Statistical Distribution of the Normalized Difference Polarization Index for 3-D Layered Structures With Slightly Rough Interfaces

We determine the probability distribution of the normalized difference polarization index (NDPI) for a stack of slightly rough interfaces illuminated by a plane wave. For infinite surface areas and Gaussian-centered height distributions, we show within the framework of the first-order small perturbation method that the co- and cross-polarized NDPIs are random variables, the probability laws of which depend only on two parameters. Contrary to the intensity ratio which follows a heavy-tailed distribution, the NDPI has finite mean and variance. For a sand layer covering a granite surface and Gaussian correlation functions, we analyze the combined influence of the anisotropy and cross correlation upon the probability laws.