Abstract

Non-Gaussian speckle generated by the coherent superposition of a small number of random complex amplitudes can be physically realized through the elastic scattering of incident monochromatic radiation with a thin, nondepolarizing random phase screen. Under conditions of Gaussian-distributed phase fluctuations whose rms is much greater than 2π rad, whose lateral autocorrelation is Gaussian in shape with a correlation length z greater than twice the wavelength of the illumination, and for which the ratio of these two is less than 0.05, the Kirchhoff diffraction integral approximation can be applied. A series solution for the first and second intensity moments for the far field is derived and presented. The steepest-descents solution given by Jakeman and McWhirter [ Appl. Phys. B 26, 125 ( 1981)] converges to the given series solution for the mean intensity. With improved experimental technique, measurements of the normalized second moment are shown to agree with Jakeman and McWhirter’s approximation over a wide range of illuminated scattering centers. A computer simulation of this experiment for phase objects of up to 50-rad rms phase deviations is shown to agree well with predictions of the mean intensity. The second moments agree well in the low- and high-illumination limits but systematically overestimate the normalized second moment near the peak of each curve.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984).
  2. J. C. Dainty, “An introduction to ‘Gaussian’ speckle,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 2–8 (1980).
  3. J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics XIV, E. Wolf, ed. (North-Holland, New York, 1976).
  4. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  5. J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984).
  6. R. K. Erf, ed., Speckle Metrology (Academic, New York, 1978); K. Creath, “Digital-speckle pattern interferometry,” J. Opt. Soc. Am. A 1, 1222 (A) (1984).
  7. J. C. Dainty, D. Newman, “Detection of gratings hidden by diffusers using photon-correlation techniques,” Opt. Lett. 8, 608–610 (1983); D. Newman, J. C. Dainty, “Detecting gratings hidden by diffusers using intensity interferometry,” J. Opt. Soc. Am. A 1, 403–411 (1984).
    [CrossRef] [PubMed]
  8. J. Marron, G. M. Morris, “Image-plane speckle form rotating rough objects,” J. Opt. Soc. Am. A 2, 1395–1402 (1985).
    [CrossRef]
  9. E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
    [CrossRef]
  10. J. K. Jao, M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
    [CrossRef]
  11. C. J. Oliver, “A model for non-Rayleigh scattering statistics,” Opt. Acta 31, 701–722 (1984).
    [CrossRef]
  12. R. L. Phillips, L. C. Andrews, “Universal statistical model for irradiance fluctuations in a turbulent medium,” J. Opt. Soc. Am. 72, 864–870 (1982).
    [CrossRef]
  13. L. C. Andrews, R. L. Phillips, “I–K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985).
    [CrossRef]
  14. J. C. Kluyver, “A local probability problem,” Proc. R. Acad. Sci. (Amsterdam) 8, 341–350 (1905).
  15. E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. 23, 453–461 (1984).
    [CrossRef]
  16. E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
    [CrossRef]
  17. R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. 58, 382–400 (1962).
    [CrossRef]
  18. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1962).
  19. E. N. Bramley, “Diffraction of an angular spectrum of waves by a phase-changing screen,” J. Atmos. Terr. Phys. 29, 1–28 (1967).
    [CrossRef]
  20. H. M. Escamilla, “Speckle contrast in the diffraction field of a weak random-phase screen when the illuminated region contains a few correlation areas,” Opt. Acta 30, 1655–1664 (1983).
    [CrossRef]
  21. B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements from diffusers of known statistics,” Opt. Commun. 45, 252–257 (1983).
    [CrossRef]
  22. J. B. Thomas, An Introduction to Applied Probability and Random Processes (Wiley, New York, 1971), p. 131, defines the multivariate characteristic function for Gaussian distributed random variables asChϕ(t1,t2,…,tN)=〈exp[i∑j−1Ntjϕ(ξj,ηj)]〉=exp[−12σϕ2∑j=1N∑k=1NtjtkC′ϕ(ξj,ηj;ξk,ηk)], on which all moments of the complex amplitudes are based.
  23. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
    [CrossRef]
  24. I. S. Gradshtyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 307.
  25. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 228.
  26. J. Ohtsubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
    [CrossRef]
  27. M. Deka, S. P. Almeida, H. Fujii, “Root-mean-square difference between the intensities of non-Gaussian speckle at two different wavelengths,” J. Opt. Soc. Am. 71, 155–163 (1981).
    [CrossRef]
  28. P. J. Chandley, H. M. Escamilla, “Speckle form a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151–154 (1979).
    [CrossRef]
  29. M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
    [CrossRef]
  30. E. Jakeman, W. J. Welford, “Speckle statistics in imaging systems,” Opt. Comm. 21, 72–79 (1977).
    [CrossRef]
  31. J. Ohtsubo, “Non-Gaussian speckle: a computer simulation,” Appl. Opt. 21, 4167–4175 (1982).
    [CrossRef] [PubMed]
  32. J. Ohtsubo, “Non-Gaussian speckle produced by a random phase screen,” ICO-13 Conference Digest (ICO-13 Conference Committee, Sapporo, Japan, 1984), paper A7-5.
  33. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 504.
  34. B. M. Levine, “The characterization and measurement of non-Gaussian speckle,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1985) (unpublished).
  35. J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 105.

