Abstract

Propagation through a generalized thick diffuser and a spaced cascade of two thin diffusers is analyzed using methods of statistical optics. Second- and fourth-order moments are derived for the scalar component of the far-zone field. In the formulation two states a and b are used, permitting one to consider variations of speckle with angle of illumination and wavelength of an input plane wave and with observation direction as well as with in-plane and longitudinal motion between the two thin diffusers. This two-state correlation function of intensity, i.e., fourth-order moment, is evaluated without the assumption of circularity, and explicit calculations of decorrelation with respect to the state variables are presented for a cascade of two paraboloidal diffusers. Decorrelation occurs with slight transverse motion of one of the diffusers that is essentially independent of the separation H. Decorrelation of the on-axis speckle occurs for a change in spacing ΔH that is also independent of H. For single thin diffusers or a closely spaced diffuser pair, decorrelation with changes of angle of illumination is quite slow. However, rapid angular decorrelation is obtained for large spacings. Wavelength decorrelation of the diffuser pair arises from two unrelated effects, i.e., the dependence of the individual diffusers and of the propagation between diffuser planes on wavelength. Plots are given to illustrate the different speckle dependencies. This analysis of speckle decorrelation can be used in various remote-sensing applications.

© 1989 Optical Society of America

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References

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  1. I. Leifer, C. J. D. Spencer, W. T. Welford, C. N. Richmond, “Grainless screens for projection microscopy,”J. Opt. Soc. Am. 51, 1422–1423 (1961).
    [Crossref]
  2. F. A. MacAdamTaylor, Taylor & Hobson, Ltd., British patent592,815 (May25, 1945).
  3. S. Lowenthal, D. Joyeux, “Speckle removal by a slowly moving diffuser associated with a motionless diffuser,”J. Opt. Soc. Am. 61, 847–851 (1971).
    [Crossref]
  4. N. George, A. Jain, “Speckle from a cascade of two diffusers,” Opt. Commun. 15, 71–75 (1975).
    [Crossref]
  5. E. G. Rawson, A. B. Nafarrate, R. E. Norton, “Speckle-free rear-projection screen using two close screens in slow relative motion,”J. Opt. Soc. Am. 66, 1290–1294 (1976).
    [Crossref]
  6. N. Barakat, T. El Dessouki, M. El Nicklawy, M. Abdel Sadek, “Interference from two identical diffusers,” Acta Phys. Acad. Sci. Hung. 51, 341–347 (1981).
    [Crossref]
  7. H. J. Tiziani, “Vibration analysis and deformation measurement,” in Speckle Metrology, R. K. Erf, ed. (Academic, New York, 1979), pp. 73–110.
  8. D. L. Fried, “Laser eye safety: the implications of ordinary speckle statistics and of speckled-speckle statistics,”J. Opt. Soc. Am. 71, 914–916 (1981).
    [Crossref] [PubMed]
  9. K. A. O’Donnell, “Speckle statistics of doubly scattered light,”J. Opt. Soc. Am. 72, 1459–1463 (1982).
    [Crossref]
  10. R. Barakat, “The brightness distribution of the product of two partially correlated speckle patterns,” Opt. Commun. 52, 1–4 (1984).
    [Crossref]
  11. R. Barakat, R. J. Salawitch, “Second- and fourth-order statistics of double scattered speckle,” Opt. Acta 33, 79–89 (1986).
    [Crossref]
  12. D. Newman, “K distributions from doubly scattered light,” J. Opt. Soc. Am. A 2, 22–26 (1985).
    [Crossref]
  13. L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,”J. Opt. Soc. Am. 55, 247–253 (1965).
    [Crossref]
  14. L. H. Enloe, “Noise-like structure in the image of diffusely reflecting objects in coherent illumination,” Bell Syst. Tech. J. 46, 1479–1489 (1967).
  15. C. B. Burckhardt, “Laser speckle pattern—a narrowband noise model,” Bell Syst. Tech. J. 49, 309–316 (1970).
  16. S. Lowenthal, H. Arsenault, “Image formation for coherent diffuse objects: statistical properties,”J. Opt. Soc. Am. 60, 1478–1483 (1970).
    [Crossref]
  17. J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
    [Crossref]
  18. N. George, A. Jain, “Speckle reduction using multiple tones of illumination,” Appl. Opt. 12, 1202–1212 (1973).
    [Crossref] [PubMed]
  19. N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
    [Crossref]
  20. N. George, A. Jain, R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
    [Crossref]
  21. L. G. Shirley, N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am. A 4, 734–745 (1987).
    [Crossref]
  22. P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, Oxford, 1953).
  23. A. Papoulis, “Ambiguity function in Fourier optics,”J. Opt. Soc. Am. 64, 779–788 (1974).
    [Crossref]
  24. K. -H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
    [Crossref]
  25. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 484–487.
  26. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 120.
  27. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,”J. Opt. Soc. Am. 67, 785–796 (1977).
    [Crossref]
  28. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 338.
  29. L. G. Shirley, “Laser speckle from thin and cascaded diffusers,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1988).
  30. L. G. Shirley, N. George, “Diffuser radiation patterns over a large dynamic range. Part 1: Strong diffusers,” Appl. Opt. 27, 1850–1861 (1988).
    [Crossref] [PubMed]
  31. J. W. Goodman, “Dependence of image speckle on surface roughness,” Opt. Commun. 14, 324–327 (1975).
    [Crossref]
  32. H. M. Pederson, “Theory of speckle dependence on surface roughness,”J. Opt. Soc. Am. 66, 1204–1210 (1976).
    [Crossref]
  33. J. Ohtsubo, T. Asakura, “Statistical properties of laser speckle produced in the diffraction field,” Appl. Opt. 16, 1742–1753 (1977).
    [Crossref] [PubMed]
  34. J. Uozumi, T. Asakura, “First-order intensity and phase statistics of Gaussian speckle produced in the diffraction region,” Appl. Opt. 20, 1454–1466 (1981).
    [Crossref] [PubMed]
  35. E. Lukacs, Characteristic Functions, 2nd ed. (Griffin, London, 1970), p. 68.
  36. E. J. Kelly, I. S. Reed, “Some properties of stationary Gaussian processes,” (MIT Lincoln Laboratory, Lexington, Mass., 1957).
  37. I. S. Reed, “On a moment theorem for complex Gaussian processes,”IRE Trans. Inf. Theory IT-8, 194–195 (1962).
    [Crossref]
  38. L. Mandel, “Fluctuations of light beams,” in Progress in Optics II, E. Wolf, ed. (North-Holland, Amsterdam, 1963), pp. 181–248.
    [Crossref]
  39. C. L. Mehta, “Coherence and statistics of radiation,” in Lectures in Theoretical Physics VII C, W. E. Brittin, ed. (University of Colorado, Boulder, Colo., 1964), pp. 345–401.
  40. K. S. Miller, “Complex Gaussian processes,” SIAM Rev. 11, 544–567 (1969).
    [Crossref]
  41. W. F. McGee, “Complex Gaussian noise moments,”IEEE Trans. Inf. Theory IT-17, 149–157 (1971).
    [Crossref]
  42. K. S. Miller, Complex Stochastic Processes (Addison-Wesley, London, 1974).

