Structure of polarimetric purity of three-dimensional polarization states

In general, the middle component R_m of the characteristic decomposition of a 3D polarization matrix exhibits nonzero circular and linear degrees of polarization.

“it is found that, in general, R m exhibits nonzero circular and linear degrees of polarization.”

Polarization states can be classified into regular and nonregular states depending on whether R_m is a real matrix or not.

“In fact, polarization states can be classified into two types depending on whether ˆ R m is a real matrix ( regular states ) or not ( nonregular states ) [10].”

For nonregular states, R_m corresponds to an equiprobable mixture of two mutually orthogonal states whose polarization ellipses lie in different planes, and therefore R_m does not represent a 2D unpolarized state and exhibits nonzero contribution to Pc and Pl.

“In the case of nonregular states, ˆ R m corresponds to an equiprobable mixture of two mutually orthogonal states whose polarization ellipses lie in different planes, and therefore ˆ R m does not represent a 2D unpolarized state, and exhibits nonzero contribution to Pc and Pl [10].”

For regular states, R_m always represents a 2D unpolarized state, i.e., an equiprobable mixture of two mutually orthogonal states whose polarization ellipses lie in the same plane, and therefore R_m does not contribute to either Pl or Pc.

“In the case of regular states ˆ R m always represents a 2D unpolarized state. In other words, it can be considered an equiprobable mixture of two mutually orthogonal states whose polarization ellipses lie in the same plane, and therefore the polarization ellipse of ˆ R m evolves fully randomly but remains in a fixed plane. Thus, for regular states, ˆ R m does not contribute to either Pl or Pc .”

The degree of elliptical purity Pe summarizes all contributions to polarimetric purity derived from the degrees of linear and circular purity, i.e., all contributions except the purity due to the degree of directionality Pd.

“where the degree of elliptical purity Pe ≡ √ P 2 l + P 2 c [10] summarizes all the contributions to polarimetric purity derived from the degrees of linear and circular purity (that is, all the contributions except for the purity due to the degree of directionality Pd ).”

A privileged view of the structure of polarimetric purity of R is given by its characteristic (or trivial) decomposition.

“among the different possible representations of R as compositions of mutually incoherent states, a privileged view of the structure of polarimetric purity of R is given by its characteristic (or trivial ) decomposition [2,10,18],”

P1 does not satisfy all the required properties to be associated to a 3D degree of polarimetric purity because in general P1 ≠ Pe (nonregular states) and the equality P1=0 does not imply that R=R_u-3D.

“Fromtheaboveanalysisweconcludethat P 1 doesnotsatisfy all the required properties to be associated to a 3D degree of polarimetric purity, because (a) in general P 1 ̸= Pe (nonregular states) and therefore the contributions of ˆ R m to linear and circular degrees of polarization are not included in P 1 [10], and (b) the equality P 1 = 0 does not imply that necessarily R = R u -3D.”

An example of a nonregular state is any state R whose component R_m is given by an equiprobable mixture R_cl of a circularly polarized state and a linearly polarized state whose electric field vibrates in a direction orthogonal to the plane containing the polarization ellipse of the circular component.

“An example of a nonregular state (see Fig. 1) is any state R whose component ˆ R m is given by an equiprobable mixture ˆ R cl of a circularly polarized state and a linearly polarized state whose electric field vibrates in a direction that is orthogonal to the plane containing the polarization ellipse (circle) of the circular component [10].”