ABSTRACT
Transformations of Jones matrices into sums of Pauli matrices separate the Jones matrix into polarization components associated with (1) x and y-oriented diattenuation and retardance, (2) 45° and 135° oriented diattenuation and retardance, and (3) the left and right circular diattenuation and retardance. When polarization elements are cascaded, new forms of polarization, not present in the original polarization elements, are generated. With the Pauli basis, these interactions of polarization effects becomes clear, for example how a sequence of x-diattenuation and 45° retardance create a circular diattenuation component, and how that circular diattenuation component changes sign when the two elements are reversed. Taking the matrix exponential of a linear combination of Pauli matrices leads directly to an expression for the retardance components, and, with a bit of further manipulation, to the diattenuation components. The first order Taylor series expansion of the matrix exponential provides equations for weak polarization elements. Unlike the polar decomposition, this Pauli matrix exponential provides an order independent decomposition into polarization properties. Thus, taking the matrix logarithm of a homogeneous Jones matrix with orthogonal eigenpolarizations leads to simple algorithms for the retardance and diattenuation components. Unfortunately, the Baker-Hausdorf-Campbell relationship for matrix exponentials prevents the matrix logarithm algorithm from being extended to the simple calculation of diattenuation and retardance for all inhomogeneous Jones matrices.
