Abstract

A generalized Rician probability density function is used to model the first-order statistics of the optical field amplitudes scattered from an optically rough object containing a number of glints, under monochromatic, spatially coherent illumination. Numerical evaluations of analytic results for the special case of a rough object containing two glints of constant and equal strength compare favorably with computer simulations.

© 1977 Optical Society of America

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References

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  1. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York, 1975).
    [Crossref]
  2. D. L. Fried, “Statistics of The Laser Radar Cross Section of Randomly Rough Target,” J. Opt. Soc. Am. 66, 1150–1160 (1976).
    [Crossref]
  3. M. Elbaum and P. Diament, “SNR in Photocounting Images of Rough Objects in Partially Coherent Light,” Appl. Opt. 15, 2268–2274 (1976).
    [Crossref] [PubMed]
  4. H. Fujii and T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35–38 (1974).
    [Crossref]
  5. J. W. Goodman, “Dependence of Image Speckle Contrast on Surface Roughness,” Opt. Commun. 14, 324–327 (1975).
    [Crossref]
  6. E. Jakeman, J. G. McWhirter, and P. N. Pusey, “Enhanced Fluctuation in Radiation Scattered by a Moving Random Phase Screen,” J. Opt. Soc. Am. 66, 1175–1182 (1976).
    [Crossref]
  7. J. Blake and R. Barakat, “Time Interval Photoelectron Statistics of Non-Gaussian Scattered Light,” Opt. Commun. 20, 10–13 (1977).
    [Crossref]
  8. E. Jakeman and P. N. Pusey, “A Model for Non-Rayleigh Sea Echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
    [Crossref]
  9. K. Bullington, “Phase and Amplitude Variations in Multipath Fading of Microwave Signals,” Bell Syst. Tech. J. 50, 2039–2053 (1971).
    [Crossref]
  10. S. H. Lin, “Statistical Behavior of a Fading Signal,” Bell Syst. Tech. J. 50, 3211–3271 (1971).
    [Crossref]
  11. S. O. Rice, “Mathematical Analysis of Random Noise,” in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954).
  12. W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), pp. 158–168.
  13. M. Slack, “The Probability Distributions of Sinusoidal Oscillations Combined in Random Phase,” J. Inst. Elec. Eng. Part 3,  93, 76–86 (1946).
  14. J. A. Greenwood and D. Durand, “The Distribution of Length and Components of the Sum of N Random Unit Vectors,” Ann. Math. Sta. 26, 233–246 (1955).
    [Crossref]
  15. R. Barakat, “First-Order Statistics of Combined Random Sinusoidal Waves with Applications to Laser Speckle Patterns,” Opt. Acta. 21, 903–921 (1974).
    [Crossref]
  16. K. A. Norton, L. E. Vogler, W. V. Mansfield, and P. J. Short, “The Probability Distribution of the Amplitude of a Constant Vector,” Proc. IRE 43, 1354–1361 (1955).
    [Crossref]
  17. R. Esposito and L. R. Wilson, “Statistical Properties of Two Sine Waves in Gaussian Noise,” IEEE Trans. Inform. Theory IT-19, 176–183 (1973).
    [Crossref]
  18. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956), p. 330.

1977 (1)

J. Blake and R. Barakat, “Time Interval Photoelectron Statistics of Non-Gaussian Scattered Light,” Opt. Commun. 20, 10–13 (1977).
[Crossref]

1976 (4)

1975 (1)

J. W. Goodman, “Dependence of Image Speckle Contrast on Surface Roughness,” Opt. Commun. 14, 324–327 (1975).
[Crossref]

1974 (2)

H. Fujii and T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35–38 (1974).
[Crossref]

R. Barakat, “First-Order Statistics of Combined Random Sinusoidal Waves with Applications to Laser Speckle Patterns,” Opt. Acta. 21, 903–921 (1974).
[Crossref]

1973 (1)

R. Esposito and L. R. Wilson, “Statistical Properties of Two Sine Waves in Gaussian Noise,” IEEE Trans. Inform. Theory IT-19, 176–183 (1973).
[Crossref]

1971 (2)

K. Bullington, “Phase and Amplitude Variations in Multipath Fading of Microwave Signals,” Bell Syst. Tech. J. 50, 2039–2053 (1971).
[Crossref]

S. H. Lin, “Statistical Behavior of a Fading Signal,” Bell Syst. Tech. J. 50, 3211–3271 (1971).
[Crossref]

1955 (2)

J. A. Greenwood and D. Durand, “The Distribution of Length and Components of the Sum of N Random Unit Vectors,” Ann. Math. Sta. 26, 233–246 (1955).
[Crossref]

K. A. Norton, L. E. Vogler, W. V. Mansfield, and P. J. Short, “The Probability Distribution of the Amplitude of a Constant Vector,” Proc. IRE 43, 1354–1361 (1955).
[Crossref]

1946 (1)

M. Slack, “The Probability Distributions of Sinusoidal Oscillations Combined in Random Phase,” J. Inst. Elec. Eng. Part 3,  93, 76–86 (1946).