1988 (1)

1987 (1)

1986 (1)

R. Barakat, R. J. Salawitch, “Second- and fourth-order statistics of double scattered speckle,” Opt. Acta 33, 79–89 (1986).
[Crossref]

1985 (1)

1984 (2)

K. -H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[Crossref]

R. Barakat, “The brightness distribution of the product of two partially correlated speckle patterns,” Opt. Commun. 52, 1–4 (1984).
[Crossref]

1982 (1)

1981 (3)

1977 (2)

1976 (2)

1975 (3)

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

N. George, A. Jain, “Speckle from a cascade of two diffusers,” Opt. Commun. 15, 71–75 (1975).
[Crossref]

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

1974 (2)

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

A. Papoulis, “Ambiguity function in Fourier optics,”J. Opt. Soc. Am. 64, 779–788 (1974).
[Crossref]

1973 (1)

1971 (2)

1970 (3)

S. Lowenthal, H. Arsenault, “Image formation for coherent diffuse objects: statistical properties,”J. Opt. Soc. Am. 60, 1478–1483 (1970).
[Crossref]

C. B. Burckhardt, “Laser speckle pattern—a narrowband noise model,” Bell Syst. Tech. J. 49, 309–316 (1970).

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

1969 (1)

K. S. Miller, “Complex Gaussian processes,” SIAM Rev. 11, 544–567 (1969).
[Crossref]

1967 (1)

L. H. Enloe, “Noise-like structure in the image of diffusely reflecting objects in coherent illumination,” Bell Syst. Tech. J. 46, 1479–1489 (1967).

1965 (1)

1962 (1)

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

1961 (1)

Abdel Sadek, M.

N. Barakat, T. El Dessouki, M. El Nicklawy, M. Abdel Sadek, “Interference from two identical diffusers,” Acta Phys. Acad. Sci. Hung. 51, 341–347 (1981).
[Crossref]

Arsenault, H.

Asakura, T.

Barakat, N.

N. Barakat, T. El Dessouki, M. El Nicklawy, M. Abdel Sadek, “Interference from two identical diffusers,” Acta Phys. Acad. Sci. Hung. 51, 341–347 (1981).
[Crossref]

Barakat, R.

R. Barakat, R. J. Salawitch, “Second- and fourth-order statistics of double scattered speckle,” Opt. Acta 33, 79–89 (1986).
[Crossref]

R. Barakat, “The brightness distribution of the product of two partially correlated speckle patterns,” Opt. Commun. 52, 1–4 (1984).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 484–487.

Brenner, K. -H.

K. -H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[Crossref]

Burckhardt, C. B.

C. B. Burckhardt, “Laser speckle pattern—a narrowband noise model,” Bell Syst. Tech. J. 49, 309–316 (1970).

Carter, W. H.

Dainty, J. C.

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

El Dessouki, T.

N. Barakat, T. El Dessouki, M. El Nicklawy, M. Abdel Sadek, “Interference from two identical diffusers,” Acta Phys. Acad. Sci. Hung. 51, 341–347 (1981).
[Crossref]

El Nicklawy, M.

N. Barakat, T. El Dessouki, M. El Nicklawy, M. Abdel Sadek, “Interference from two identical diffusers,” Acta Phys. Acad. Sci. Hung. 51, 341–347 (1981).
[Crossref]

Enloe, L. H.