1985 (2)

1984 (2)

C. J. Oliver, “A model for non-Rayleigh scattering statistics,” Opt. Acta 31, 701–722 (1984).
[CrossRef]

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. 23, 453–461 (1984).
[CrossRef]

1983 (3)

H. M. Escamilla, “Speckle contrast in the diffraction field of a weak random-phase screen when the illuminated region contains a few correlation areas,” Opt. Acta 30, 1655–1664 (1983).
[CrossRef]

B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements from diffusers of known statistics,” Opt. Commun. 45, 252–257 (1983).
[CrossRef]

J. C. Dainty, D. Newman, “Detection of gratings hidden by diffusers using photon-correlation techniques,” Opt. Lett. 8, 608–610 (1983); D. Newman, J. C. Dainty, “Detecting gratings hidden by diffusers using intensity interferometry,” J. Opt. Soc. Am. A 1, 403–411 (1984).
[CrossRef] [PubMed]

1982 (2)

1981 (3)

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
[CrossRef]

M. Deka, S. P. Almeida, H. Fujii, “Root-mean-square difference between the intensities of non-Gaussian speckle at two different wavelengths,” J. Opt. Soc. Am. 71, 155–163 (1981).
[CrossRef]

1979 (1)

P. J. Chandley, H. M. Escamilla, “Speckle form a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151–154 (1979).
[CrossRef]

1978 (2)

J. Ohtsubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
[CrossRef]

J. K. Jao, M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
[CrossRef]

1977 (1)

E. Jakeman, W. J. Welford, “Speckle statistics in imaging systems,” Opt. Comm. 21, 72–79 (1977).
[CrossRef]

1976 (1)

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

1967 (1)

E. N. Bramley, “Diffraction of an angular spectrum of waves by a phase-changing screen,” J. Atmos. Terr. Phys. 29, 1–28 (1967).
[CrossRef]

1962 (2)

R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. 58, 382–400 (1962).
[CrossRef]

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

1905 (1)

J. C. Kluyver, “A local probability problem,” Proc. R. Acad. Sci. (Amsterdam) 8, 341–350 (1905).

Almeida, S. P.

Andrews, L. C.

Asakura, T.

J. Ohtsubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1962).

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 504.

Bramley, E. N.

E. N. Bramley, “Diffraction of an angular spectrum of waves by a phase-changing screen,” J. Atmos. Terr. Phys. 29, 1–28 (1967).
[CrossRef]

Chandley, P. J.

P. J. Chandley, H. M. Escamilla, “Speckle form a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151–154 (1979).
[CrossRef]

Dainty, J. C.

B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements from diffusers of known statistics,” Opt. Commun. 45, 252–257 (1983).
[CrossRef]

J. C. Dainty, D. Newman, “Detection of gratings hidden by diffusers using photon-correlation techniques,” Opt. Lett. 8, 608–610 (1983); D. Newman, J. C. Dainty, “Detecting gratings hidden by diffusers using intensity interferometry,” J. Opt. Soc. Am. A 1, 403–411 (1984).
[CrossRef] [PubMed]

J. C. Dainty, “An introduction to ‘Gaussian’ speckle,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 2–8 (1980).

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics XIV, E. Wolf, ed. (North-Holland, New York, 1976).

J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984).

Deka, M.

Elbaum, M.

J. K. Jao, M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
[CrossRef]

Escamilla, H. M.

