A complete characterization of pre-Mueller and Mueller matrices in polarization optics

Abstract

The Mueller-Stokes formalism that governs conventional polarization optics is formulated for plane waves, and thus the only qualification one could require of a 4 x 4 real matrix M in order that it qualify to be the Mueller matrix of some physical system would be that M map Omega((pol)), the positive solid light cone of Stokes vectors, into itself. In view of growing current interest in the characterization of partially coherent partially polarized electromagnetic beams, there is a need to extend this formalism to such beams wherein the polarization and spatial dependence are generically inseparably intertwined. This inseparability brings in additional constraints that a pre-Mueller matrix M mapping Omega((pol)) into itself needs to meet in order to be an acceptable physical Mueller matrix. These additional constraints are motivated and fully characterized.

Physical model of differential Mueller matrix for depolarizing uniform media

“…Being d [7][8][9][10][11][12][13][14][15] coefficients not independent is obvious since they are related to correlation between the different fluctuations. So put all these coefficients (with the exception of d [13][14][15] ) to zero except one, will result in an invalid Mueller matrix.…”

“…m m (11) Regardless to Eq. (11) of [13] an overall isotropic absorption factor -0.5(k 1 +k 4 ) is introduced in m (2) in order to be consistent with the present notation where left upper corner entry of the matrix can be non zero.…”

“…[4], Ossikovski provides a physical interpretation of this symmetry breaking in the matrix m. He proposes to consider the d 0-6 parameters as mean values of the non-depolarizing properties (m ND part) and d [7][8][9][10][11][12][13][14][15] as their respective uncertainties resulting from the depolarization (m D part). The elementary optical non-depolarizing properties are [1] the amplitude or isotropic absorption (d 0-), the linear dichroism along the x-y laboratory axes (d 1 ), the linear dichroism along the 45° axes (d 2 ), the circular dichroism (d 3 ), the linear birefringence along the x-y axes (d 4 ) , the linear birefringence along the 45° axes (d 5 ) and the circular birefringence (d 6 ).…”

“…So there must be a relationship between the off-diagonal coefficients d [7][8][9][10][11][12] physically representing the uncertainties of the respective elementary polarization properties d [1][2][3][4][5][6] and the diagonal anisotropic depolarizations d [13][14][15] . In order to solve this problem Germer [10] proposes a very interesting approach to ensure that the parameterization will only lead to physical Mueller matrices.…”

“…Nevertheless, this approach suffers from two main drawbacks: a) Hypothesis is only valid for the definition of Mueller matrix as a matrix which transforms the space of valid Stokes vectors into a subspace of valid Stokes vectors but is not valid for the correct definition of Mueller matrix as a convex sum of MuellerJones [11].…”