L. H. Enloe, “Noise-like structure in the image of diffusely reflecting objects in coherent illumination,” Bell Syst. Tech. J. 46, 1479–1489 (1967).

Fried, D. L.

George, N.

L. G. Shirley, N. George, “Diffuser radiation patterns over a large dynamic range. Part 1: Strong diffusers,” Appl. Opt. 27, 1850–1861 (1988).
[Crossref] [PubMed]

L. G. Shirley, N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am. A 4, 734–745 (1987).
[Crossref]

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

N. George, A. Jain, “Speckle from a cascade of two diffusers,” Opt. Commun. 15, 71–75 (1975).
[Crossref]

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

N. George, A. Jain, “Speckle reduction using multiple tones of illumination,” Appl. Opt. 12, 1202–1212 (1973).
[Crossref] [PubMed]

Goldfischer, L. I.

Goodman, J. W.

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 120.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 338.

Jain, A.

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

N. George, A. Jain, “Speckle from a cascade of two diffusers,” Opt. Commun. 15, 71–75 (1975).
[Crossref]

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

N. George, A. Jain, “Speckle reduction using multiple tones of illumination,” Appl. Opt. 12, 1202–1212 (1973).
[Crossref] [PubMed]

Joyeux, D.

Kelly, E. J.

E. J. Kelly, I. S. Reed, “Some properties of stationary Gaussian processes,” (MIT Lincoln Laboratory, Lexington, Mass., 1957).

Leifer, I.

Lowenthal, S.

Lukacs, E.

E. Lukacs, Characteristic Functions, 2nd ed. (Griffin, London, 1970), p. 68.

MacAdam, F. A.

F. A. MacAdamTaylor, Taylor & Hobson, Ltd., British patent592,815 (May25, 1945).

Mandel, L.

L. Mandel, “Fluctuations of light beams,” in Progress in Optics II, E. Wolf, ed. (North-Holland, Amsterdam, 1963), pp. 181–248.
[Crossref]

McGee, W. F.

W. F. McGee, “Complex Gaussian noise moments,”IEEE Trans. Inf. Theory IT-17, 149–157 (1971).
[Crossref]

Mehta, C. L.

C. L. Mehta, “Coherence and statistics of radiation,” in Lectures in Theoretical Physics VII C, W. E. Brittin, ed. (University of Colorado, Boulder, Colo., 1964), pp. 345–401.

Melville, R. D. S.

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

Miller, K. S.

K. S. Miller, “Complex Gaussian processes,” SIAM Rev. 11, 544–567 (1969).
[Crossref]

K. S. Miller, Complex Stochastic Processes (Addison-Wesley, London, 1974).

Nafarrate, A. B.

Newman, D.

Norton, R. E.

O’Donnell, K. A.

Ohtsubo, J.

Ojeda-Castaneda, J.

K. -H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[Crossref]

Papoulis, A.

Pederson, H. M.

Rawson, E. G.

Reed, I. S.

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

E. J. Kelly, I. S. Reed, “Some properties of stationary Gaussian processes,” (MIT Lincoln Laboratory, Lexington, Mass., 1957).

Richmond, C. N.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 338.

Salawitch, R. J.

R. Barakat, R. J. Salawitch, “Second- and fourth-order statistics of double scattered speckle,” Opt. Acta 33, 79–89 (1986).
[Crossref]

Shirley, L. G.

Spencer, C. J. D.

Tiziani, H. J.

H. J. Tiziani, “Vibration analysis and deformation measurement,” in Speckle Metrology, R. K. Erf, ed. (Academic, New York, 1979), pp. 73–110.

Uozumi, J.

Welford, W. T.

Wolf, E.

Woodward, P. M.

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, Oxford, 1953).

Acta Phys. Acad. Sci. Hung. (1)

N. Barakat, T. El Dessouki, M. El Nicklawy, M. Abdel Sadek, “Interference from two identical diffusers,” Acta Phys. Acad. Sci. Hung. 51, 341–347 (1981).
[Crossref]

Appl. Opt. (4)

Appl. Phys. (2)

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

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

Bell Syst. Tech. J. (2)

L. H. Enloe, “Noise-like structure in the image of diffusely reflecting objects in coherent illumination,” Bell Syst. Tech. J. 46, 1479–1489 (1967).

C. B. Burckhardt, “Laser speckle pattern—a narrowband noise model,” Bell Syst. Tech. J. 49, 309–316 (1970).

IEEE Trans. Inf. Theory (1)

W. F. McGee, “Complex Gaussian noise moments,”IEEE Trans. Inf. Theory IT-17, 149–157 (1971).
[Crossref]

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. (10)

H. M. Pederson, “Theory of speckle dependence on surface roughness,”J. Opt. Soc. Am. 66, 1204–1210 (1976).
[Crossref]

S. Lowenthal, H. Arsenault, “Image formation for coherent diffuse objects: statistical properties,”J. Opt. Soc. Am. 60, 1478–1483 (1970).
[Crossref]

E. G. Rawson, A. B. Nafarrate, R. E. Norton, “Speckle-free rear-projection screen using two close screens in slow relative motion,”J. Opt. Soc. Am. 66, 1290–1294 (1976).
[Crossref]