H. M. Escamilla, “Speckle contrast in the diffraction field of a weak random-phase screen when the illuminated region contains a few correlation areas,” Opt. Acta 30, 1655–1664 (1983).
[CrossRef]

P. J. Chandley, H. M. Escamilla, “Speckle form a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151–154 (1979).
[CrossRef]

Fujii, H.

Garcia, N.

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

Goodman, J. W.

J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 105.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Gradshtyn, I. S.

I. S. Gradshtyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 307.

Jakeman, E.

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. 23, 453–461 (1984).
[CrossRef]

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
[CrossRef]

E. Jakeman, W. J. Welford, “Speckle statistics in imaging systems,” Opt. Comm. 21, 72–79 (1977).
[CrossRef]

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

Jao, J. K.

J. K. Jao, M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
[CrossRef]

Kluyver, J. C.

J. C. Kluyver, “A local probability problem,” Proc. R. Acad. Sci. (Amsterdam) 8, 341–350 (1905).

Levine, B. M.

B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements from diffusers of known statistics,” Opt. Commun. 45, 252–257 (1983).
[CrossRef]

B. M. Levine, “The characterization and measurement of non-Gaussian speckle,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1985) (unpublished).

Marron, J.

McWhirter, J. G.

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
[CrossRef]

Mercier, R. P.

R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. 58, 382–400 (1962).
[CrossRef]

Morris, G. M.

Newman, D.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

Ohtsubo, J.

J. Ohtsubo, “Non-Gaussian speckle: a computer simulation,” Appl. Opt. 21, 4167–4175 (1982).
[CrossRef] [PubMed]

J. Ohtsubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
[CrossRef]

J. Ohtsubo, “Non-Gaussian speckle produced by a random phase screen,” ICO-13 Conference Digest (ICO-13 Conference Committee, Sapporo, Japan, 1984), paper A7-5.

Oliver, C. J.

C. J. Oliver, “A model for non-Rayleigh scattering statistics,” Opt. Acta 31, 701–722 (1984).
[CrossRef]

Phillips, R. L.

Pusey, P. N.

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshtyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 307.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1962).

Thomas, J. B.

J. B. Thomas, An Introduction to Applied Probability and Random Processes (Wiley, New York, 1971), p. 131, defines the multivariate characteristic function for Gaussian distributed random variables asChϕ(t1,t2,…,tN)=〈exp[i∑j−1Ntjϕ(ξj,ηj)]〉=exp[−12σϕ2∑j=1N∑k=1NtjtkC′ϕ(ξj,ηj;ξk,ηk)], on which all moments of the complex amplitudes are based.

Welford, W. J.

E. Jakeman, W. J. Welford, “Speckle statistics in imaging systems,” Opt. Comm. 21, 72–79 (1977).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 504.

Appl. Opt. (1)

Appl. Phys. B (1)

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

IRE Trans. Inf. Theory (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

J. Atmos. Terr. Phys. (1)

E. N. Bramley, “Diffraction of an angular spectrum of waves by a phase-changing screen,” J. Atmos. Terr. Phys. 29, 1–28 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Acta (3)

C. J. Oliver, “A model for non-Rayleigh scattering statistics,” Opt. Acta 31, 701–722 (1984).
[CrossRef]

H. M. Escamilla, “Speckle contrast in the diffraction field of a weak random-phase screen when the illuminated region contains a few correlation areas,” Opt. Acta 30, 1655–1664 (1983).
[CrossRef]

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

Opt. Comm. (1)

E. Jakeman, W. J. Welford, “Speckle statistics in imaging systems,” Opt. Comm. 21, 72–79 (1977).
[CrossRef]

Opt. Commun. (3)

P. J. Chandley, H. M. Escamilla, “Speckle form a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151–154 (1979).
[CrossRef]

J. Ohtsubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
[CrossRef]

B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements from diffusers of known statistics,” Opt. Commun. 45, 252–257 (1983).
[CrossRef]

Opt. Eng. (1)

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. 23, 453–461 (1984).
[CrossRef]

Opt. Lett. (1)

Proc. Cambridge Philos. Soc. (1)

R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. 58, 382–400 (1962).
[CrossRef]

Proc. IEEE (1)

J. K. Jao, M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
[CrossRef]

Proc. R. Acad. Sci. (Amsterdam) (1)

J. C. Kluyver, “A local probability problem,” Proc. R. Acad. Sci. (Amsterdam) 8, 341–350 (1905).

Other (14)

J. B. Thomas, An Introduction to Applied Probability and Random Processes (Wiley, New York, 1971), p. 131, defines the multivariate characteristic function for Gaussian distributed random variables asChϕ(t1,t2,…,tN)=〈exp[i∑j−1Ntjϕ(ξj,ηj)]〉=exp[−12σϕ2∑j=1N∑k=1NtjtkC′ϕ(ξj,ηj;ξk,ηk)], on which all moments of the complex amplitudes are based.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984).

