Mueller matrix cone and its application to filtering

Tim Zander,Juergen Beyerer

doi:10.1364/osac.383317

Abstract

We show that there is an isometry between the real ambient space of all Mueller matrices and the space of all Hermitian matrices that maps the Mueller matrices onto the positive semidefinite matrices. We use this to establish an optimality result for the filtering of Mueller matrices, which roughly says that it is always enough to filter the eigenvalues of the corresponding “coherency matrix.” Then we further explain how the knowledge of the cone of Hermitian positive semidefinite matrices can be transferred to the cone of Mueller matrices with a special emphasis towards optimisation. In particular, we suggest that means of Mueller matrices should be computed within the corresponding Riemannian geometry.

Highlights

In polarisation optics Mueller matrices are of great importance, as they describe the change of polarisation of light after interacting with a medium in a linear fashion
In order to be a Mueller matrix the matrix has to satisfy the Stokes criterion, which states that every Stokes vector has to be mapped onto a Stokes vector
It was shown that any coherency matrix of a non-depolarising Mueller matrix known as a Stokes-Mueller matrix has only one non-zero eigenvalue

Summary

Introduction

In polarisation optics Mueller matrices are of great importance, as they describe the change of polarisation of light after interacting with a medium in a linear fashion. Cloude showed in [9] that Mueller matrices can be associated with Hermitian matrices with non-negative eigenvalues the so called coherency or covariance matrices. This was used for filtering measured matrices in order to make them physically meaningful, i.e. satisfying the Stokes criterion. We will show how this can be used to prove a more general theorem about the optimality of filtering of Mueller matrices This simplifies and generalises part of the results of [14].we review the mathematical theory about the Hermitian positive semidefinite cone and explain, along with reviewing existing results, how this gives rise to the differential geometry of the manifold of all Mueller matrices

Isometry of the ambient space

Optimal filtering revisited

Geometry of the semidefinite cone

Euclidean

Conclusion