L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,”J. Opt. Soc. Am. 55, 247–253 (1965).
[Crossref]

D. L. Fried, “Laser eye safety: the implications of ordinary speckle statistics and of speckled-speckle statistics,”J. Opt. Soc. Am. 71, 914–916 (1981).
[Crossref] [PubMed]

K. A. O’Donnell, “Speckle statistics of doubly scattered light,”J. Opt. Soc. Am. 72, 1459–1463 (1982).
[Crossref]

I. Leifer, C. J. D. Spencer, W. T. Welford, C. N. Richmond, “Grainless screens for projection microscopy,”J. Opt. Soc. Am. 51, 1422–1423 (1961).
[Crossref]

S. Lowenthal, D. Joyeux, “Speckle removal by a slowly moving diffuser associated with a motionless diffuser,”J. Opt. Soc. Am. 61, 847–851 (1971).
[Crossref]

A. Papoulis, “Ambiguity function in Fourier optics,”J. Opt. Soc. Am. 64, 779–788 (1974).
[Crossref]

W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,”J. Opt. Soc. Am. 67, 785–796 (1977).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Acta (3)

R. Barakat, R. J. Salawitch, “Second- and fourth-order statistics of double scattered speckle,” Opt. Acta 33, 79–89 (1986).
[Crossref]

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

K. -H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[Crossref]

Opt. Commun. (3)

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

N. George, A. Jain, “Speckle from a cascade of two diffusers,” Opt. Commun. 15, 71–75 (1975).
[Crossref]

R. Barakat, “The brightness distribution of the product of two partially correlated speckle patterns,” Opt. Commun. 52, 1–4 (1984).
[Crossref]

SIAM Rev. (1)

K. S. Miller, “Complex Gaussian processes,” SIAM Rev. 11, 544–567 (1969).
[Crossref]

Other (12)

L. Mandel, “Fluctuations of light beams,” in Progress in Optics II, E. Wolf, ed. (North-Holland, Amsterdam, 1963), pp. 181–248.
[Crossref]

C. L. Mehta, “Coherence and statistics of radiation,” in Lectures in Theoretical Physics VII C, W. E. Brittin, ed. (University of Colorado, Boulder, Colo., 1964), pp. 345–401.

K. S. Miller, Complex Stochastic Processes (Addison-Wesley, London, 1974).

H. J. Tiziani, “Vibration analysis and deformation measurement,” in Speckle Metrology, R. K. Erf, ed. (Academic, New York, 1979), pp. 73–110.

F. A. MacAdamTaylor, Taylor & Hobson, Ltd., British patent592,815 (May25, 1945).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 338.

L. G. Shirley, “Laser speckle from thin and cascaded diffusers,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1988).

E. Lukacs, Characteristic Functions, 2nd ed. (Griffin, London, 1970), p. 68.

E. J. Kelly, I. S. Reed, “Some properties of stationary Gaussian processes,” (MIT Lincoln Laboratory, Lexington, Mass., 1957).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 484–487.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 120.

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, Oxford, 1953).

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Figures (8)

Fig. 1
Fig. 1

Coordinate system for the analysis of far-zone speckle from a generalized thick diffuser or (inset) a cascade of two thin diffusers.

Fig. 2
Fig. 2

Wave-vector notation for two states of input illumination k0a and k0b and corresponding normalized far-zone observation directions ka and kb. The wave-vector differences perpendicular to the z axis are used in Eqs. (12) and (13) for defining components Δkab and kab.

Fig. 3
Fig. 3

Amplitude of the aperture ambiguity function for a rectangular aperture plotted versus x/wx and kxwx for Eq. (25)

Fig. 4
Fig. 4

Dependence of Eq. (A11) for Rt, solid curves, and Eq. (70) for R t , dashed curves, on the offset ρ for Gaussian Rh, Eq. (72), and for S of 0.5, 1.0, and 2.0.

Fig. 5
Fig. 5

Correlation coefficient ρI for the diffuser pair versus longitudinal displacement ΔH by Eq. (90).

Fig. 6
Fig. 6

Correlation coefficient ρI for the double diffuser versus angle of rotation at σt/w = 0.03 for spacings H ranging from 0.1 to 10 mm by Eq. (91).

Fig. 7
Fig. 7

Wavelength decorrelation of speckle from a cascade of two diffusers. Equation (92) is plotted against βpp for Q =0.5, 1.0, 2.0, and 5.0 in the solid curve and for Q = ∞ in the dashed curve.

Fig. 8
Fig. 8

Wavelength decorrelation of Eq. (92) for a cascade of two diffusers plotted against λb for λa = 0.5 μm, spacing H ranging between 0.1 and 10 mm, σt1/w1 = σt2 = 0.03, and σt1 = σt2 = 0.5 μm.

