Abstract

The statistical properties of the spatial derivatives of the phase of a monochromatic speckle pattern are studied. Initially, a one-dimensional probability density function for the derivative of the phase is obtained and compared with the solution for the analogous problem concerning instantaneous frequency of narrow-band Gaussian noise. Subsequently, a two-dimensional probability density function is derived that depends on the two first and three second spatial moments of the illumination intensity distribution of the scattering object. Some sample intensity distributions are considered for which explicit expressions for the probability density function are given.

© 1983 Optical Society of America

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  1. J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975).
  2. J. C. Dainty, "The statistics of speckle patterns," Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. 14, pp. 1–46.
  3. K. J. Ebeling, "Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns," Opt. Acta 26, 1505 (1979).
  4. J. Ohtsubo, "Exact solution of the zero crossing rate of a differentiated speckle pattern," Opt. Commun. 42, 13 (1982).
  5. M. S. Longuet-Higgins, "The statistical analysis of a random, moving surface," Philos. Trans. Ser. A 249, 321 (1957).
  6. N. M. Blachman, Noise and Its Effect on Communication (McGraw-Hill, New York, 1966).
  7. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  8. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  9. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).
  10. An alternative way of setting up the problem presented in this paper is to define the axes of the system in terms of the intensity distribution I(u, Ʋ). If we choose its centroid as our origin, Eqs. (47) tell us that cx and cy are zero. In addition, d will be zero if the x and y axes are aligned with the principal axes of the scattering surface, i.e., the directions along the maximum and minimum of the second-moment ellipse. The resulting covariance matrix and probability density function are simpler in form. We appreciate the comments of N. Blachman (GTE Products Corporation, Sylvania Systems Group, P.O. Box 188, Mountain View, California 94042; personal communication) on this point.

1982

J. Ohtsubo, "Exact solution of the zero crossing rate of a differentiated speckle pattern," Opt. Commun. 42, 13 (1982).

1979

K. J. Ebeling, "Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns," Opt. Acta 26, 1505 (1979).

1957

M. S. Longuet-Higgins, "The statistical analysis of a random, moving surface," Philos. Trans. Ser. A 249, 321 (1957).

Blachman, N. M.

N. M. Blachman, Noise and Its Effect on Communication (McGraw-Hill, New York, 1966).

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).

Dainty, J. C.

J. C. Dainty, "The statistics of speckle patterns," Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. 14, pp. 1–46.

Ebeling, K. J.

K. J. Ebeling, "Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns," Opt. Acta 26, 1505 (1979).

Goodman, J. W.

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975).

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, "The statistical analysis of a random, moving surface," Philos. Trans. Ser. A 249, 321 (1957).

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Ohtsubo, J.

J. Ohtsubo, "Exact solution of the zero crossing rate of a differentiated speckle pattern," Opt. Commun. 42, 13 (1982).

Papoulis, A.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).

Opt. Acta

K. J. Ebeling, "Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns," Opt. Acta 26, 1505 (1979).

Opt. Commun.

J. Ohtsubo, "Exact solution of the zero crossing rate of a differentiated speckle pattern," Opt. Commun. 42, 13 (1982).

Philos. Trans. Ser.A

M. S. Longuet-Higgins, "The statistical analysis of a random, moving surface," Philos. Trans. Ser. A 249, 321 (1957).

Other

N. M. Blachman, Noise and Its Effect on Communication (McGraw-Hill, New York, 1966).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

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

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).

An alternative way of setting up the problem presented in this paper is to define the axes of the system in terms of the intensity distribution I(u, Ʋ). If we choose its centroid as our origin, Eqs. (47) tell us that cx and cy are zero. In addition, d will be zero if the x and y axes are aligned with the principal axes of the scattering surface, i.e., the directions along the maximum and minimum of the second-moment ellipse. The resulting covariance matrix and probability density function are simpler in form. We appreciate the comments of N. Blachman (GTE Products Corporation, Sylvania Systems Group, P.O. Box 188, Mountain View, California 94042; personal communication) on this point.

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975).

J. C. Dainty, "The statistics of speckle patterns," Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. 14, pp. 1–46.

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