A Directional-Entropy Framework for Polarization Disorder: Information-Theoretic Insights from von Mises–Fisher Statistics

The degree of polarization is usually obtained from the coherency matrix or, equivalently, from the mean Stokes vector of a partially polarized optical field. Here, we adopt a complementary geometric viewpoint by representing normalized local Stokes vectors as random directions on the Poincaré sphere. When these directions are described by an effective unimodal von Mises–Fisher distribution, the concentration parameter gives a direct one-to-one description of the degree of polarization through the mean resultant length. This formulation does not define a new independent polarization observable. Instead, it gives the degree of polarization a rotation-invariant information-theoretic meaning, expressed in terms of directional concentration and angular disorder. Within this framework, we derive closed-form expressions for the differential entropy of the von Mises–Fisher distribution and for the Kullback–Leibler divergence between two directional polarization states. The symmetrized divergence further incorporates both differences in concentration and relative orientation on the Poincaré sphere. We also discuss the assumptions, range of validity, and limitations of the single-vMF model, particularly in relation to Gaussian-field statistics and to more general directional models needed for anisotropic or multimodal polarization fluctuations. Overall, this formalism establishes a model-based theoretical framework for entropy and divergence descriptors of unimodal directional polarization and suggests natural extensions toward mixtures of vMF, Bingham, or Kent distributions.