Equations (125)

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v ( k ; k 0 ) = i k 2 π R exp ( - i k R ) cos θ v 2 + [ ρ ; k 0 ] exp ( i k · ρ ) d 2 ρ .
s ^ = e ^ x sin θ cos ϕ + e ^ y sin θ sin ϕ + e ^ z cos θ .
v 2 + [ ρ ; k 0 ] = exp ( - i k 0 · ρ ) t 12 ( ρ ; k 0 ) a ( ρ ) .
v ( k ; k 0 ) = i k 2 π R exp ( - i k R ) cos θ × t 12 ( ρ ; k 0 ) a ( ρ ) exp [ - i ( k 0 - k ) · ρ ] d 2 ρ .
u a b = R 2 A e v * ( k a ; k 0 a ) v ( k b ; k 0 b ) .
A e = a ( ρ ) d 2 ρ ,
u a b = k a k b ( 2 π ) 2 1 A e exp [ - i R ( k b - k a ) ] cos θ a cos θ b × R t 12 ( ρ 2 - ρ 1 ; k 0 a , k 0 b ) a * ( ρ 1 ) a ( ρ 2 ) × exp { i [ ( k 0 a - k a ) · ρ 1 - ( k 0 b - k b ) · ρ 2 ] } d 2 ρ 1 d 2 ρ 2
R t 12 ( ρ ; k 0 a , k 0 b ) = t 12 * ( ρ ; k 0 a ) t 12 ( ρ + ρ ; k 0 b )
ρ 1 = ρ - ρ 2
ρ 2 = ρ + ρ 2
A ¯ ( ρ ; k ) = 1 A e a * ( ρ - ρ 2 ) a ( ρ + ρ 2 ) exp ( - i k · ρ ) d 2 ρ ,
u a b = k a k b ( 2 π ) 2 exp [ - i R ( k b - k a ) ] cos θ a cos θ b × R t 12 ( ρ ; k 0 a , k 0 b ) A ¯ ( ρ ; Δ k a b ) exp ( i k a b · ρ ) d 2 ρ .
Δ k a b = k a - k 0 a - k b + k 0 b
k a b = 1 2 ( k a - k 0 a + k b - k 0 b ) .
u a b = k a k b ( 2 π ) 2 exp [ - i R ( k b - k a ) ] cos θ a cos θ b × R t ( ρ ; k a , k b ) A ¯ ( ρ ; Δ k a b ) exp ( i k a b · ρ ) d 2 ρ .
v ( k ; k 0 ) = i k 2 π R exp ( - i k R ) cos θ × t ( ρ ; k ) a ( ρ ) exp [ i ( k - k 0 ) · ρ ] d 2 ρ
Δ k a b = 0.
k a b = k a - k 0 a = k b - k 0 b ,
a ( ρ ) = exp ( - ρ 2 w a 2 ) ;
a ( ρ ) = rect ( x 2 w x ) rect ( y 2 w y ) ;
a ( ρ ) = circ ( ρ w a ) .
A e = π 2 w a 2 .
A e = 4 w x w y ,
A e = π w a 2 .
A ¯ ( ρ ; k ) = exp ( - ρ 2 2 w a 2 ) exp ( - 1 8 w a 2 k 2 ) .
A ¯ ( ρ ; k ) = 1 w x k x rect ( x 4 w x ) sin [ ( 1 - x 2 w x ) w x k x ] × 1 w y k y rect ( y 4 w y ) sin [ ( 1 - y 2 w y ) w y k y ] .
A ¯ ( ρ ; 0 ) = 2 π circ ( ρ 2 w a ) { cos - 1 ( ρ 2 w a ) - ρ 2 w a [ 1 - ( ρ 2 w a ) 2 ] 1 / 2 } .
u a b = R 2 A e v * ( k a ; k 0 a ; ρ 0 a ; H a ) v ( k b ; k 0 b ; ρ 0 b ; H b ) .
v 1 - [ ρ ; k 0 ] = exp ( - i k 0 · ρ ) .
v 1 + [ ρ ; k 0 ] = v 1 - [ ρ ; k 0 ] t 1 ( ρ ; k ) .
v 2 - [ ρ ; k 0 ] = i k 2 π H exp ( - i k H ) t 1 ( ρ - ρ 0 ; k ) × exp ( - i k 0 · ρ ] exp ( - i k 2 H ρ - ρ 2 ) d 2 ρ ,
H 3 > w t 1 4 2 λ ,
v ( k ; k 0 ; ρ 0 ; H ) = - k 2 ( 2 π ) 2 exp [ - i k ( H + R ) ] H R cos θ × d 2 ρ a ( ρ ) t 2 ( ρ ; k ) exp ( i k · ρ ) × exp ( - i k 0 · ρ ) t 1 ( ρ - ρ 0 ; k ) × exp ( - i k 2 H ρ - ρ 2 ) d 2 ρ .
t 12 ( ρ ; k 0 ; ρ 0 ; H ) = i k 2 π exp ( - i k H ) H t 2 ( ρ ; k ) × exp [ - i ( k 2 H ρ 2 + k 0 · ρ ) ] × t 1 ( ρ + ρ - ρ 0 ; k ) d 2 ρ .
R t 12 ( ρ ; k 0 a , k 0 b ; ρ 0 a , ρ 0 b ; H a , H b ) = exp [ i ( k a H a - k b H b ) ] R t 2 ( ρ ; k a , k b ) × R t 1 ( ρ + ρ + ρ 0 a - ρ 0 b ; k a , k b ) × B ( ρ ; k 0 a - k 0 a ; k a H a , k b H b ) × exp [ - i 2 ( k 0 b + k 0 b ) · ρ ] d 2 ρ ,
