Abstract

A speckle pattern formed in polarized monochromatic light may be regarded as resulting from a classical random walk in the complex plane. The resulting irradiance fluctuations obey negative exponential statistics, with ratio of standard deviation to mean (i.e., contrast) of unity. Reduction of this contrast, or smoothing of the speckle, requires diversity in polarization, space, frequency, or time. Addition of M uncorrelated speckle patterns on an intensity basis can reduce the contrast by 1/√M. However, addition of speckle patterns on a complex amplitude basis provides no reduction of contrast. The distribution of scale sizes in a speckle pattern (i.e., the Wiener spectrum) is investigated from a physical point of view.

© 1976 Optical Society of America

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References

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  1. J. D. Rigden and E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367–2368 (1962).
  2. B. M. Oliver, “Sparkling spots and random diffraction,” Proc. IEEE 51, 220–221 (1963).
    [Crossref]
  3. E. Verdet, Ann. Scientif. l’Ecole Normal Superieure 2, 291 (1865).
  4. J. W. Strutt (Lord Rayleigh), “On the resultant of a large number of vibrations of the same pitch and of arbitrary phase,” Philos. Mag. 10, 73–78 (1880).
  5. M. von Laue, Sitzungsber. Akad. Wiss. (Berlin) 44, 1144 (1914); Mitt. Physik Ges. (Zurich) 18, 90 (1916); Verhandl. Deut. Phys. Ges. 19, 19 (1917).
  6. P. E. Green, “Radar measurements of target scattering properties,” in Radar Astronomy, edited by J. V. Evans and T. Hagfors (McGraw-Hill, New York, 1968), pp. 1–77.
  7. E. N. Leith, “Quasi-holographic techniques in the microwave region,” Proc. IEEE 59, 1305–1318 (1971).
    [Crossref]
  8. Acoustical Holography, edited by P. S. Green (Plenum, New York, 1974), Vol. 5.
    [Crossref]
  9. J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” in Reports on Progress in Physics, edited by A. C. Strickland (The Physical Society, London, 1956), Vol. 19, pp. 188–267.
    [Crossref]
  10. L. Mandel, “Fluctuations of photon beams: the distribution of photoelectrons,” Proc. Phys. Soc. Lond. 74, 233–243 (1959).
    [Crossref]
  11. D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  12. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958).
  13. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser speckle and related phenomena, edited by J. C. Dainty (Springer-Verlag, Heidelberg, 1975), Vol. 9 (Topics in Applied Physics), pp. 9–75.
    [Crossref]
  14. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon/Macmillan, London, New York, 1963).
  15. K. Pearson, A Mathematical Theory of Random Migration (Draper’s Company Research Memoirs, London, 1906), Biometric Ser. III.
  16. J. W. Strutt (Lord Rayleigh), “On the problem of random vibrations, and of random flights in one, two, or three dimensions,” Philos. Mag. 37, 321–347 (1919).
  17. M. A. Condie, “An Experimental Investigation of the Statistics of Diffusely Reflected Coherent Light,” thesis (Dept. of Electrical Engineering, Stanford University, Stanford, Calif., 1966) (unpublished).
  18. J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
    [Crossref]
  19. T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optik 39, 258–267 (1974), “Statistics of coherent light speckle produced by stationary and moving apertures,” Ph. D. thesis (Dept. of Physics, Imperial College, London1974) (unpublished).
  20. D. Leger, E. Mathieu, and J. C. Perin, “Optical surface roughness determination using speckle correlation techniques,” Appl. Opt. 14, 872–877 (1975).
    [Crossref]
  21. N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
    [Crossref]
  22. G. Parry, “Some effects of surface roughness on the appearance of speckle in polychromatic light,” Opt. Commun. 12, 75–78 (1974).
    [Crossref]
  23. L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,” J. Opt. Soc. Am. 55, 247–253 (1965).
    [Crossref]

1975 (1)

1974 (3)

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

G. Parry, “Some effects of surface roughness on the appearance of speckle in polychromatic light,” Opt. Commun. 12, 75–78 (1974).
[Crossref]

T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optik 39, 258–267 (1974), “Statistics of coherent light speckle produced by stationary and moving apertures,” Ph. D. thesis (Dept. of Physics, Imperial College, London1974) (unpublished).

