S. Wolfram, Mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).

The generalization of the results derived here to systems that are not cylindrically symmetric is straightforward.

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

A. S. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.

Note that the complex cross covariance is proportional to exp[ikJ-ik(J-vZτ)]=exp[iωDτ], where J is the path length along the optic axis and ωD=kvZ is the Doppler shift caused by the longitudinal motion of the object.

See, for example, J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIV, Chap. 1; S. M. Kozel, G. R. Lokshin, “Longitudinal correlation properties of coherent radiation scattered from a rough surface,” Opt. Spectrosc. 33, 89–90 (1972); T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981); T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. A 3, 1032–1054 (1986); B. Rose, H. Imam, S. G. Hanson, H. T. Yura, R. S. Hansen, “Laser-speckle angular-displacement sensor: theoretical and experimental study,” Appl. Opt. 37, 2119–2129 (1998); T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1995), Vol. XXXIV, Chap. 3; H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. Am. A 10, 316–323 (1993).

[CrossRef]

Because each scatterer appears in the same position in the illuminated region once with each revolution of the object, the speckle pattern exhibits a periodicity with a frequency equal to the frequency of revolution.

Substituting Eqs. (4.6a) and (4.6b) into Eq. (4.1) yields that the maximum value of the space–time cross covariance equals unity.

This because the orientation of the (static) speckles is independent of the motion of the object.