B ( ρ ; k ; k a H a , k b H b ) = 1 ( 2 π ) 2 k a k b H a H b × exp [ i 2 ( k a H a | ρ - ρ 2 | 2 - k b H b | ρ + ρ 2 | 2 ) ] × exp ( - i k + ρ ) d 2 ρ .
u a b = k a k b ( 2 π ) 2 exp ( - i ϕ a b ) cos θ a cos θ b × B ( ρ ; k 0 b - k 0 a ; k a H a , k b H b ) × exp [ - i 2 ( k 0 a + k 0 b ) · ρ ] × R t 1 ( ρ + ρ - Δ ρ a b ; k a , k b ) R t 2 ( ρ ; k a , k b ) A ¯ ( ρ ; Δ k a b ) × exp ( i k a b · ρ ) d 2 ρ d 2 ρ .
ϕ a b = k b H b - k a H a + ( k b - k a ) R ,
Δ ρ a b = ρ 0 b - ρ 0 a .
D ( ρ ; k a , k b ; k , Δ k ) = R t 1 ( ρ + ρ ; k a , k b ) R t 2 ( ρ ; k a , k b ) × A ¯ ( ρ ; Δ k ) exp ( i k · ρ ) d 2 ρ .
u a b = k a k b ( 2 π ) 2 exp ( - i ϕ a b ) cos θ a cos θ b × B ( ρ ; k 0 b - k 0 a ; k a H a , k b H b ) × D ( ρ - Δ ρ a b ; k a , k b ; k a b , Δ k a b ) × exp [ - i 2 ( k 0 a + k 0 b ) · ρ ] d 2 ρ .
exp [ i ( C ρ 2 - k · ρ ) ] d 2 ρ = i π C exp ( - i k 2 4 C )
B ( ρ ; k ; C a , C b ) = - i 2 π C a C b C b - C a exp { i 2 1 C b - C a × [ k 2 + ( C a + C b ) k · ρ + C a C b ρ 2 ] } .
B ( ρ ; k ; C a , C a ) = δ ( ρ + k C a ) .
B ( ρ ; k 0 b - k 0 a ; k a H a , k b H b ) = - i 2 π k a k b k b H a - k a H b × exp ( i 2 H a H b k b H a - k a H b k 0 b - k 0 a 2 ) × exp [ i 2 k b H a + k a H b k b H a - k a H b ( k 0 b - k 0 a ) · ρ ] × exp ( i 2 k a k b ρ 2 k b H a - k a H b ) .
k a H a = k b H b
B ( ρ ; k 0 b - k 0 a ; k a H a , k b H b = k a H b ) = δ ( ρ + Δ H a b ) ,
Δ H a b = H b s ^ 0 b - H a s ^ 0 a .
u a b = - i k a 2 k b 2 ( 2 π ) 3 exp ( - i ϕ a b ) k b H a - k a H b × exp [ i 2 ( H b k b k 0 b 2 - H a k a k 0 a 2 ) ] cos θ a cos θ b × D ( ρ - Δ ρ a b - Δ H a b ; k a , k b ; k a b , Δ k a b ) × exp ( i 2 k a k b k b H a - k a H b ρ 2 ) d 2 ρ .
u a b = k a k b ( 2 π ) 2 exp ( - i ϕ a b ) exp [ i 2 ( k b H b - k a H a ) ] cos θ a cos θ b × D ( - Δ ρ a b - Δ H a b ; k a , k b ; k a b , Δ k a b ) .
u a b = - i k a 2 k b 2 ( 2 π ) 3 exp ( - i ϕ a b ) k b H a - k a H b cos θ a cos θ b × exp { i 2 1 k b H a - k a H b [ k a k b Δ ρ a b 2 + 2 ( k a H b k 0 b - k b H a k 0 a ) · Δ ρ a b + H a H b k 0 b - k 0 a 2 ] } × D ( ρ ; k a , k b ; k a b , Δ k a b ) exp { i k a k b k b H a - k a H b × [ 1 2 ρ 2 + ( Δ ρ a b + H b s ^ 0 b - H a s ^ 0 a ) · ρ ] } d 2 ρ .
u a a = I a ,
I a = d P d Ω 1 P 0 .
I a = u a a = ( k a 2 π ) 2 cos 2 θ a D ( 0 ; k a , k a ; k a - k 0 a , 0 ) .
I a = ( k a 2 π ) 2 cos 2 θ a R t 1 ( ρ ; k a , k a ) R t 2 ( ρ ; k a , k a ) A ¯ ( ρ ; 0 ) × exp [ i ( k a - k 0 a ) · ρ ] d 2 ρ .
I a = ( k a 2 π ) 2 cos 2 θ a R t ( ρ ; k a , k a ) A ¯ ( ρ ; 0 ) × exp [ i ( k a - k 0 a ) · ρ ] d 2 ρ .