J. C. Dainty, “An introduction to ‘Gaussian’ speckle,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 2–8 (1980).

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics XIV, E. Wolf, ed. (North-Holland, New York, 1976).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984).

R. K. Erf, ed., Speckle Metrology (Academic, New York, 1978); K. Creath, “Digital-speckle pattern interferometry,” J. Opt. Soc. Am. A 1, 1222 (A) (1984).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1962).

I. S. Gradshtyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 307.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 228.

J. Ohtsubo, “Non-Gaussian speckle produced by a random phase screen,” ICO-13 Conference Digest (ICO-13 Conference Committee, Sapporo, Japan, 1984), paper A7-5.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 504.

B. M. Levine, “The characterization and measurement of non-Gaussian speckle,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1985) (unpublished).

J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 105.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

The geometry for describing the generation of non-Gaussian speckle.

Fig. 2
Fig. 2

Comparison of the series solution and the steepest-descents solution for the mean intensity of non-Gaussian speckle as a function of the phase standard deviation and inverse scale. The solid lines denote the series (exact) solution, and the dashed lines denote the approximation.

Fig. 3
Fig. 3

Evaluation of Jakeman and McWhirter’s solution for the normalized second moment of intensity for non-Gaussian speckle. (a) Normalized second moment for phase standard deviations: 2π, 10, 20, 30, 40, 50. (b) Normalized second moments for phase standard deviations: 2π, 7, 8, 9, 10.

Fig. 4
Fig. 4

Experimental schematic of non-Gaussian generation and measurement (after Levine and Dainty21).

Fig. 5
Fig. 5

Remeasurement of diffuser D in transmission after careful focus of the full-aperture non-Gaussian speckle pattern.

Fig. 6
Fig. 6

Comparison of the simulated mean intensity to the theory given by Eq. (15).

Fig. 7
Fig. 7

Comparison of the simulated normalized second moment of intensity for an rms phase deviation of σϕ = 10 rad. The solid line represents the theory of Jakeman and McWhirter [Eq. (21)].

Fig. 8
Fig. 8

Comparison of the simulated normalized second moment of intensity for an rms phase deviation of σϕ = 20 rad. The solid line represents the theory of Jakeman and McWhirter [Eq. (21)].

Fig. 9
Fig. 9

Comparison of the simulated normalized second moment of intensity for an rms phase deviation of σϕ = 30 rad. The solid line represents the theory of Jakeman and McWhirter [Eq. (21)].

Fig. 10
Fig. 10

Comparison of the simulated normalized second moment of intensity for an rms phase deviation of σϕ = 40 rad. The solid line represents the theory of Jakeman and McWhirter [Eq. (21)].

Fig. 11
Fig. 11

Comparison of the simulated normalized second moment of intensity for an rms phase deviation of σϕ = 50 rad. The solid line represents the theory of Jakeman and McWhirter [Eq. (21)].

Equations (46)

