Structure of polarimetric purity of three-dimensional polarization states

It has recently been demonstrated that a general three-dimensional (3D) polarization state cannot be considered an incoherent superposition of (1) a pure state, (2) a two-dimensional unpolarized state, and (3) a 3D unpolarized state [J. J. Gil, Phys. Rev. A 90, 043858 (2014)]. This fact is intimately linked to the existence of 3D polarization states with fluctuating directions of propagation, but whose associated polarization matrices R satisfy rank $\mathbf{R}=2$. In this work, such peculiar states are analyzed and characterized, leading to a meaningful general classification and interpretation of 3D polarization states. Within this theoretical framework, the interrelations among the more significant polarization descriptors presented in the literature, as well as their respective physical interpretations, are studied and illustrated with examples, providing a better understanding of the structure of polarimetric purity of any kind of polarization state.