I a = R 2 A e v a * v a
I a I b = R 4 A e 2 v a * v a v b * v b .
ρ I = I a I b - I a I b σ I a σ I b ,
σ I = I 2 - I 2 .
I a I b = u a a u b b + u a b 2 + u a b 2 - 2 u a u b ,
u a b = R 2 A e v ( k a ; k 0 a ) v ( k b ; k 0 b )
u a = R 2 A e v ( k a ; k 0 a ) 2 .
ρ I = u a b 2 I a I b .
u a b = - k a k b ( 2 π ) 2 exp [ - i R ( k a + k b ) ] cos θ a cos θ b × R t 12 ( ρ ; k 0 a , k 0 b ) A ¯ ( ρ ; 2 k a b ) × exp ( - i 2 Δ k a b · ρ ) d 2 ρ ,
R t 12 ( ρ ; k 0 a , k 0 b ) = t 12 ( ρ ; k 0 a ) t 12 ( ρ + ρ ; k 0 b )
A ¯ ( ρ ; k ) = 1 A e a ( ρ - ρ 2 ) a ( ρ + ρ 2 ) exp ( - i k · ρ ) d 2 ρ .
u a b = - k a k b ( 2 π ) 2 exp [ - i R ( k a + k b ) ] cos θ a cos θ b × R t ( ρ ; k a , k b ) A ¯ ( ρ ; - 2 k a b ) × exp ( - i 2 Δ k a b · ρ ) d 2 ρ ,
R t ( ρ b - ρ a ; k a k b ) = t ( ρ a ; k a ) t ( ρ b ; k b ) .
R t ( ρ ; k a , k b ) = R t ( ρ ; - k a , k b ) .
R t ( ρ ; k a , k b ) = exp [ - σ t 2 2 ( k a + k b ) 2 ] × exp { k a k b σ t 2 [ 1 - R h ( ρ ) ] } .
R t ( ; k a , k b ) = R t ( ; k a , k b ) = exp [ - σ t 2 2 ( k a 2 + k b 2 ) ] .
R h ( ρ ) = exp ( - ρ 2 w 2 )
R t 12 ( ρ ; k 0 a , k 0 b ; ρ 0 a , ρ 0 b ; H a , H b ) = - exp [ - i ( k a H a + k b H b ) ] R t 2 ( ρ ; k a , k b ) × R t 12 ( ρ + ρ - Δ ρ a b ; k a , k b ) × B ( ρ ; k 0 a + k 0 b ; - k a H a , k b H b ) × exp [ - i 2 ( k 0 b - k 0 a ) · ρ ] d 2 ρ .
D ( ρ ; k a , k b ; Δ k , k ) = R t 12 ( ρ + ρ ; k a , k b ) R t 12 ( ρ ; k a , k b ) A ¯ ( ρ ; k ) exp ( i Δ k · ρ ) d 2 ρ ,
u a b = - i k a 2 k b 2 ( 2 π ) 3 exp ( - i ϕ a b ) k b H a + k a H b × exp [ i 2 ( H a k a k 0 a 2 + H b k b k 0 b 2 ) ] cos θ a cos θ b × D ( ρ - Δ ρ a b - Δ H a b ; k a , k b ; - ½ Δ k a b , - 2 k a b ) exp ( - i 2 k a k b k b H a + k a H b ρ 2 ) d 2 ρ ,
ϕ a b = k a H a + k b H b + ( k a + k b ) R .
v ( k ; k 0 ; H ) = i k 2 π R exp ( - i k R ) exp ( - i k z H ) cos θ × t 1 ( 0 ; k ) t 2 ( 0 ; k ) A ( k 0 - k ) ,
A ( k ) = a ( ρ ) exp ( - i k · ρ ) d 2 ρ .
u a = k a 2 ( 2 π ) 2 cos 2 θ a 1 A e t 1 ( 0 ; k a ) × t 2 ( 0 ; k a ) A ( k 0 a - k a ) 2 .
t ( 0 , k ) = R t ( 0 , 0 , k ) .
u a 2 u a a 2 = u a a 2 u a a 2 = ( 2 w a w p ) 4 exp ( - 4 S 2 ) .
S = k σ h ( n - 1 ) 1 ,
D ( ρ ; k a , k b ; 0 , 0 ) = π k a k b ( σ t 1 2 w 1 2 + σ t 2 2 w 2 2 ) - 1 × exp [ - 1 2 ( σ t 1 2 + σ t 2 2 ) ( k b - k a ) 2 ] × exp [ - ( w 1 2 σ t 1 2 + w 2 2 σ t 2 2 ) - 1 k a k b ρ 2 ] ,
exp [ - ( a 2 ρ 2 + i k · ρ ) ] d 2 ρ = π a 2 exp ( - k 2 4 a 2 ) ,
u a b = 1 4 π exp ( i - ϕ a b ) cos θ a cos θ b × exp { i 2 1 k b H a - k a H b [ k a k b Δ ρ a b 2 + 2 ( k a H b k 0 b - k b H a k 0 a ) · Δ ρ a b + H a H b k 0 b - k 0 a 2 ] } ( σ t 1 2 w 1 2 + σ t 2 2 w 2 2 ) - 1 × exp [ - 1 2 ( σ t 1 2 + σ t 2 2 ) ( k b - k a ) 2 ] 1 1 + i β p p × exp [ - k a k b ( w 1 2 σ t 1 2 + w 2 2 σ t 2 2 ) - 1 1 + i β p p - 1 1 + β p p 2 × Δ ρ a b + H b s ^ 0 b - H a s ^ 0 a 2 ] ,
