Construction of Stokes probability clouds from the polarization matrices for thermal light and polarization speckle

Utilizing the eigendecomposition of the Hermitian polarization matrix, we present a geometric framework for the study of polarization speckles and partially polarized thermal light. The proposed methodology is founded on the concept of the Stokes probability cloud that provides a geometric representation of the joint probability distribution of instantaneous (or spatially localized) random Stokes vectors. In addition to the eigenvalues that specify the shape and size of the Stokes probability cloud, we will find the associated eigenvectors that define the orientation of the Stokes probability cloud. The relationship between the joint probability density and the marginal probability densities of the Stokes parameters is illustrated by projecting the Stokes probability cloud onto the axes in the three-dimensional Hilbert space. This geometrical approach provides intuitive insights into the involved mathematical relations of stochastic polarization phenomena in optics.