OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 30, Iss. 5 — May. 1, 2013
  • pp: 969–978

K speckle: space–time correlation function of doubly scattered light in an imaging system

Dayan Li, Damien P. Kelly, and John T. Sheridan  »View Author Affiliations

JOSA A, Vol. 30, Issue 5, pp. 969-978 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (793 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The scattering of coherent monochromatic light at an optically rough surface, such as a diffuser, produces a speckle field, which is usually described by reference to its statistical properties. For example, the real and imaginary parts of a fully developed speckle field can be modeled as a random circular Gaussian process. When such a speckle field is used to illuminate a second diffuser, the statistics of the resulting doubly scattered field are in general no longer Gaussian, but rather follow a K distribution. In this paper we determine the space–time correlation function of such a doubly scattered speckle field that has been imaged by a single lens system. A space–time correlation function is derived that contains four separate terms; similar to the Gaussian case it contains an average DC term and a fluctuating AC term. However, in addition there are two terms that are related to contributions from each of the diffusers independently. We examine how our space–time correlation function varies as the diffusers are rotated at different speeds and as the point spread function of the imaging system is changed. A series of numerical simulations are used to confirm different aspects of the theoretical analysis. We then finish with a discussion of our results and some potential applications, including controlling spatial coherence and speckle reduction.

© 2013 Optical Society of America

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(030.6140) Coherence and statistical optics : Speckle
(030.6600) Coherence and statistical optics : Statistical optics
(110.6150) Imaging systems : Speckle imaging

ToC Category:
Imaging Systems

Original Manuscript: January 30, 2013
Revised Manuscript: March 28, 2013
Manuscript Accepted: March 30, 2013
Published: April 23, 2013

Dayan Li, Damien P. Kelly, and John T. Sheridan, "K speckle: space–time correlation function of doubly scattered light in an imaging system," J. Opt. Soc. Am. A 30, 969-978 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts, 2006), pp. 47–53.
  2. S. G. Hanson, T. F. Q. Iversen, and R. S. Hansen, “Dynamic properties of speckled speckles,” Proc. SPIE 7387, 738716 (2010). [CrossRef]
  3. J. H. Churnside, “Speckle from a rotating diffuse object,” J. Opt. Soc. Am. 72, 1464–1469 (1982). [CrossRef]
  4. L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A 7, 827–832 (1990). [CrossRef]
  5. T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. A 10, 324–328 (1993). [CrossRef]
  6. H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. A 16, 1402–1412 (1999). [CrossRef]
  7. D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields: part I. Theory and a numerical investigation,” J. Opt. Soc. Am. A 28, 1896–1903 (2011). [CrossRef]
  8. D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields: part II. Experimental investigation,” J. Opt. Soc. Am. A 28, 1904–1908 (2011). [CrossRef]
  9. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962). [CrossRef]
  10. K. A. O’Donnell, “Speckle statistics of doubly scattered light,” J. Opt. Soc. Am. 72, 1459–1463 (1982). [CrossRef]
  11. D. Newman, “K distributions from doubly scattered light,” J. Opt. Soc. Am. A 2, 22–26 (1985). [CrossRef]
  12. T. Yoshimura and K. Fujiwara, “Statistical properties of doubly scattered image speckle,” J. Opt. Soc. Am. A 9, 91–95 (1992). [CrossRef]
  13. R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986). [CrossRef]
  14. L. G. Shirley, “Laser speckle from thin and cascaded diffusers,” Ph.D. dissertation (University of Rochester, 1988).
  15. L. G. Shirley and N. George, “Speckle from a cascade of two thin diffusers,” J. Opt. Soc. Am. A 6, 765–781 (1989). [CrossRef]
  16. T. Okamoto and T. Asakura, “Detection of the object velocity using doubly-scattered dynamic speckles under Gaussian beam illumination,” J. Mod. Opt. 38, 1821–1839 (1991). [CrossRef]
  17. A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008). [CrossRef]
  18. D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009). [CrossRef]
  19. Personal communication, Prof. S. G. Hanson (during a research visit at the Technische Universität Ilmenau, Germany, November2012).
  20. W. Wang, S. Hanson, and M. Takeda, “Complex amplitude correlations of dynamic laser speckle in complex ABCD optical systems,” J. Opt. Soc. Am. A 23, 2198–2207 (2006). [CrossRef]
  21. V. S. R. Gudimetla, “Moments of the intensity of a non-circular Gaussian laser speckle in the diffraction field,” Opt. Commun. 130, 348–356 (1996). [CrossRef]
  22. D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006). [CrossRef]
  23. Q. B. Li and F. P. Chiang, “Three-dimensional dimension of laser speckle,” Appl. Opt. 31, 6287–6291 (1992). [CrossRef]
  24. D. P. Kelly and D. Claus, “Filtering role of the sensor pixel in Fourier and Fresnel digital holography,” Appl. Opt. 52, A336–A345 (2013). [CrossRef]
  25. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1966).
  26. D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Fundamental diffraction limits in a paraxial 4-f imaging system with coherent and incoherent illumination,” J. Opt. Soc. Am. A 24, 1911–1919 (2007). [CrossRef]
  27. D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009). [CrossRef]
  28. D. A. Tichenor and J. W. Goodman, “Coherent transfer function,” J. Opt. Soc. Am. 62, 293–295 (1972). [CrossRef]
  29. J. Ohtsubo, “Non-Gaussian speckle: a computer simulation,” Appl. Opt. 21, 4167–4175 (1982). [CrossRef]
  30. T. Fricke-Begemann, “Optical measurement of deformation fields and surface processes with digital speckle correlation,” Ph.D. dissertation (Carl von Ossietzky Universitat Oldenburg, 2002).
  31. J. E. Ward, D. P. Kelly, and J. T. Sheridan, “An alignment technique based on the speckle correlation properties of Fresnel transforming optical systems,” Proc. SPIE 7068, 70680L (2008). [CrossRef]
  32. The Mathworks Ins., http://www.mathworks.co.uk/help/images/ref/imrotate.html (Data retrieved: November2012).
  33. S. Lowenthal and D. Joyeux, “Speckle removal by a slowly moving diffuser associated with a motionless diffuser,” J. Opt. Soc. Am. 61, 847–851 (1971). [CrossRef]
  34. Y. Kuratomi, K. Sekiya, H. Satoh, T. Tomiyama, T. Kawakami, B. Katagiri, Y. Suzuki, and T. Uchida, “Speckle reduction mechanism in laser rear projection displays using a small moving diffuser,” J. Opt. Soc. Am. A 27, 1812–1817 (2010). [CrossRef]
  35. X. Xiao and D. Voelz, “Wave optics simulation approach of partially coherent beams,” Opt. Express 14, 6986–6992(2006). [CrossRef]
  36. D. L. Fried, “Laser eye safety: the implications of ordinary speckle statistics and of speckled-speckle statistics,” J. Opt. Soc. Am. 71, 914–916 (1981). [CrossRef]

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.

CrossCheck Deposited