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. 35, 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, Amsterdam1976), Vol. 14.
  3. J. C. Dainty, “Coherent addition of a uniform beam to a speckle pattern,” J. Opt. Soc. Am. 62, 595–596 (1972).
    [CrossRef]
  4. J. Ohtsubo and T. Asakura, “Statistical properties of the sum of partially developed speckle patterns,” Opt. Lett. 1, 98–100 (1977).
    [CrossRef] [PubMed]
  5. J. Ohtsubo and T. Asakura, “Statistical properties of laser speckle produced in the diffraction field,” Appl. Opt. 16, 1742–1752 (1977).
    [CrossRef] [PubMed]
  6. J. C. Erdmann and R. I. Gellert, “Recurrence rate correlation in scattered light intensity,” J. Opt. Soc. Am. 68, 787–795 (1978)
    [CrossRef]
  7. K. J. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1523 (1979).
    [CrossRef]
  8. K. J. Ebeling, “Experimental investigation of some statistical properties of monochromatic speckle patterns,” Opt. Acta 26, 1345–1349 (1979).
    [CrossRef]
  9. K. J. Ebeling, “K-distributed spatial intensity derivatives in monochromatic speckle patterns,” Opt. Commun. 35, 323–326 (1980).
    [CrossRef]
  10. N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Zero crossing study on dynamic properties of speckle,” J. Opt. 11, 93–101 (1980).
    [CrossRef]
  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 xn+1/2Kn+1/2(x). The K distribution obtained in this paper is somewhat different from previous K distributions.
    [CrossRef]
  12. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

1980 (3)

K. J. Ebeling, “K-distributed spatial intensity derivatives in monochromatic speckle patterns,” Opt. Commun. 35, 323–326 (1980).
[CrossRef]

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

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 xn+1/2Kn+1/2(x). The K distribution obtained in this paper is somewhat different from previous K distributions.
[CrossRef]

1979 (2)

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

K. J. Ebeling, “Experimental investigation of some statistical properties of monochromatic speckle patterns,” Opt. Acta 26, 1345–1349 (1979).
[CrossRef]

1978 (1)

1977 (2)

1972 (1)

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).
[CrossRef]

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

J. C. Dainty, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam1976), Vol. 14.

Ebeling, K. J.

K. J. Ebeling, “K-distributed spatial intensity derivatives in monochromatic speckle patterns,” Opt. Commun. 35, 323–326 (1980).
[CrossRef]

K. J. Ebeling, “Experimental investigation of some statistical properties of monochromatic speckle patterns,” Opt. Acta 26, 1345–1349 (1979).
[CrossRef]

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

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).
[CrossRef]

Jakeman, E.

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 xn+1/2Kn+1/2(x). The K distribution obtained in this paper is somewhat different from previous K distributions.
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

Appl. Opt. (1)

J. Opt. (1)

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

J. Opt. Soc. Am. (2)

J. Phys. A (1)

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 xn+1/2Kn+1/2(x). The K distribution obtained in this paper is somewhat different from previous K distributions.
[CrossRef]

Opt. Acta (2)

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

K. J. Ebeling, “Experimental investigation of some statistical properties of monochromatic speckle patterns,” Opt. Acta 26, 1345–1349 (1979).
[CrossRef]

Opt. Commun. (1)

K. J. Ebeling, “K-distributed spatial intensity derivatives in monochromatic speckle patterns,” Opt. Commun. 35, 323–326 (1980).
[CrossRef]

Opt. Lett. (1)

Other (3)

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

J. C. Dainty, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam1976), Vol. 14.

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

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Figures (3)

Fig. 1
Fig. 1

Probability density functions of the intensity derivative of a partially developed speckle pattern. The parameter r is the beam ratio I0/2σ2.

Fig. 2
Fig. 2

Probability density functions of the integrated intensity of a differentiated partially developed speckle pattern plotted for beam ratios r = 0, 2, 4, 10. The number N of speckles to be integrated is fixed at N = 3 in this figure.