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

I 2 I 2 = 2 ,
a s ( ξ , η ) = 1 π w 2 exp [ ( ξ 2 + η 2 ) w 2 ] ,
a ( ξ , η ) = exp [ i ϕ ( ξ , η ) ] .
ϕ ( ξ , η ) = 2 π λ ( n 1 ) h ( ξ , η ) ,
σ ϕ 2 = [ 2 π λ ( n 1 ) ] 2 σ h 2 ,
ϕ * ( ξ 1 , η 1 ) ϕ ( ξ 2 , η 2 ) = σ ϕ 2 C ϕ ( ξ 1 ξ 2 ; η 1 η 2 )
C ϕ ( ξ 1 ξ 2 ; η 1 η 2 ) = exp [ ( ξ 1 ξ 2 ) 2 + ( η 1 η 2 ) 2 z 2 ] .
C a ( ξ 1 , η 1 ; ξ 2 , η 2 ) = a * ( ξ 1 , η 1 ) a ( ξ 2 , η 2 ) = exp { i [ ϕ ( ξ 2 , η 2 ) ϕ ( ξ 1 , η 1 ) ] } = exp { σ ϕ 2 [ 1 C ϕ ( ξ 1 ξ 2 ; η 1 η 2 ) ] } ,
C I ( ξ 1 , η 1 ; ξ 2 , η 2 ; ξ 3 , η 3 ; ξ 4 , η 4 ) = a * ( ξ 1 , η 1 ) a ( ξ 2 , η 2 ) a * ( ξ 3 , η 3 ) a ( ξ 4 , η 4 ) = exp { i [ ϕ ( ξ 2 , η 2 ) ϕ ( ξ 1 , η 1 ) + ϕ ( ξ 4 , η 4 ) ϕ ( ξ 3 , η 3 ) ] } = exp { σ ϕ 2 [ 2 C ϕ ( ξ 1 ξ 2 ; η 1 η 2 ) + C ϕ ( ξ 1 ξ 3 ; η 1 η 3 ) C ϕ ( ξ 1 ξ 4 ; η 1 η 4 ) C ϕ ( ξ 2 ξ 3 ; η 2 η 3 ) + C ϕ ( ξ 2 ξ 4 ; η 2 η 4 ) C ϕ ( ξ 3 ξ 4 ; η 3 η 4 ) ] } .
A ( x , y ) = i cos θ λ R 0 a s ( ξ , η ) a ( ξ , η ) × exp [ i 2 π ( u ξ + υ η ) ] d ξ d η ,
u = x λ R 0 , υ = y λ R 0 .
I ( x , y ) = cos 2 θ ( λ l ) 2 d ξ 1 d ξ 2 d η 1 d η 2 a * ( ξ 1 , η 1 ) a s ( ξ 2 , η 2 ) × C a ( ξ 1 , η 1 ; ξ 2 , η 2 ) × exp { i 2 π [ u ( ξ 1 ξ 2 ) + υ ( η 1 η 2 ) ] } .
τ = ξ 1 ξ 2 , τ = η 1 η 2 , a = ξ 2 , a = η 2 ,
I ( x , y ) = cos 2 θ ( λ l ) 2 ( 2 π w 2 ) d τ d τ exp [ ( τ 2 + τ 2 2 w 2 ) ] × exp ( σ ϕ 2 { 1 exp [ ( τ 2 + τ 2 z 2 ) ] } ) × exp [ i 2 π ( u τ + υ τ ) ] .
I ( x , y ) = cos 2 θ ( λ l ) 2 ( 2 π w 2 ) exp ( σ ϕ 2 ) j = 0 σ ϕ 2 j j ! × { d τ exp [ τ 2 ( 1 2 w 2 + j z 2 ) ] × exp ( i 2 π u τ ) } { × d τ exp [ τ 2 ( 1 2 w 2 + j z 2 ) ] × exp ( i 2 π υ τ ) } .
d x exp ( p 2 x 2 ± q x ) = π 1 / 2 p exp ( q 2 4 p 2 ) , p > 0 ,
I ( θ ) = cos 2 θ ( λ l ) 2 exp ( σ ϕ 2 ) j = 0 σ ϕ 2 j j ! ( 1 + 2 w 2 σ ϕ 2 j z 2 ) 1 × exp [ k 2 w 2 sin 2 θ / 2 ( 1 + 2 w 2 σ ϕ 2 j z 2 ) ] .
I ( θ ) = cos 2 θ ( λ l ) 2 1 ( 1 + 2 w 2 σ ϕ 2 z 2 ) × exp [ k 2 w 2 sin 2 θ / ( 1 + 2 w 2 σ ϕ 2 z 2 ) ] .
I 2 ( x , y ) = A * ( x 1 , y 1 ) A ( x 2 , y 2 ) A * ( x 3 , y 3 ) × A ( x 4 , y 4 ) | x 1 = x 2 = x 3 = x 4 = x ; y 1 = y 2 = y 3 = y 4 = y
