Error propagation in polarimetric demodulation

TL;DR

This paper analyzes how uncertainties in the modulation matrix affect the accuracy of polarimetric measurements, providing formulas to quantify the propagation of these errors through the demodulation process.

Contribution

It introduces analytical expressions for the covariance propagation from modulation matrix uncertainties to Stokes parameters in polarimetric analysis.

Findings

Covariance matrix of demodulation is generally non-diagonal due to matrix inversion.

Uncertainties in modulation matrix induce correlations in inferred Stokes parameters.

Analytical formulas enable better error estimation in polarimetric measurements.

Abstract

The polarization analysis of the light is typically carried out using modulation schemes. The light of unknown polarization state is passed through a set of known modulation optics and a detector is used to measure the total intensity passing the system. The modulation optics is modified several times and, with the aid of such several measurements, the unknown polarization state of the light can be inferred. How to find the optimal demodulation process has been investigated in the past. However, since the modulation matrix has to be measured for a given instrument and the optical elements can present problems of repeatability, some uncertainty is present in the elements of the modulation matrix and/or covariances between these elements. We analyze in detail this issue, presenting analytical formulae for calculating the covariance matrix produced by the propagation of such uncertainties on the demodulation matrix, on the inferred Stokes parameters and on the efficiency of the modulation process. We demonstrate that, even if the covariance matrix of the modulation matrix is diagonal, the covariance matrix of the demodulation matrix is, in general, non-diagonal because matrix inversion is a nonlinear operation. This propagates through the demodulation process and induces correlations on the inferred Stokes parameters.