On the correct mathematical proof of the polarization mode dispersion equation

TL;DR

This paper addresses the lack of a rigorous mathematical proof for the polarization mode dispersion equation by proposing a corrected approach using a Euclidean pseudo-scalar product on Poincare's sphere.

Contribution

It introduces a mathematically sound proof of the polarization mode dispersion equation by redefining the metric on Poincare's sphere using a Euclidean pseudo-scalar product.

Findings

A correct proof of the polarization mode dispersion equation is established.

The use of a Euclidean pseudo-scalar product on Poincare's sphere is justified.

The approach resolves previous mathematical inconsistencies.

Abstract

The fundamental equation that describes polarization mode dispersion does not have a mathematically correct and convincing proof. This problem stems from the fact that Poincare's sphere, where Stokes vectors are represented, is just a manifold (a representation space) devoid of metric. In this "space" orthogonal vectors are antiparallel and, therefore, it is hard to justify the use of a Euclidean metric. However, if one realizes that in Poincare's sphere only three-dimensional rotations are represented, a Euclidean pseudo-scalar product can be defined and a mathematically correct proof for the polarization mode dispersion can be given.