I 2 ( x , y ) = cos 4 θ exp ( 2 σ ϕ 2 ) ( λ l ) 4 × j = 0 n = 0 j ( 1 ) n 5 + n 6 σ ϕ 2 j n 1 ! n 2 ! n 3 ! n 4 ! n 5 ! n 6 ! × exp ( k 2 q 2 sin 2 θ p 2 ) p 2 ,
n = n 1 + n 2 + n 3 + n 4 + n 5 + n 6 .
q 2 = [ 1 + w 2 ( n 1 + n 2 + n 5 + n 6 ) z 2 ] × [ 1 + w 2 ( n 3 + n 4 + n 5 + n 6 ) z 2 ] [ w 2 ( n 5 n 6 ) z 2 ] 2
p 2 = [ 1 + w 2 ( n 1 + n 2 + n 3 + n 4 ) z 2 ] × [ 1 + w 2 ( n 3 + n 4 + n 5 + n 6 ) z 2 ] × [ 1 + w 2 ( n 1 + n 2 + n 5 + n 6 ) z 2 ] [ w 2 ( n 1 n 2 ) z 2 ] 2 × [ 1 + w 2 ( n 3 + n 4 + n 5 + n 6 ) z 2 ] [ w 2 ( n 3 n 4 ) z 2 ] 2 × [ 1 + w 2 ( n 1 + n 2 + n 5 + n 6 ) z 2 ] [ w 2 ( n 5 n 6 ) z 2 ] 2 × [ 1 + w 2 ( n 1 + n 2 + n 3 + n 4 ) z 2 ] + 2 [ w 2 ( n 1 n 2 ) z 2 ] 2 [ w 2 ( n 3 n 4 ) z 2 ] 2 [ w 2 ( n 5 n 6 ) z 2 ] 2 .
I 2 ( θ ) I ( θ ) = 2 e s + s t ( s + t ) 2 ( s + 2 t ) [ E 1 ( s 2 t ) 2 E 1 ( s 2 t + s ) ] × exp [ s 2 t + k 2 w 2 sin 2 θ ( s + t ) ( s + 2 t ) ] ,
E 1 ( x ) = z e t t d t
s = z 2 2 w 2
t = σ ϕ 2 .
ϕ ( ξ , η ) = h ( ξ , η ) n ( ξ ξ , η η ) d ξ d η = F 1 [ h ( u , υ ) n ( u , υ ) ] ,
F 1 the inverse Fourier transform ,
h ( u , υ ) = F [ C ϕ ( ξ , η ) ] 1 / 2 = π 1 / 2 z exp [ π 2 z 2 ( u 2 + υ 2 ) 2 ] .
A ( x , y ) = a ( ξ , η ) h ( x ξ , y η ) d ξ d η ,
A ( 0 , 0 ) = a ( ξ , η ) h ( ξ , y ) d ξ d η ,
h ( ξ , η ) = π ( D / 2 ) 2 exp { [ π ( D / 2 ) ] 2 [ ( ξ / λ f ) 2 + ( η / λ f ) 2 ] } .
I ( 0 , 0 ) = A ( 0 , 0 ) A * ( 0 , 0 ) .
I 2 ( x , y ) = A * ( x 1 , y 1 ) A ( x 2 , y 2 ) × A * ( x 3 , y 3 ) A ( x 4 , y 4 ) | x 1 = x 2 = x 3 = x 4 = x ; y 1 = y 2 = y 3 = y 4 = y ,
I 2 ( x , y ) = cos 4 θ ( λ l ) 4 ( π w 2 ) 4 d ξ 1 d ξ 2 d ξ 3 d ξ 4 d η 1 d η 2 d η 3 d η 4 × exp [ ( ξ 1 2 + ξ 2 2 + ξ 3 2 + ξ 4 2 w 2 ) ( η 1 2 + η 2 2 + η 3 2 + η 4 2 w 2 ) ] × exp { σ ϕ 2 [ 2 C ϕ ( ξ 1 ξ 2 ; η 1 η 2 ) + C ϕ ( ξ 1 ξ 3 ; η 1 η 3 ) C ϕ ( ξ 1 ξ 4 ; η 1 η 4 ) C ϕ ( ξ 2 ξ 3 ; η 2 η 3 ) + C ϕ ( ξ 2 η 2 ; ξ 4 η 4 ) C ϕ ( ξ 3 ξ 4 ; η 3 η 4 ) ] } × exp { i 2 π [ u ( ξ 1 ξ 2 + ξ 3 ξ 4 ) + υ ( η 1 η 2 + η 3 η 4 ) ] } .
