Direct derivation of intensity and phase statistics of speckle produced by a weak scatterer from the random sinusoid model

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Abstract
The complex amplitude at a point in a speckle pattern that is due to a weak scatterer is modeled as the superposition of N sinusoidal waves of random phase, with the probability density of these phases given by the nonuniform von Mises rather than by the uniform one that characterizes a strong scatterer. Explicit formulas are obtained for both intensity and total phase statistics in terms of a single parameter directly related to the density function of the constituent phasors. The case in which, in addition, N itself is random (governed by a Poisson distribution with mean value 〈N〉) is also studied.