Fig. 3
Fig. 3

Probability density functions of the integrated intensity of a differentiated partially developed speckle pattern plotted for N = 1, 3, 10, 20. The beam ratio r is fixed at r = 5 in this figure.

Equations (25)

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P ( A r , A i , A ˙ r , A ˙ i ) = 1 4 π 2 σ 2 μ 2 × exp ( - [ ( A r - I 0 ) 2 + A i 2 2 σ 2 + A ˙ r 2 + A ˙ i 2 2 μ 2 ] ) ,
σ 2 = ( A r - I 0 ) 2 = A i 2 ,
μ 2 = A ˙ r 2 = A ˙ i 2 ,
I = A r 2 + A i 2 = 2 σ 2 + I 0 ,
I ˙ = 2 A r A ˙ r + A i A ˙ i = 0.
A r = I cos θ ,
A i = I sin θ .
A ˙ r = I ˙ 2 I cos θ - I θ ˙ sin θ ,
A i = I ˙ 2 I sin θ + I θ ˙ cos θ .
p ( I , I ˙ ) = 1 4 σ 2 μ 2 π I I 0 ( I 0 I σ 2 ) exp ( I + I 0 2 σ 2 - 1 2 μ 2 I ˙ 2 4 I ) ,
p ( I ˙ ) = e - r 2 σ μ π r n = 0 1 ( n ! ) 2 ( r 4 σ μ I ˙ ) n + 1 / 2 K n + 1 / 2 ( I ˙ 2 σ μ ) ,
I ˙ 2 k - 1 = 0             ( k = 1 , 2 , , ) .
I ˙ 2 k = ( 8 σ 2 μ 2 ) k ( 2 k - 1 ) ! ! exp ( - r ) n = 0 ( k + n ) ! ( n ! ) 2 r n             ( k = 1 , 2 , , ) .
I ˙ 2 = 8 σ 2 μ 2 ( 1 + r ) .
M I ˙ ( i v ) = exp ( i v I ˙ ) = - p ( I ˙ ) exp ( i v I ˙ ) d I ˙ = 1 1 + 4 σ 2 μ 2 v 2 exp ( - r + r 1 + 4 σ 2 μ 2 v 2 ) .
I ˙ i = - S ( X ) I ˙ ( X ) d 2 X - S ( X ) d 2 X ,
I ˙ i = 1 N m = 1 N I ˙ m ,
p ( I ˙ i ) = 1 2 π - exp ( i v I ˙ i ) exp ( - i v I ˙ i ) d v .
M I ˙ ( i v ) = exp ( i v I ˙ i ) = m = 1 N exp ( i v I ˙ m N ) = 1 ( 1 + 4 σ 2 μ 2 N 2 v 2 ) N exp ( - N r + N r 1 + 4 σ 2 μ 2 N 2 v 2 ) .
p ( I ˙ i ) = N e - N r 2 π σ μ ( N r ) - N + 1 / 2 n = 0 1 n ! ( N + n - 1 ) ! × ( N 2 r I ˙ i 4 σ μ ) N + n - 1 / 2 K N + n - 1 / 2 ( N I ˙ i 2 σ μ ) .
I ˙ i 2 k - 1 = 0 ,
I ˙ i 2 k = ( 8 σ 2 μ 2 N 2 ) k ( 2 k - 1 ) ! ! exp ( - N r ) × n = 0 ( k + N + n - 1 ) ! n ! ( N + n - 1 ) ! ( N r ) n             ( k = 1 , 2 , , ) .
I ˙ i 2 = I ˙ 2 N = 8 σ 2 μ 2 N ( 1 + r ) .
p ( I ˙ ) = 1 4 σ μ exp ( - I ˙ 2 σ μ ) .
p ( I ˙ i ) = N 2 π σ μ ( N - 1 ) ! ( N I ˙ i 4 σ μ ) N - 1 / 2 K N - 1 / 2 ( N I ˙ i 2 σ μ ) .