Classification of degenerate 4-dimensional matrices with semi-group structure and polarization optics

TL;DR

This paper classifies degenerate 4x4 Mueller matrices in polarization optics, focusing on rank 1, 2, and 3 matrices, using a parameterization technique and group-theoretic restrictions.

Contribution

It develops a systematic classification of degenerate Mueller matrices of various ranks using a novel parameterization and algebraic restrictions.

Findings

Classified rank 1, 2, and 3 degenerate matrices.

Identified sub-groups and semigroups among these matrices.

Specified 16 cases for rank 3 matrices with zero determinant.

Abstract

In polarization optics, an important role play Mueller matrices -- real four-dimensional matrices which describe the effect of action of optical elements on the polarization state of the light, described by 4-dimensional Stokes vectors. An important issue is to classify possible classes of the Mueller matrices. In particular, of special interest are degenerate Mueller matrices with vanishing determinants. Earlier, it was developed a special technique of parameterizing arbitrary 4-dimensional matrices with the use of four 4-dimensional vector (k, m, l, n). In the paper, a classification of degenerate 4-dimensional real matrices of rank 1, 2, 3. is elaborated. To separate possible classes of degenerate matrices of ranks 1 and 2, we impose linear restrictions on (k, m, l, n), which are compatible with the group multiplication law. All the subsets of matrices obtained by this method, are either sub-groups or semigroups. To obtain singular matrices of rank 3, we specify 16 independent possibilities to get 4-dimensional matrices with zero determinant.