β p p = 2 ( k b H a - k a H b ) ( w 1 2 σ t 1 2 + w 2 2 σ t 2 2 ) - 1 .
ρ I = exp [ - ( σ t 1 2 + σ t 2 2 ) ( k b - k a ) 2 ] 1 1 + β p p 2 × exp [ - k a k b γ 12 2 1 1 + β p p 2 Δ ρ a b + H b s ^ 0 b - H a s ^ 0 a 2 ] ,
1 γ 12 2 = 1 2 ( w 1 2 σ t 1 2 + w 2 2 σ t 2 2 ) .
ρ I = exp [ - ( k γ 12 Δ ρ a b ) 2 ] .
ρ I = 1 1 + ( γ 12 2 k Δ H ) 2 .
ρ I = exp [ - ( γ 12 k H sin Δ θ 0 ) 2 ] .
ρ I = exp [ - ( σ t 1 2 + σ t 2 2 ) ( k b - k a ) 2 ] 1 1 + β p p 2 ,
β p p = ( k b - k a ) H γ 12 2 .
Q = H γ 12 2 ( σ t 1 2 + σ t 2 2 ) 1 / 2 .
t ( ρ ; k ) = exp [ i k ( n - 1 ) h ( ρ ) ] .
R t ( ρ ; k a , k b ) = t * ( ρ ; k a ) t ( ρ + ρ ; k b )
σ h = h 2 ( ρ ) 1 / 2 .
σ t = ( n - 1 ) σ h .
S = k σ t = k σ h ( n - 1 ) .
R h ( ρ b - ρ a ) = h ( ρ a ) h ( ρ b ) σ h 2 .
R t ( ρ 2 - ρ 1 ; k a k b ) = exp { i [ η a h 1 ( ρ 1 ) - η b h 2 ( ρ 2 ) ] } ,
η = k ( n - 1 ) .
R t ( ρ ; k a , k b ) = exp { - σ h 2 2 [ η a 2 - 2 R h ( ρ ) η a η b + η b 2 ] } .
R t ( ρ ; k a , k b ) = exp [ - σ h 2 2 ( η b - η a ) 2 ] × exp { - σ h 2 η a η b [ 1 - R h ( ρ ) ] } .
R t ( ρ ; k a , k b ) = exp [ - σ t 2 2 ( k b - k a ) 2 ] × exp { - k a k b σ t 2 [ 1 - R h ( ρ ) ] } .
R t ( ρ ; k a , k b ) 1 - σ t 2 2 ( k a 2 + k b 2 ) + k a k b σ t 2 R h ( ρ ) .
R h ( ρ ) = 1 - ρ w + ,
R t ( ρ ; k a , k b ) = exp [ - σ t 2 2 ( k b - k a ) 2 ] × exp ( - k a k b σ t 2 ρ w ) .
R h ( ρ ) = 1 - ( ρ w ) 2 + .
R t ( ρ ; k a , k b ) = exp [ - σ t 2 2 ( k b - k a ) 2 ] × exp ( - k a k b σ t 2 ρ 2 w 2 ) .
w c = w k a k b σ t 2 ,
w p = w k a k b σ t .
z = x + i y ,
x 1 x 2 x 3 x 4 = x 1 x 1 x 3 x 4 + x 1 x 3 x 2 x 4 + x 1 x 4 x 2 x 3 .
x 1 x 2 x 3 = 0.
x 1 x 2 x 3 = x 2 x 3 x 1 + x 1 x 3 x 2 + x 1 x 2 x 3 - 2 x 1 x 2 x 3 .
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
z 1 z 2 z 3 z 4 = z 1 z 2 z 3 z 4 + z 1 z 3 z 2 z 4 + z 1 z 4 z 2 z 3
z 1 z 2 z 3 = 0.
z * ( ρ 1 ) z * ( ρ 2 ) z ( ρ 3 ) z ( ρ 4 ) = z * ( ρ 1 ) z * ( ρ 2 ) z ( ρ 3 ) z ( ρ 4 ) + z * ( ρ 1 ) z ( ρ 3 ) z * ( ρ 2 ) z ( ρ 4 ) + z * ( ρ 1 ) z ( ρ 4 ) z * ( ρ 2 ) z ( ρ 3 ) - 2 z * ( ρ 1 ) z * ( ρ 2 ) z ( ρ 3 ) z ( ρ 4 ) .
z ( ρ 1 ) z ( ρ 2 ) = z ( ρ 1 ) z ( ρ 2 )
x ( ρ 1 ) x ( ρ 2 ) - x ( ρ 1 ) x ( ρ 2 ) = y ( ρ 1 ) y ( ρ 2 ) - y ( ρ 1 ) y ( ρ 2 )
x ( ρ 1 ) y ( ρ 2 ) - x ( ρ 1 ) y ( ρ 2 ) = - x ( ρ 2 ) y ( ρ 1 ) + x ( ρ 2 ) y ( ρ 1 ) .
z * ( ρ 1 ) z * ( ρ 2 ) z ( ρ 3 ) z ( ρ 4 ) = z * ( ρ 1 ) z ( ρ 3 ) z * ( ρ 2 ) z ( ρ 4 ) + z * ( ρ 1 ) z ( ρ 4 ) z * ( ρ 2 ) z ( ρ 3 ) - z * ( ρ 1 ) z * ( ρ 2 ) z ( ρ 3 ) z ( ρ 4 ) .

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