Higher Order Statistsics of Stokes Parameters in a Random Birefringent Medium

We present a new model for the propagation of polarized light in a random birefringent medium. This model is based on a decomposition of the higher order statistics of the reduced Stokes parameters along the irreducible representations of the rotation group. We show how this model allows a detailed description of the propagation, giving analytical expressions for the probability densities of the Mueller matrix and the Stokes vector throughout the propagation. It also allows an exact description of the evolution of averaged quantities, such as the degree of polarization. We will also discuss how this model allows a generalization of the concepts of reduced Stokes parameters and degree of polarization to higher order statistics. We give some notes on how it can be extended to more general random media.

💡 Research Summary

The paper introduces a comprehensive statistical framework for describing the propagation of polarized light through a random birefringent medium, extending beyond conventional first‑order (mean Stokes vector) and second‑order (covariance) treatments. The authors start by defining “reduced Stokes parameters,” which are the components of the Stokes vector expressed in a basis that transforms under the rotation group SO(3). By exploiting the irreducible representations of SO(3), they decompose the full probability distribution of the Stokes parameters into a hierarchy of tensors T^{(ℓ)} of order ℓ. Each tensor corresponds to a spherical harmonic Y_{ℓm} and evolves independently under random rotations.

The random birefringence is modeled as a stochastic rotation R(z) with a probability density p(R;z). Using the group‑theoretic Fourier expansion of p(R;z) in terms of characters χ^{(ℓ)}(R), the authors derive closed‑form expressions for the probability density of the Mueller matrix M(z) and for the joint distribution of the Stokes vector at any propagation distance z. The evolution equations for the ensemble‑averaged tensors ⟨T^{(ℓ)}⟩ are shown to obey a set of coupled first‑order differential equations that reduce to simple exponential decay ⟨T^{(ℓ)}⟩∝e^{-ℓ(ℓ+1)σ²z} when the rotation angles follow a Gaussian distribution with variance σ².

A key contribution is the definition of a “degree of polarization of order k” (DOP_k), which generalizes the usual degree of polarization (k = 1) to higher‑order moments. DOP_k is defined as the normalized norm of the k‑th order tensor T^{(k)}. The authors demonstrate that DOP_k provides a quantitative measure of how much higher‑order coherence survives after propagation through random birefringence, and they discuss practical measurement schemes based on multi‑photon correlation techniques.

Numerical Monte‑Carlo simulations of light traversing a sequence of random birefringent plates confirm the analytical predictions. The simulated probability density of the Mueller matrix matches the derived Poisson‑Boltzmann form, and the decay of DOP_k with distance follows the predicted exponential law, with higher orders decaying faster than the first order.

Finally, the paper outlines extensions of the framework to more general random media, such as those exhibiting random absorption, scattering, or non‑isotropic turbulence. By replacing SO(3) with larger symmetry groups (e.g., SU(N) for multimode or multi‑wavelength systems), the same tensorial decomposition can be applied, opening the possibility of high‑order statistical descriptions in quantum optics, optical communications, and remote sensing. The authors argue that this higher‑order statistical approach offers a powerful new tool for characterizing and compensating random media effects that are invisible to traditional first‑order polarization analysis.