## Non-Gaussian speckle caused by thin phase screens of large root-mean-square phase variations and long single-scale autocorrelations

JOSA A, Vol. 3, Issue 8, pp. 1283-1292 (1986)

http://dx.doi.org/10.1364/JOSAA.3.001283

Enhanced HTML Acrobat PDF (1013 KB)

### 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 2*π* 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
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

**History**

Original Manuscript: January 6, 1986

Manuscript Accepted: April 9, 1986

Published: August 1, 1986

**Citation**

Bruce Martin Levine, "Non-Gaussian speckle caused by thin phase screens of large root-mean-square phase variations and long single-scale autocorrelations," J. Opt. Soc. Am. A **3**, 1283-1292 (1986)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-3-8-1283

Sort: Year | Journal | Reset

### References

- 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).
- 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. Marron, G. M. Morris, “Image-plane speckle form rotating rough objects,” J. Opt. Soc. Am. A 2, 1395–1402 (1985). [CrossRef]
- E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976). [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]
- C. J. Oliver, “A model for non-Rayleigh scattering statistics,” Opt. Acta 31, 701–722 (1984). [CrossRef]
- 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]
- 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]
- J. C. Kluyver, “A local probability problem,” Proc. R. Acad. Sci. (Amsterdam) 8, 341–350 (1905).
- 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]
- R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. 58, 382–400 (1962). [CrossRef]
- P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1962).
- E. N. Bramley, “Diffraction of an angular spectrum of waves by a phase-changing screen,” J. Atmos. Terr. Phys. 29, 1–28 (1967). [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]
- 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. 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.
- I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962). [CrossRef]
- 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, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978). [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]
- 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]
- 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, W. J. Welford, “Speckle statistics in imaging systems,” Opt. Comm. 21, 72–79 (1977). [CrossRef]
- J. Ohtsubo, “Non-Gaussian speckle: a computer simulation,” Appl. Opt. 21, 4167–4175 (1982). [CrossRef] [PubMed]
- 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 |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.