Abstract

It is shown that, because of the effects of the turbulent atmosphere, the probability density function of the intensity for a monochromatic, fully developed speckle pattern changes from an exponential distribution at low turbulence levels to a K distribution as the turbulence level increases. A physical model that leads to the K distribution is proposed, and the parameters of the K distribution are derived in terms of the strength of turbulence, path length, wavelength, and beam size. The work is then extended to polychromatic and partially developed speckle patterns and to speckle with a coherent background. Good agreement is obtained between the theoretical predictions and experimental measurements.

© 1982 Optical Society of America

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References

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  1. E. Jakeman and P. N. Pusey, “Significance of K-distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
    [CrossRef]
  2. G. Parry and P. N. Pusey, “K-distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
    [CrossRef]
  3. S. F. Clifford and R. J. Hill, “Relation between irradiance and log-amplitude variance for optical scintillation described by the K distribution,” J. Opt. Soc. Am. 71, 112–114 (1981).
    [CrossRef]
  4. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 2.
    [CrossRef]
  5. P. A. Pincus, M. E. Fossey, J. F. Holmes, and J. R. Kerr, “Speckle propagation through turbulence; experimental,” J. Opt. Soc. Am. 68, 760–762 (1978).
    [CrossRef]
  6. J. W. Strohbehn, T.-i Wang, and J. P. Speck, “On the probability distribution of line of sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
    [CrossRef]
  7. J. F. Holmes and V. S. Rao Gudimetla, “Variance of intensity for a discrete spectrum polychromatic speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 71, 1176–1179 (1981).
    [CrossRef]
  8. M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, New York, 1960), pp. 3–36.
  9. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 340.
  10. J. F. Holmes, M. H. Lee, and J. R. Kerr, “Effect of log-amplitude covariance function on the statistics of speckle propagation through turbulent atmosphere,” J. Opt. Soc. Am. 70, 355–360 (1980).
    [CrossRef]
  11. G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 3.
    [CrossRef]
  12. V. S. Rao Gudimetla and J. F. Holmes, “Use of dominant eigenvalues in evaluating the probability density function of the intensity for a polychromatic speckle pattern,” J. Opt. Soc. Am. 70, 1015–1017 (1980).
    [CrossRef]
  13. J. W. Goodman, “Probability density function of the sum of N partially correlated speckle patterns,” Opt. Commun. 13, 244–247 (1975).
    [CrossRef]

1981 (2)

1980 (2)

1979 (1)

1978 (2)

P. A. Pincus, M. E. Fossey, J. F. Holmes, and J. R. Kerr, “Speckle propagation through turbulence; experimental,” J. Opt. Soc. Am. 68, 760–762 (1978).
[CrossRef]

E. Jakeman and P. N. Pusey, “Significance of K-distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

1975 (2)

J. W. Strohbehn, T.-i Wang, and J. P. Speck, “On the probability distribution of line of sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

J. W. Goodman, “Probability density function of the sum of N partially correlated speckle patterns,” Opt. Commun. 13, 244–247 (1975).
[CrossRef]

Clifford, S. F.

Fossey, M. E.

Goodman, J. W.

J. W. Goodman, “Probability density function of the sum of N partially correlated speckle patterns,” Opt. Commun. 13, 244–247 (1975).
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 2.
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 340.

Hill, R. J.

Holmes, J. F.

Jakeman, E.

E. Jakeman and P. N. Pusey, “Significance of K-distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

Kerr, J. R.

Lee, M. H.

Nakagami, M.

M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, New York, 1960), pp. 3–36.

Parry, G.

G. Parry and P. N. Pusey, “K-distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
[CrossRef]

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 3.
[CrossRef]

Pincus, P. A.

Pusey, P. N.

G. Parry and P. N. Pusey, “K-distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
[CrossRef]

E. Jakeman and P. N. Pusey, “Significance of K-distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

Rao Gudimetla, V. S.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 340.

Speck, J. P.

J. W. Strohbehn, T.-i Wang, and J. P. Speck, “On the probability distribution of line of sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

Strohbehn, J. W.

J. W. Strohbehn, T.-i Wang, and J. P. Speck, “On the probability distribution of line of sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

Wang, T.-i

J. W. Strohbehn, T.-i Wang, and J. P. Speck, “On the probability distribution of line of sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

J. Opt. Soc. Am. (6)

Opt. Commun. (1)

J. W. Goodman, “Probability density function of the sum of N partially correlated speckle patterns,” Opt. Commun. 13, 244–247 (1975).
[CrossRef]

Phys. Rev. Lett. (1)

E. Jakeman and P. N. Pusey, “Significance of K-distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

Radio Sci. (1)

J. W. Strohbehn, T.-i Wang, and J. P. Speck, “On the probability distribution of line of sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

Other (4)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 2.
[CrossRef]

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 3.
[CrossRef]

M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, New York, 1960), pp. 3–36.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 340.

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Figures (4)

Fig. 1
Fig. 1

Comparison of experimental cumulative PDF with theory for a monochromatic speckle pattern, 300-m path length.

