Statistics of polarization and Stokes parameters: Multiple orthonormal wave populations

A stochastic analysis is developed for the superposition of multiple, fully polarized, electric-field vectors. Each vector is described by a polarization ellipse, with fixed axial ratio and polarization angle, and probability distribution functions (pdfs) for the field strength and phase. These wave populations are then superposed in orthonormal modes of polarization, representing the normal modes of a medium. Central results of this work include analytic and Monte Carlo methods to calculate the pdfs of the measurable Stokes parameters I, U, Q, and V, and degrees of polarization, of the superposed waves. Predictions are computed for the superposition of some characteristic wave populations, and several striking and counterintuitive results produced. These include nonzero probabilities for U, Q, and V at U=0, Q=0, and V=0, irrespective of the constituent wave polarizations and field distributions. For wave populations with identical polarization ellipses, a power-law enhancement of the pdf of the intensity I at low I is found, which is independent of the constituent electric-field distributions. Generation of elliptically polarized light from components which each have an opposite sense of polarization is shown to be possible. A description of the asymptotic limits of the pdfs of the Stokes parameters is obtained, and the appearance of fine structure in the pdfs of the degrees of polarization is demonstrated. Together, these results demonstrate the necessity of systematic analysis when predicting pdfs for the Stokes parameters and degrees of polarization: qualitative results cannot be correctly inferred from intuition alone.