1971 (1)

E. N. Leith, “Quasi-holographic techniques in the microwave region,” Proc. IEEE 59, 1305–1318 (1971).
[Crossref]

1970 (1)

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[Crossref]

1965 (1)

1963 (1)

B. M. Oliver, “Sparkling spots and random diffraction,” Proc. IEEE 51, 220–221 (1963).
[Crossref]

1962 (1)

J. D. Rigden and E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367–2368 (1962).

1959 (1)

L. Mandel, “Fluctuations of photon beams: the distribution of photoelectrons,” Proc. Phys. Soc. Lond. 74, 233–243 (1959).
[Crossref]

1919 (1)

J. W. Strutt (Lord Rayleigh), “On the problem of random vibrations, and of random flights in one, two, or three dimensions,” Philos. Mag. 37, 321–347 (1919).

1914 (1)

M. von Laue, Sitzungsber. Akad. Wiss. (Berlin) 44, 1144 (1914); Mitt. Physik Ges. (Zurich) 18, 90 (1916); Verhandl. Deut. Phys. Ges. 19, 19 (1917).

1880 (1)

J. W. Strutt (Lord Rayleigh), “On the resultant of a large number of vibrations of the same pitch and of arbitrary phase,” Philos. Mag. 10, 73–78 (1880).

1865 (1)

E. Verdet, Ann. Scientif. l’Ecole Normal Superieure 2, 291 (1865).

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon/Macmillan, London, New York, 1963).

Condie, M. A.

M. A. Condie, “An Experimental Investigation of the Statistics of Diffusely Reflected Coherent Light,” thesis (Dept. of Electrical Engineering, Stanford University, Stanford, Calif., 1966) (unpublished).

Dainty, J. C.

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[Crossref]

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958).

George, N.

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

Goldfischer, L. I.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser speckle and related phenomena, edited by J. C. Dainty (Springer-Verlag, Heidelberg, 1975), Vol. 9 (Topics in Applied Physics), pp. 9–75.
[Crossref]

Gordon, E. I.

J. D. Rigden and E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367–2368 (1962).

Green, P. E.

P. E. Green, “Radar measurements of target scattering properties,” in Radar Astronomy, edited by J. V. Evans and T. Hagfors (McGraw-Hill, New York, 1968), pp. 1–77.

Jain, A.

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

Leger, D.

Leith, E. N.

E. N. Leith, “Quasi-holographic techniques in the microwave region,” Proc. IEEE 59, 1305–1318 (1971).
[Crossref]

Mandel, L.

L. Mandel, “Fluctuations of photon beams: the distribution of photoelectrons,” Proc. Phys. Soc. Lond. 74, 233–243 (1959).
[Crossref]

Mathieu, E.

McKechnie, T. S.

T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optik 39, 258–267 (1974), “Statistics of coherent light speckle produced by stationary and moving apertures,” Ph. D. thesis (Dept. of Physics, Imperial College, London1974) (unpublished).

Middleton, D.

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

Oliver, B. M.

B. M. Oliver, “Sparkling spots and random diffraction,” Proc. IEEE 51, 220–221 (1963).
[Crossref]

Parry, G.

G. Parry, “Some effects of surface roughness on the appearance of speckle in polychromatic light,” Opt. Commun. 12, 75–78 (1974).
[Crossref]

Pearson, K.

K. Pearson, A Mathematical Theory of Random Migration (Draper’s Company Research Memoirs, London, 1906), Biometric Ser. III.

Perin, J. C.

Ratcliffe, J. A.

J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” in Reports on Progress in Physics, edited by A. C. Strickland (The Physical Society, London, 1956), Vol. 19, pp. 188–267.
[Crossref]

Rigden, J. D.

J. D. Rigden and E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367–2368 (1962).

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958).

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon/Macmillan, London, New York, 1963).

Strutt, J. W.

J. W. Strutt (Lord Rayleigh), “On the problem of random vibrations, and of random flights in one, two, or three dimensions,” Philos. Mag. 37, 321–347 (1919).

J. W. Strutt (Lord Rayleigh), “On the resultant of a large number of vibrations of the same pitch and of arbitrary phase,” Philos. Mag. 10, 73–78 (1880).

Verdet, E.

E. Verdet, Ann. Scientif. l’Ecole Normal Superieure 2, 291 (1865).

von Laue, M.

M. von Laue, Sitzungsber. Akad. Wiss. (Berlin) 44, 1144 (1914); Mitt. Physik Ges. (Zurich) 18, 90 (1916); Verhandl. Deut. Phys. Ges. 19, 19 (1917).

Ann. Scientif. l’Ecole Normal Superieure (1)

E. Verdet, Ann. Scientif. l’Ecole Normal Superieure 2, 291 (1865).

Appl. Opt. (1)

Appl. Phys. (1)

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[Crossref]

Opt. Commun. (1)

G. Parry, “Some effects of surface roughness on the appearance of speckle in polychromatic light,” Opt. Commun. 12, 75–78 (1974).
[Crossref]

Optik (1)

T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optik 39, 258–267 (1974), “Statistics of coherent light speckle produced by stationary and moving apertures,” Ph. D. thesis (Dept. of Physics, Imperial College, London1974) (unpublished).