Asakura, T.

H. Fujii and T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35–38 (1974).
[Crossref]

Barakat, R.

J. Blake and R. Barakat, “Time Interval Photoelectron Statistics of Non-Gaussian Scattered Light,” Opt. Commun. 20, 10–13 (1977).
[Crossref]

R. Barakat, “First-Order Statistics of Combined Random Sinusoidal Waves with Applications to Laser Speckle Patterns,” Opt. Acta. 21, 903–921 (1974).
[Crossref]

Blake, J.

J. Blake and R. Barakat, “Time Interval Photoelectron Statistics of Non-Gaussian Scattered Light,” Opt. Commun. 20, 10–13 (1977).
[Crossref]

Bullington, K.

K. Bullington, “Phase and Amplitude Variations in Multipath Fading of Microwave Signals,” Bell Syst. Tech. J. 50, 2039–2053 (1971).
[Crossref]

Davenport, W. B.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), pp. 158–168.

Diament, P.

Durand, D.

J. A. Greenwood and D. Durand, “The Distribution of Length and Components of the Sum of N Random Unit Vectors,” Ann. Math. Sta. 26, 233–246 (1955).
[Crossref]

Elbaum, M.

Esposito, R.

R. Esposito and L. R. Wilson, “Statistical Properties of Two Sine Waves in Gaussian Noise,” IEEE Trans. Inform. Theory IT-19, 176–183 (1973).
[Crossref]

Fried, D. L.

Fujii, H.

H. Fujii and T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35–38 (1974).
[Crossref]

Goodman, J. W.

J. W. Goodman, “Dependence of Image Speckle Contrast on Surface Roughness,” Opt. Commun. 14, 324–327 (1975).
[Crossref]

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York, 1975).
[Crossref]

Greenwood, J. A.

J. A. Greenwood and D. Durand, “The Distribution of Length and Components of the Sum of N Random Unit Vectors,” Ann. Math. Sta. 26, 233–246 (1955).
[Crossref]

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956), p. 330.

Jakeman, E.

Lin, S. H.

S. H. Lin, “Statistical Behavior of a Fading Signal,” Bell Syst. Tech. J. 50, 3211–3271 (1971).
[Crossref]

Mansfield, W. V.

K. A. Norton, L. E. Vogler, W. V. Mansfield, and P. J. Short, “The Probability Distribution of the Amplitude of a Constant Vector,” Proc. IRE 43, 1354–1361 (1955).
[Crossref]

McWhirter, J. G.

Norton, K. A.

K. A. Norton, L. E. Vogler, W. V. Mansfield, and P. J. Short, “The Probability Distribution of the Amplitude of a Constant Vector,” Proc. IRE 43, 1354–1361 (1955).
[Crossref]

Pusey, P. N.

Rice, S. O.

S. O. Rice, “Mathematical Analysis of Random Noise,” in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954).

Root, W. L.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), pp. 158–168.

Short, P. J.

K. A. Norton, L. E. Vogler, W. V. Mansfield, and P. J. Short, “The Probability Distribution of the Amplitude of a Constant Vector,” Proc. IRE 43, 1354–1361 (1955).
[Crossref]

Slack, M.

M. Slack, “The Probability Distributions of Sinusoidal Oscillations Combined in Random Phase,” J. Inst. Elec. Eng. Part 3,  93, 76–86 (1946).

Vogler, L. E.

K. A. Norton, L. E. Vogler, W. V. Mansfield, and P. J. Short, “The Probability Distribution of the Amplitude of a Constant Vector,” Proc. IRE 43, 1354–1361 (1955).
[Crossref]

Wilson, L. R.

