Intrinsic Stokes parameters for 3D and 2D polarization states

Abstract

The second-order characterization of a three-dimensional (3D) state of polarization is provided either by the corresponding 3D coherency matrix or (equivalently) by the associated 3D Stokes parameters. The analysis of the polarization properties that are invariant under orthogonal transformations of the laboratory reference frame allows to define a set of six intrinsic Stokes parameters which provides a simplified interpretation of 3D states of polarization in terms of meaningful physical properties. The rotationally invariant properties of 2D states of polarization are straightforwardly retrieved in a consistent way, so that the 2D intrinsic Stokes parameters are constituted by the intensity, the degree of linear polarization and the degree of circular polarization.

Structure of polarimetric purity of three-dimensional polarization states

“…From the sole point of view of the structure of polarimetric purity of a given 3D state R, such structure is fully characterized by the corresponding indices of polarimetric purity, P 1 and P 2 , which are insensitive to specific polarization features of R, which in turn are determined by the components of purity P d , P l , and P c . The degree of polarimetric purity P 3D constitutes an overall measure of the closeness of R to a polarimetrically pure state and can be calculated either from the IPP or from the CP through the following expressions [19]:…”

“…As pointed out in previous works [18,19], the only unitary transformations U † RU that can be physically realized as rotations of the laboratory reference frame XYZ are those where there exists an orthogonal matrix Q (i.e., Q satisfying Q T = Q −1 and det Q = +1) that satisfies U † RU = Q T RQ. Therefore, given a polarization matrix R satisfying rank R = 2 there is not always an orthogonal transformation Q T RQ such that one of its diagonal elements is zero.…”

“…and give complete information about the polarimetric purity (randomness) of R. The expression of the characteristic decomposition in terms of the eigenvalues of 3 × 3 polarization matrices was examined for the first time by Samson [4] through considering their expansion in a set of nondisjoint idempotent matrices, and later it was analyzed under the scope of radar polarimetry by Holm and Barnes [13] and by Cloude and Pottier [14], while Ellis, Dogariu, Brosseau, and co-workers [3,15,16] dealt with it for electromagnetic fields. It is worth recalling now that a simplified and significant view of the features of a 3D polarization state is obtained through its corresponding intrinsic polarization matrix R O , defined as [17][18][19]…”

Dimensionality of random light fields

“…where the eigenvalues a 1 ≥ a 2 ≥ a 3 ≥ 0 of (r, ω) are the principal intensities and the vector n = (n 1 , n 2 , n 3 ) is the angular-momentum vector [16][17][18]. In addition, a 1 is the largest while a 3 is the smallest diagonal element of (r, ω) that can be obtained by an orthogonal transformation [19].…”

Polarimetric purity and the concept of degree of polarization

“…Given a polarization matrix R it always can be represented by its corresponding intrinsic polarization matrix R O , defined as [27][28][29] …”