ( exp { σ ϕ 2 [ c ϕ ( x 4 - x 1 ) + c ϕ ( x 3 - x 2 ) ] } - 1 ) ] ,
R ( x 1 , x 2 , x 3 , x 4 ) = exp { - σ ϕ 2 [ 2 - c ϕ ( x 2 - x 1 ) - c ϕ ( x 4 - x 3 ) ] } × ( exp { - σ ϕ 2 [ c ϕ ( x 3 - x 1 ) + c ϕ ( x 4 - x 2 ) - c ϕ ( x 4 - x 1 ) - c ϕ ( x 3 - x 2 ) ] } - 1 ) .
C I ( δ ) = C ˜ t ( ξ 1 ) C ˜ t * ( ξ 2 ) V ( x 1 , x 2 ) V ( x 1 + δ - ξ 1 , x 2 + δ - ξ 2 ) d 2 ξ 1 d 2 ξ 2 d 2 x 1 d 2 x 2 + 2 Re [ t 0 * 2 C ˜ t ( ξ ) V ( x 1 , x 2 ) V ( x 1 + δ - ξ , x 3 ) d 2 ξ d 2 x 1 d 2 x 2 d 2 x 3 ] + C t ( ξ 1 ) C t * ( ξ 2 ) V ( x 1 , x 2 ) V * ( x 1 + δ - ξ 1 , x 2 + δ - ξ 2 ) d 2 ξ 1 d 2 ξ 2 x 1 d 2 x 2 + 2 t 0 2 Re [ C t ( ξ ) V ( x 1 , x 2 ) V * ( x 1 + δ - ξ , x 3 ) d 2 ξ d 2 x 1 d 2 x 2 d 2 x 3 ] .
I = C t ( ξ ) V ( x , x + ξ ) d 2 ξ d 2 x + t 0 2 V ( x 1 , x 2 ) d 2 x 1 d 2 x 2 .
T ( f 1 , f 2 ) = F [ V ( x 1 , x 2 ) ] = s ( - f ) H ( f + f 1 ) H * ( f - f 2 ) d 2 f = T * ( - f 2 , - f 1 ) ,
W I ( f ) = F [ C I ( δ ) ] = W ˜ t ( f ) W ˜ t * ( f - f ) T ( f , f - f ) T * ( f - f , f ) d 2 f + 2 Re [ t 0 * 2 W ˜ t ( f ) T ( f , 0 ) T * ( 0 , f ) ] + W t ( f ) W t ( f - f ) T ( f - f , f ) 2 d 2 f + [ W t ( f ) + W t ( - f ) ] t 0 T ( 0 , f ) 2 .
C I ( δ ) W t ( 0 ) 2 V ( x 1 , x 2 ) V ( x 1 + δ , x 2 + δ ) d 2 x 1 d 2 x 2 + 2 Re [ t 0 * 2 W ˜ t ( 0 ) V ( x 1 , x 2 ) V ( x 1 + δ , x 3 ) d 2 x 1 d 2 x 2 d 2 x 3 ] + W t ( 0 ) 2 V ( x 1 , x 2 ) V * ( x 1 + δ , x 2 + δ ) d 2 x 1 d 2 x 2 + 2 t 0 2 W t ( 0 ) Re [ V ( x 1 , x 2 ) V * ( x 1 + δ , x 3 ) d 2 x 1 d 2 x 2 d 2 x 3 ] ,
I W t ( 0 ) V ( x , x ) d 2 x + t 0 2 V ( x 1 , x 2 ) d 2 x 1 d 2 x 2 ,
W I ( f ) W ˜ t ( 0 ) 2 T ( f , f - f ) T * ( f - f , f ) d 2 f + 2 Re [ t 0 * 2 W ˜ t ( 0 ) T ( f , 0 ) T * ( 0 , f ) ] + W t ( 0 ) 2 T ( f - f , f ) 2 d 2 f + 2 W t ( 0 ) t 0 T ( 0 , f ) 2 .
C I ( δ ) C t ( ξ 1 ) C t * ( ξ 2 ) V ( x 1 , x 2 ) × V * ( x 1 + δ - ξ 1 , x 2 + δ - ξ 2 ) d 2 ξ 1 d 2 ξ 2 d 2 x 1 d 2 x 2 W t ( 0 ) 2 V ( x 1 , x 2 ) V * ( x 1 + δ , x 2 + δ ) d 2 x 1 d 2 x 2 ,
W I ( f ) W t ( f ) W t ( f - f ) T ( f - f , f ) 2 d 2 f W t ( 0 ) 2 T ( f - f , f ) 2 d 2 f .
σ I 2 = C I ( 0 ) = W ˜ t ( 0 ) 2 [ γ ( x ) ] 2 [ [ h ( x ) ] 2 * [ h * ( - x ) ] 2 ] d 2 x + 2 Re ( t 0 * 2 W ˜ t ( 0 ) { h ( x ) [ γ ( x ) * h ( x ) ] * } 2 d 2 x ) + W t ( 0 ) 2 γ ( x ) 2 [ h ( x ) 2 * h ( - x ) 2 ] d 2 x + 2 t 0 2 W t ( 0 ) h ( x ) [ γ ( x ) * h ( x ) ] 2 d 2 x ,
I = W t ( 0 ) h ( x ) 2 d 2 x + t 0 2 γ ( x ) [ h ( x ) * h * ( - x ) ] d 2 x ,
H ( f ) = exp ( - 2 π 2 σ h 2 f 2 ) ,
h ( x , z ) = F - 1 [ H ( f ) exp ( - π i λ z f 2 ) ] = F - 1 { exp [ - 2 π 2 ( σ h 2 + i z k ) f 2 ] } = 1 2 π σ ( z ) 2 exp ( - x 2 2 σ ( z ) 2 ) ,
σ ( z ) 2 = σ h 2 + i z k .
γ ( x ) = exp ( - x 2 2 σ c 2 ) ,
β = ( σ h σ c ) 2 = number of incoherent contributions in focused image point ;
ζ = z / ( k σ h 2 ) = defocusing in focal depth units ,
σ I 2 = | W ˜ t ( 0 ) 4 π σ h 2 | 2 1 1 + 2 β 1 1 + ζ 2 + 2 1 + 2 β Re ( t 0 * 2 W ˜ t ( 0 ) 4 π σ h 2 1 1 + β ( 1 + ζ 2 ) + i ζ ) + ( W t ( 0 ) 4 π σ h 2 ) 2 1 1 + 2 β ( 1 + ζ 2 ) + 2 1 + 2 β t 0 2 W t ( 0 ) 4 π σ h 2 1 1 + β ( 1 + ζ 2 ) ,
I = W t ( 0 ) 4 π σ h 2 + t 0 2 1 1 + 2 β .