a 1 = 1 2 ( ξ 1 ξ 2 + ξ 3 ξ 4 ) , ξ 1 = 1 2 ( a 1 + a 2 + a 3 + a 4 ) , a 2 = 1 2 ( ξ 1 ξ 2 ξ 3 + ξ 4 ) , ξ 2 = 1 2 ( a 1 a 2 + a 3 + a 4 ) , a 3 = 1 2 ( ξ 1 + ξ 2 ξ 3 ξ 4 ) , ξ 3 = 1 2 ( a 1 a 2 a 3 + a 4 ) , a 4 = 1 2 ( ξ 1 + ξ 2 + ξ 3 + ξ 4 ) , ξ 4 = 1 2 ( a 1 + a 2 a 3 + a 4 ) ,
I 2 ( x , y ) = cos 4 θ ( λ l ) 4 ( π w 2 ) 3 d a 1 d a 2 d a 3 d γ 1 d γ 2 d γ 3 × exp [ i 2 π ( u a 1 + υ γ 1 ) ] × exp [ ( a 1 2 + a 2 2 + a 3 2 / w 2 ) ] exp [ ( γ 1 2 + γ 2 2 + γ 3 2 / w 2 ) ] × exp { σ ϕ 2 [ 2 C ϕ ( a 1 + a 2 ; γ 1 + γ 2 ) C ϕ ( a 2 a 1 ; γ 2 + γ 1 ) ] } + [ C ϕ ( a 2 + a 3 , γ 2 + γ 3 ) + C ϕ ( a 3 a 2 , γ 3 γ 2 ) C ϕ ( a 1 + a 3 , γ 1 + γ 3 ) C ϕ ( a 3 a 1 ; γ 3 γ 1 ) ] } .
I 2 ( x , y ) = cos 4 θ exp ( 2 σ ϕ 2 ) ( λ l ) 4 ( π w 2 ) 3 d a 1 d a 2 d a 3 d γ 1 d γ 2 d γ 3 × exp [ i 2 π ( u a 1 + υ γ 1 ) ] × exp [ ( a 1 2 + a 2 2 + a 3 2 / w 2 ) ] exp [ ( γ 1 2 + γ 2 2 + γ 3 2 / w 2 ) ] × j = 0 σ ϕ 2 j j ! [ C 12 + C 21 + C 23 + C 32 C 13 C 31 ] j .
C j k = { C ϕ ( a j + a k , γ j + γ k ) for j < k C ϕ ( a j a k , γ j γ k ) for j > k .
( C 12 + C 21 + C 23 + C 32 C 13 C 31 ) j = n = 0 j j ! ( 1 ) n 5 + n 6 n 1 ! n 2 ! n 3 ! n 4 ! n 5 ! n 6 ! C 12 n 1 C 21 n 2 C 23 n 3 C 32 n 4 C 13 n 5 C 31 n 6
n = n 1 + n 2 + n 3 + n 4 + n 5 + n 6 .
I 2 ( x , y ) = cos 4 θ exp ( 2 σ ϕ 2 ) ( λ l ) 4 × j = 0 n = 0 j ( 1 ) n 5 + n 6 σ θ 2 j n 1 ! n 2 ! n 3 ! n 4 ! n 5 ! n 6 ! exp ( k 2 q 2 sin 2 θ p 2 ) p 2 .
q 2 = [ 1 + w 2 ( n 1 + n 2 + n 5 + n 6 ) z 2 ] × [ 1 + w 2 ( n 3 + n 4 + n 5 + n 6 ) z 2 ] [ w 2 ( n 5 n 6 ) z 2 ] 2
p 2 = [ 1 + w 2 ( n 1 + n 2 + n 3 + n 4 ) z 2 ] × [ 1 + w 2 ( n 3 + n 4 + n 5 + n 6 ) z 2 ] [ 1 + w 2 ( n 1 + n 2 + n 5 + n 6 ) z 2 ] [ w 2 ( n 1 n 2 ) z 2 ] 2 [ 1 + w 2 ( n 3 + n 4 + n 5 + n 6 ) z 2 ] [ w 2 ( n 3 n 4 ) z 2 ] 2 [ 1 + w 2 ( n 1 + n 2 + n 5 + n 6 ) z 2 ] [ w 2 ( n 5 n 6 ) z 2 ] 2 [ 1 + w 2 ( n 1 + n 2 + n 3 + n 4 ) z 2 ] + 2 [ w 2 ( n 1 n 2 ) z 2 ] 2 [ w 2 ( n 3 n 4 ) z 2 ] 2 [ w 2 ( n 5 n 6 ) z 2 ] 2 .
Chϕ(t1,t2,,tN)=exp[ij1Ntjϕ(ξj,ηj)]=exp[12σϕ2j=1Nk=1NtjtkCϕ(ξj,ηj;ξk,ηk)],

Metrics