Polarimetric purity and the concept of degree of polarization

The concept of degree of polarization for electromagnetic waves, in its general three-dimensional version, is revisited in the light of the implications of the recent findings on the structure of polarimetric purity and of the existence of nonregular states of polarization [J. J. Gil et al., Phys Rev. A 95, 053856 (2017)]. From the analysis of the characteristic decomposition of a polarization matrix R into an incoherent convex combination of (1) a pure state ${\mathbf{R}}_p$, (2) a middle state ${\mathbf{R}}_m$ given by an equiprobable mixture of two eigenstates of R, and (3) a fully unpolarized state ${\mathbf{R}}_{u-3\mathrm{D}}$, it is found that, in general, ${\mathbf{R}}_m$ exhibits nonzero circular and linear degrees of polarization. Therefore, the degrees of linear and circular polarization of R cannot always be assigned to the single totally polarized component ${\mathbf{R}}_p$. It is shown that the parameter $P_{3\mathrm{D}}$ proposed formerly by Samson [J. C. Samson, Geophys. J. R. Astron. Soc. 34, 403 (1973)] takes into account, in a proper and objective form, all the contributions to polarimetric purity, namely, the contributions to the linear and circular degrees of polarization of R as well as to the stability of the plane containing its polarization ellipse. Consequently, $P_{3\mathrm{D}}$ constitutes a natural representative of the degree of polarimetric purity. Some implications for the common convention for the concept of two-dimensional degree of polarization are also analyzed and discussed.

Figure 1

A particular example of a nonregular state (with intensity $I_m$) is given by an equiprobable incoherent mixture ${\mathbf{R}}_{cl}$ of a circularly polarized state ${\mathbf{R}}_c$ and a linearly polarized state ${\mathbf{R}}_l$ whose electric field vibrates in the direction orthogonal to the plane containing the electric field of ${\mathbf{R}}_c$.