Light propagation equation for a double Stokes-Mueller polarimetry

The Stokes-Mueller polarimetry is a well-established measurement technique to provide both purely polarized and partially polarized optical signals. However, this polarimetry is not applicable for the nonlinear optical process second harmonic generation (SHG). To solve the problem involved, a generalized double Stokes-Mueller polarimetry (DSMP) formalism is employed to describe the polarized SHG signals. Instead of a 4×1 matrix for Stokes vector, the double Stokes vector is a 9×1 matrix representing the polarization of two incoming photons, which results in the double Mueller containing 36 elements rather than 16 elements for Mueller matrix. In this paper, we present a specific novel mathematical framework of the light propagation for DSMP. In this paper, all the light propagation process, including the linear optical process and the nonlinear optical process SHG in the sample. Firstly, we introduce the expressions for the Stokes vector of the incoming, outgoing radiations and the Mueller matrix of the linear optical elements. Secondly, the double Stokes vector and the double Mueller matrix similar to the linear Stoke-Mueller formalism is present. Thirdly, to combine a train of linear optical elements in front of SHG and the nonlinear optical process, we design a transition matrix T connecting the Stokes vector and the double Stokes vector. Finally, we get the mathematical framework of the light propagation for DSMP, which has the potential to be part of the error analysis of a polarization-resolved Second-harmonic Generation (PSHG) microscopy.