Abstract

The statistics of the derivative of intensity for partially developed speckle patterns have been investigated. The probability density function of the intensity derivative is given by an infinite series of modified Bessel functions of the second kind. The effect of integrating over space and time has also been discussed. The approximate form for the probability density function of the integrated intensity of the differentiated partially developed speckle pattern is also given by an infinite series of modified Bessel functions of the second kind. In the special case of fully developed speckle patterns, the probability density functions derived in this paper reduce to those given by Ebeling [Opt. Commun. <b>35</b>, 323 (1980)].

© 1982 Optical Society of America

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References

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  1. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
  2. J. C. Dainty, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam 1976), Vol. 14.
  3. J. C. Dainty, "Coherent addition of a uniform beam to a speckle pattern," J. Opt. Soc. Am. 62, 595–596 (1972).
  4. J. Ohtsubo and T. Asakura, "Statistical properties of the sum of partially developed speckle patterns," Opt. Lett. 1, 98–100 (1977).
  5. J. Ohtsubo and T. Asakura, "Statistical properties of laser speckle produced in the diffraction field," Appl. Opt. 16, 1742–1752 (1977).
  6. J. C. Erdmann and R. I. Gellert, "Recurrence rate correlation in scattered light intensity," J. Opt. Soc. Am. 68, 787–795 (1978)
  7. K. J. Ebeling, "Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns," Opt. Acta 26, 1505–1523 (1979).
  8. K. J. Ebeling, "Experimental investigation of some statistical properties of monochromatic speckle patterns," Opt. Acta 26, 1345–1349 (1979).
  9. K. J. Ebeling, "K-distributed spatial intensity derivatives in monochromatic speckle patterns," Opt. Commun. 35, 323–326 (1980).
  10. N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, "Zero crossing study on dynamic properties of speckle," J. Opt. 11, 93–101 (1980).
  11. Each term of the summations in Eqs. (7) and (16) is a K distribution when it has a unit area. The K distribution quoted in several places [for example, E. Jakeman, "On the statistics of K-distributed noise" J. Phys. A 13, 31–48 (1980)] is a type of xn+1Kn(x) (n is an integer), whereas the K distribution derived here is a type of xnKn(x). The K distribution obtained in this paper is somewhat different from previous K distributions.
  12. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

1980

K. J. Ebeling, "K-distributed spatial intensity derivatives in monochromatic speckle patterns," Opt. Commun. 35, 323–326 (1980).

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, "Zero crossing study on dynamic properties of speckle," J. Opt. 11, 93–101 (1980).

1979

K. J. Ebeling, "Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns," Opt. Acta 26, 1505–1523 (1979).

K. J. Ebeling, "Experimental investigation of some statistical properties of monochromatic speckle patterns," Opt. Acta 26, 1345–1349 (1979).

1978

1977

1972

Asakura, T.

Dainty, J. C.

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

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

J. C. Dainty, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam 1976), Vol. 14.

Ebeling, K. J.

K. J. Ebeling, "K-distributed spatial intensity derivatives in monochromatic speckle patterns," Opt. Commun. 35, 323–326 (1980).

K. J. Ebeling, "Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns," Opt. Acta 26, 1505–1523 (1979).

K. J. Ebeling, "Experimental investigation of some statistical properties of monochromatic speckle patterns," Opt. Acta 26, 1345–1349 (1979).

Erdmann, J. C.

Gellert, R. I.

Gradshteyn, I. S.

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

Iwai, T.

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, "Zero crossing study on dynamic properties of speckle," J. Opt. 11, 93–101 (1980).

Ohtsubo, J.

Ryzhik, I. M.

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

Takai, N.

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, "Zero crossing study on dynamic properties of speckle," J. Opt. 11, 93–101 (1980).

Ushizaka, T.

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, "Zero crossing study on dynamic properties of speckle," J. Opt. 11, 93–101 (1980).

Appl. Opt.

J. Opt.

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, "Zero crossing study on dynamic properties of speckle," J. Opt. 11, 93–101 (1980).

J. Opt. Soc. Am.

Opt. Acta

K. J. Ebeling, "Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns," Opt. Acta 26, 1505–1523 (1979).

K. J. Ebeling, "Experimental investigation of some statistical properties of monochromatic speckle patterns," Opt. Acta 26, 1345–1349 (1979).

Opt. Commun.

K. J. Ebeling, "K-distributed spatial intensity derivatives in monochromatic speckle patterns," Opt. Commun. 35, 323–326 (1980).

Opt. Lett.

Other

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

J. C. Dainty, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam 1976), Vol. 14.

Each term of the summations in Eqs. (7) and (16) is a K distribution when it has a unit area. The K distribution quoted in several places [for example, E. Jakeman, "On the statistics of K-distributed noise" J. Phys. A 13, 31–48 (1980)] is a type of xn+1Kn(x) (n is an integer), whereas the K distribution derived here is a type of xnKn(x). The K distribution obtained in this paper is somewhat different from previous K distributions.

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

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