Philos. Mag. (2)

J. W. Strutt (Lord Rayleigh), “On the problem of random vibrations, and of random flights in one, two, or three dimensions,” Philos. Mag. 37, 321–347 (1919).

J. W. Strutt (Lord Rayleigh), “On the resultant of a large number of vibrations of the same pitch and of arbitrary phase,” Philos. Mag. 10, 73–78 (1880).

Proc. IEEE (2)

B. M. Oliver, “Sparkling spots and random diffraction,” Proc. IEEE 51, 220–221 (1963).
[Crossref]

E. N. Leith, “Quasi-holographic techniques in the microwave region,” Proc. IEEE 59, 1305–1318 (1971).
[Crossref]

Proc. IRE (1)

J. D. Rigden and E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367–2368 (1962).

Proc. Phys. Soc. Lond. (1)

L. Mandel, “Fluctuations of photon beams: the distribution of photoelectrons,” Proc. Phys. Soc. Lond. 74, 233–243 (1959).
[Crossref]

Sitzungsber. Akad. Wiss. (Berlin) (1)

M. von Laue, Sitzungsber. Akad. Wiss. (Berlin) 44, 1144 (1914); Mitt. Physik Ges. (Zurich) 18, 90 (1916); Verhandl. Deut. Phys. Ges. 19, 19 (1917).

Other (9)

P. E. Green, “Radar measurements of target scattering properties,” in Radar Astronomy, edited by J. V. Evans and T. Hagfors (McGraw-Hill, New York, 1968), pp. 1–77.

Acoustical Holography, edited by P. S. Green (Plenum, New York, 1974), Vol. 5.
[Crossref]

J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” in Reports on Progress in Physics, edited by A. C. Strickland (The Physical Society, London, 1956), Vol. 19, pp. 188–267.
[Crossref]

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

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser speckle and related phenomena, edited by J. C. Dainty (Springer-Verlag, Heidelberg, 1975), Vol. 9 (Topics in Applied Physics), pp. 9–75.
[Crossref]

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon/Macmillan, London, New York, 1963).

K. Pearson, A Mathematical Theory of Random Migration (Draper’s Company Research Memoirs, London, 1906), Biometric Ser. III.

M. A. Condie, “An Experimental Investigation of the Statistics of Diffusely Reflected Coherent Light,” thesis (Dept. of Electrical Engineering, Stanford University, Stanford, Calif., 1966) (unpublished).

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Figures (7)

FIG. 1
FIG. 1

Typical speckle pattern.

FIG. 2
FIG. 2

Speckle formation in the free-space geometry.

FIG. 3
FIG. 3

Speckle formation in the imaging geometry.

FIG. 4
FIG. 4

Random walk in the complex plane.

FIG. 5
FIG. 5

Probability density function of a polarized speckle pattern.

FIG. 6
FIG. 6

Single vector spacing s is embraced by the scattering spot in many different ways.

FIG. 7
FIG. 7

Cross section of the Wiener spectrum of a speckle pattern arising from a rectangular scattering spot.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

u ( x , y , z ; t ) = A ( x , y , z ) exp ( i 2 π ν t ) ,
A ( x , y , z ) = A ( x , y , z ) exp [ i θ ( x , y , z ) ] .
I ( x , y , z ) = lim T 1 T - T / 2 T / 2 u ( x , y , z ; t ) 2 d t = A ( x , y , z ) 2 .
A ( x , y , z ) = k = 1 N a k exp ( i ϕ k ) ,
p ( I ) = { ( 1 / Ī ) exp ( - I / Ī ) , I 0 , 0 , otherwise ,
P ( I > I t ) = exp ( - I t / Ī ) ,             I t 0.
C = σ I / Ī ,
C = σ I / Ī = 1 / M .
Δ ν c / 2 σ z ,
s = ( s X , s Y ) = ( λ z ν X , λ z ν Y ) ,
W ( ν ) = ( Ī ) 2 [ δ ( ν ) + - R ( ξ ) R ( ξ - λ z ν ) d ξ / ( - R ( ξ ) d ξ ) 2 ] ,
W ( ν ) = ( Ī ) 2 [ δ ( ν ) + ( λ z L ) 2 Λ ( λ z L ν X ) Λ ( λ z L ν Y ) ] ,
W ( ν ) = ( Ī ) 2 ( δ ( ν ) + 2 ( λ z D ) 2 · 2 π { cos - 1 ( λ z D ν ) - ( λ z D ν ) [ 1 - ( λ z D ν ) 2 ] 1 / 2 } )