R. Esposito and L. R. Wilson, “Statistical Properties of Two Sine Waves in Gaussian Noise,” IEEE Trans. Inform. Theory IT-19, 176–183 (1973).
[Crossref]

Ann. Math. Sta. (1)

J. A. Greenwood and D. Durand, “The Distribution of Length and Components of the Sum of N Random Unit Vectors,” Ann. Math. Sta. 26, 233–246 (1955).
[Crossref]

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

K. Bullington, “Phase and Amplitude Variations in Multipath Fading of Microwave Signals,” Bell Syst. Tech. J. 50, 2039–2053 (1971).
[Crossref]

S. H. Lin, “Statistical Behavior of a Fading Signal,” Bell Syst. Tech. J. 50, 3211–3271 (1971).
[Crossref]

IEEE Trans. Antennas Propag. (1)

E. Jakeman and P. N. Pusey, “A Model for Non-Rayleigh Sea Echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[Crossref]

IEEE Trans. Inform. Theory (1)

R. Esposito and L. R. Wilson, “Statistical Properties of Two Sine Waves in Gaussian Noise,” IEEE Trans. Inform. Theory IT-19, 176–183 (1973).
[Crossref]

J. Inst. Elec. Eng. Part 3 (1)

M. Slack, “The Probability Distributions of Sinusoidal Oscillations Combined in Random Phase,” J. Inst. Elec. Eng. Part 3,  93, 76–86 (1946).

J. Opt. Soc. Am. (2)

Opt. Acta. (1)

R. Barakat, “First-Order Statistics of Combined Random Sinusoidal Waves with Applications to Laser Speckle Patterns,” Opt. Acta. 21, 903–921 (1974).
[Crossref]

Opt. Commun. (3)

J. Blake and R. Barakat, “Time Interval Photoelectron Statistics of Non-Gaussian Scattered Light,” Opt. Commun. 20, 10–13 (1977).
[Crossref]

H. Fujii and T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35–38 (1974).
[Crossref]

J. W. Goodman, “Dependence of Image Speckle Contrast on Surface Roughness,” Opt. Commun. 14, 324–327 (1975).
[Crossref]

Proc. IRE (1)

K. A. Norton, L. E. Vogler, W. V. Mansfield, and P. J. Short, “The Probability Distribution of the Amplitude of a Constant Vector,” Proc. IRE 43, 1354–1361 (1955).
[Crossref]

Other (4)

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956), p. 330.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York, 1975).
[Crossref]

S. O. Rice, “Mathematical Analysis of Random Noise,” in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954).

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), pp. 158–168.

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Figures (3)

FIG. 1
FIG. 1

Probability density function of the total field amplitude for N = 2. The total average power σ s 2 = 1 and the glint-to-diffuse power ratios r are shown.

FIG. 2
FIG. 2

Probability density functions of the total field amplitude for one (N = 1, Rician) and two (N = 2 generalized Rician) glints. The total average power σ s 2 = 1 and the glint-to-diffuse power ratio r = 20.

FIG. 3
FIG. 3

Numerical evaluation of the analytical probability density function compared with the computer-simulated histogram for N = 2. The total average power σ s 2 = 1 and the glint-to-diffuse power ratio r = 100.

Equations (13)

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E = E d + E g = E exp ( j θ ) ,
E g = n = 1 N A n exp ( j ϕ n ) = E g exp ( j ϕ ) ,
E r = E g cos ϕ + E d r , E i = E g sin ϕ + E d i ,
p N = ( E ) = E 0 ω J 0 ( ω E ) Φ N ( ω ) d ω .
Φ N ( ω ) = exp ( σ d 2 ω 2 2 ) n = 1 N J 0 ( A n ω ) .
p N ( E | E g ) = E σ d 2 exp [ E g 2 + E 2 2 σ d 2 ] I 0 ( E g E σ d 2 ) ,
p N ( E ) = 0 p N ( E g ) p N ( E | E g ) d ( E g ) ,
σ g 2 = 1 2 n = 1 N A n 2 .
I 0 ( z ) 1 ( 2 π z ) 1 / 2 exp z ,
p N ( E ) 1 ( 2 π σ d 2 ) 1 / 2 0 p N ( E g ) exp [ ( E E g ) 2 2 σ d 2 ] d ( E g ) .
p 2 ( E g ) = { 1 A π ( 1 E g 2 / 4 A 2 ) 1 / 2 0 E g < 2 A 0 otherwise .
p 2 ( E ) = E σ d 2 exp ( E 2 2 σ d 2 ) 1 A π × 0 2 A 1 ( 1 E g 2 / 4 A 2 ) 1 / 2 exp ( E g 2 2 σ d 2 ) I 0 ( E g E σ d 2 ) d ( E g ) .
σ s 2 = σ g 2 + σ d 2 , r = σ g 2 + σ d 2 .