The first- and second-order space-time correlation functions of the optical field and the intensity, respectively, are calculated for the case of a rotating symmetrical curved surface with Gaussian roughness, scattering laser light into the far-field zone from a small illuminated spot. The incident beam is either collimated or spherical. The diameter of the illuminated area is small compared with the radii of curvature. The mean surface is approximated by a torus with two principal radii of curvature. The sphere, the cone, and the flat surface are special cases. Optical phase variations linear and quadratic in the surface coordinates are included in the calculation. The surface is assumed to be very rough. It is found that the correlation function of the intensity depends on the radius of the incident beam, the local surface velocity, and the radii of curvature in the illuminated region. Curvature decreases the decay time of the correlation functions. This is confirmed by experiments carried out with the flat disk, the sphere, and the cone. The temporal cross-correlation function of the outputs of two detectors in the speckle field obtained with an incident spherical wave is very sensitive to normal displacements of the surface.
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