Statistical insights of polarization speckle via von Mises-Fisher distribution on the Poincaré sphere.

Polarization speckles generated via random scattering of light are ubiquitous in natural and engineered systems. They not only manifest intensity fluctuations but also reveal a spatially fluctuating, random polarization distribution. The precise morphology of the polarization speckle pattern serves as a deterministic signature of the light's state of polarization fluctuation within a scattering medium. Given the inherent randomness of polarization speckle patterns, a statistical approach emerges as the most pragmatic method for their analysis. Stokes parameters, implemented as temporal or spatial averages, are utilized for this purpose. However, within a polarization speckle field featuring a specific spatial average of Stokes parameters, the polarization state exhibits spatial variations across the speckle pattern. These random polarization fluctuations can be effectively modeled using a particular probability density function (PDF), visually represented on the Poincaré sphere. In this work, von Mises-Fisher (vMF) distribution on the Poincaré sphere is extended and applied to demonstrate a statistical insight of polarization speckle fields. A complete theoretical basis is established to investigate the spatial fluctuation of the state of polarization in the polarization speckle using vMF distribution on the Poincaré sphere, including the spatial mean direction, and spatial concentration parameter. Behavior of the marginal vMF distribution on the axes of the Poincaré sphere and its association with the probability density function of the normalized at-the-point Stokes parameters for three different polarization speckles are examined by experiment and simulation. The experimental results are in good agreement with the simulation results and confirm the usefulness of the established theoretical framework for the analysis of the polarization speckles. Characterization of spatial polarization fluctuation offers significant applications, such as in polarimetric analysis and optical sensing, and the same analogy can be used in quantum optics.