First-order intensity and phase statistics of Gaussian speckle produced in the diffraction region.

General expressions for first-order statistics of Gaussian speckle produced in the diffraction region are derived for an arbitrary profile of the illuminating beam with a plane wave front. The statistical properties of the complex amplitude, the intensity, and the phase of speckles are studied under illumination of a Gaussian beam. The joint probability density function of the speckle field characterized by an equiprobability density ellipse is investigated in some detail with an intimate relation to the correlation coefficient between the real and imaginary parts of the complex speckle amplitude. This correlation coefficient is found to affect appreciably the statistics of speckles produced especially in the near-field diffraction region when a standard deviation of the random phase variation produced by a diffuse object under illumination is relatively small and the number of independent scatterers contributing to the formation of speckles is small.