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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 28, Iss. 9 — Sep. 1, 2011
  • pp: 1896–1903

Three-dimensional static speckle fields. Part I. Theory and numerical investigation

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

JOSA A, Vol. 28, Issue 9, pp. 1896-1903 (2011)

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When monochromatic light is scattered from an optically rough surface a complicated three-dimensional (3D) field is generated. These fields are often described by reference to the 3D volume (extent) of their speckles, leading to the definition of lateral ( x , y ) and longitudinal speckle sizes (z). For reasons of mathematical simplicity the longitudinal speckle size is often derived by examining the decorrelation of the speckle field for a single point lying on axis, i.e., x = y = 0 , and this size is generally assumed to be representative for other speckles that lie further off-axis. Some recent theoretical results, however, indicate that in fact longitudinal speckle size gets smaller as the observation position moves to off-axis spatial locations. In this paper (Part I), we review the physical argument leading to this conclusion and support this analysis with a series of robust numerical simulations. We discuss, in some detail, computational issues that arise when simulating the propagation of speckle fields numerically, showing that the spectral method is not a suitable propagation algorithm when the autocorrelation of the scattering surface is assumed to be delta correlated. In Part II [ J. Opt. Soc. Am. A 28, 1904 (2011)] of this paper, experimental results are provided that exhibit the predicted variation of longitudinal speckle size as a function of position in x and y. The results are not only of theoretical interest but have practical implications, and in Part II a method for locating the optical system axis is proposed and experimentally demonstrated.

© 2011 Optical Society of America

OCIS Codes
(030.6140) Coherence and statistical optics : Speckle
(030.6600) Coherence and statistical optics : Statistical optics
(050.1940) Diffraction and gratings : Diffraction
(200.2610) Optics in computing : Free-space digital optics
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Coherence and Statistical Optics

Original Manuscript: June 15, 2011
Manuscript Accepted: July 12, 2011
Published: August 25, 2011

Dayan Li, Damien P. Kelly, and John T. Sheridan, "Three-dimensional static speckle fields. Part I. Theory and numerical investigation," J. Opt. Soc. Am. A 28, 1896-1903 (2011)

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  1. J. Ohtsubo, “The second-order statistics of speckle patterns,” J. Opt. 12, 129–142 (1981). [CrossRef]
  2. T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. 3, 1032–1054 (1986). [CrossRef]
  3. L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. 7, 827–832 (1990). [CrossRef]
  4. Q. B. Li and F. P. Chiang, “Three-dimensional dimension of laser speckle,” Appl. Opt. 31, 6287–6291 (1992). [CrossRef] [PubMed]
  5. T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. 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. 16, 1402–1412 (1999). [CrossRef]
  7. D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology system with an aperture,” J. Opt. Soc. Am. 23, 2861–2870 (2006). [CrossRef]
  8. D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O’Neill, and J. T. Sheridan, “Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture,” Opt. Lett. 31, 3444–3446 (2006). [CrossRef] [PubMed]
  9. D. Duncan and S. Kirkpatrick, “Performance analysis of a maximum-likelihood speckle motion estimator,” Opt. Express 10, 927–941 (2002). [PubMed]
  10. R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O’Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32–34 (2006). [CrossRef] [PubMed]
  11. D. P. Kelly, J. E. Ward, U. Gopinathan, and J. T. Sheridan, “Controlling speckle using lenses and free space,” Opt. Lett. 32, 3394–3396 (2007). [CrossRef] [PubMed]
  12. S. G. Hanson, W. Wang, M. L. Jakobsen, and M. Takeda, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes through complex ABCD optical systems,” J. Opt. Soc. Am. 25, 2338–2346(2008). [CrossRef]
  13. J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. 26, 1855–1864 (2009). [CrossRef]
  14. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express 18, 13782–13787(2010). [CrossRef] [PubMed]
  15. N. Chang, N. George, and W. Chi, “Wavelength decorrelation of speckle in propagation through a thick diffuser,” J. Opt. Soc. Am. 28, 245–254 (2011). [CrossRef]
  16. G. P. Weigelt and B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).
  17. C. E. Halford, W. L. Gamble, and N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).
  18. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts, 2007).
  19. I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981). [CrossRef]
  20. D. W. Li and F. P. Chiang, “Decorrelation functions in laser speckle photography,” J. Opt. Soc. Am. 3, 1023–1031 (1986). [CrossRef]
  21. 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]
  22. G. W. Goodman, “Role of coherence concepts in the study of speckle,” Proc. SPIE 194, 86–94 (1979).
  23. D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express 19, 1408–1421(2011). [CrossRef] [PubMed]
  24. X. Zhao and Z. Gao, “Surface roughness measurement using spatial-average analysis of objective speckle pattern in specular direction,” Opt. Lasers Eng. 47, 1307–1316 (2009). [CrossRef]
  25. A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. 78, 063806 (2008). [CrossRef]
  26. D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. 79, 053831 (2009). [CrossRef]
  27. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999). [CrossRef]
  28. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques, 1st ed. (Springer, 2004).
  29. 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]
  30. H. M. Pedersen, “Intensity correlation metrology: a comparative study,” Opt. Acta 29, 105–118 (1982). [CrossRef]
  31. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IEEE Trans. Inf. Theory 8, 194–195 (1962). [CrossRef]
  32. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).
  33. H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
  34. MathWorks, http://www.mathworks.com/help/toolbox/images/ref/normxcorr2.html.
  35. MathWorks, http://www.mathworks.com/help/toolbox/images/ref/imresize.html.
  36. D. P. Kelly, N. Sabitov, T. Meinecke, and S. Sinzinger, “Some considerations when numerically calculating diffraction patterns,” in Digital Holography and Three-Dimensional Imaging, Technical Digest (CD) (Optical Society of America, 2011), paper DTuC5.

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