Fig. 2
Fig. 2

Comparison of experimental cumulative PDF with theory for a monochromatic speckle pattern, 910-m path length.

Fig. 3
Fig. 3

Comparison of experimental cumulative PDF with theory for a polychromatic speckle pattern, σIN2 = 0.638.

Fig. 4
Fig. 4

Comparison of experimental cumulative PDF with theory for a polychromatic speckle pattern, σIN2 = 0.487.

Tables (2)

Tables Icon

Table 1 Comparison of Calculated and Measured Moments of the Intensity for a Monochromatic Speckle Pattern in the Turbulent Atmosphere

Tables Icon

Table 2 Comparison of Calculated and Measured Moments of the Intensity for a Polychromatic Speckle Pattern in the Turbulent Atmospherea

Equations (27)

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p I ( I ) = 1 λ e - I / λ ,
p I ( I ) = 0 p I ( I / λ = x ) p x ( x ) d x ,
p x ( x ) = 1 β exp ( - α + x β ) I 0 ( 2 x α β ) ,
x = α + β
σ x 2 = β 2 + 2 α β .
p x ( x ) = M M x M - 1 exp ( - M x x ) Γ ( M ) x M
M = ( α + β ) 2 β 2 + 2 α β .
x n = x n Γ ( n + M ) M n Γ ( M ) .
E 2 = 0 [ ( p I ( I / λ = x ) p s w ( x ) - p I ( I / λ = x ) p M ( x ) ] 2 d x ,
p I ( I ) = M M x M Γ ( M ) 0 X M - 2 exp ( - I X - M X x ) d x .
p I ( I ) = 2 ( M x ) ( M + 1 ) / 2 I ( M - 1 ) / 2 Γ ( M ) K M - 1 [ 2 ( M x I ) 1 / 2 ] ,
I n = Γ ( n + M ) Γ ( M ) Γ ( 1 + n ) ( x M ) n .
I = x = λ ,
σ I N 2 = σ I 2 I 2 = 1 + 2 M .
F I ( I ) = 0 I p I ( I ) d I = 1 - ( M I I ) M / 2 2 Γ ( M ) K M [ 2 ( M I I ) 1 / 2 ] .
p ( I ) = i = 1 N α i N - 2 j = 1 j i N ( α i - α j ) exp ( - I / α i λ ) λ ,
λ = i = 1 N λ i
p I ( I ) = M 1 M 1 M 2 M 2 I M 1 - 1 x M 2 Γ ( M 2 ) Γ ( M 1 ) 0 x M 2 - M 1 - 1 × exp ( - M 1 I x - M 2 x x ) d x = ( M 1 M 2 ) ( M 1 + M 2 ) / 2 Γ ( M 2 ) Γ ( M 1 ) 2 x ( M 1 + M 2 ) / 2 I ( M 1 + M 2 ) / 2 - 1 × K M 2 - M 1 [ 2 ( M 1 M 2 I x ) 1 / 2 ] .
I n = x n M 1 n M 2 n Γ ( n + M 2 ) Γ ( n + M 1 ) Γ ( M 2 ) Γ ( M 1 ) ,
I = x = λ , I 2 = I 2 ( 1 + 1 M 2 ) ( 1 + 1 M 1 ) , I 3 = I 3 ( 1 + 2 M 2 ) ( 1 + 1 M 2 ) ( 1 + 2 M 1 ) ( 1 + 1 M 1 ) , I 4 = I 4 ( 1 + 3 M 2 ) ( 1 + 2 M 2 ) ( 1 + 1 M 2 ) × ( 1 + 3 M 1 ) ( 1 + 2 M 1 ) ( 1 + 1 M 1 ) ,
σ I N 2 = ( 1 + 1 M 1 ) ( 1 + 1 M 2 ) - 1.
p I ( I ) = 2 M 2 ( M 2 + 1 ) / 2 Γ ( M 2 ) x ( M 2 + 1 ) / 2 I ( M 2 - 1 ) / 2 × i = 1 N α i N - M 2 / 2 - 3 / 2 j = 1 j i N ( α i - α j ) K 1 - M 2 [ 2 ( M 2 I x α i ) 1 / 2 ] ,
I n = Γ ( n + M 2 ) Γ ( M 2 ) M 2 n I n vacuum
I n vacuum = Γ ( n + 1 ) i = 1 N λ i N - 1 + n j = 1 j i N ( λ i - λ j ) .
p I ( I ) = 1 I ( 1 - α ) exp [ - I I ( 1 - α ) - α ( 1 - α ) ] × I 0 ( 2 I α ( 1 - α ) I ) ,
I = λ + I c ,
p I ( I ) = 2 M 2 ( M 2 - 1 ) / 2 Γ ( M 2 ) ( I λ ) ( M 2 - 1 ) / 2 exp ( - I c / λ ) n = 0 1 Γ 2 ( n + 1 ) × ( I c λ ) n ( I λ ) n / 2 M 2 n / 2 K 1 - n + M 2 [ 2 ( M 2 I λ ) 1 / 2 ] .