Statistical properties of speckle patterns for a random number of scatterers and nonuniform phase distributions

TL;DR

This paper derives analytical formulas for speckle pattern amplitude distributions considering a random number of scatterers and arbitrary phase distributions, extending classical models and revealing how scatterer number variability influences fluctuations.

Contribution

It provides the first closed-form analytic results for speckle amplitude distributions with nonuniform phases and random scatterer counts, generalizing classical speckle theory.

Findings

Amplitude distribution depends on scatterer count distribution.

Phase distribution affects scale parameters but not form.

Large amplitude fluctuations increase with biased phases in negative binomial models.

Abstract

The statistical properties of speckle patterns have important applications in optics, oceanography, and transport phenomena in disordered systems. Here we obtain closed-form analytic results for the amplitude distribution of speckle patterns formed by a random number of partial waves characterized by an arbitrary phase distribution, generalizing classical results of the random walk theory of speckle patterns. We show that the functional form of the amplitude distribution is solely determined by the distribution of the number of scatterers, while the phase distribution only influences the scale parameters. In the case of a non-random number of scatterers, we find an analytic expression for the amplitude distribution that extends the Rayleigh law to non-uniform random phases. For a negative binomial distribution of the number of scatterers, our results reveal that large fluctuations of the wave amplitudes become more pronounced in the case of biased random phases. We present numerical results that fully support our analytic findings.