Symmetry Analysis of the Numerical Instabilities in the Transfer Matrix Method

This paper discusses the numerical exponential instability of the transfer matrix method (TMM) in the framework of the symmetry formalism. This numerical weakness is attributed to a series of increasingly extreme exponentials that appear in the TMM when it is applied to geometries involving total internal reflection (TIR) or very high absorption. We design a TMM formalism that identifies the internal symmetries of the multilayer geometry. These symmetries suggest particular transformations of reference system in the TMM that improve its ill-conditioned exponentials. To illustrate the numerical improvements, we present examples with calculations of electric fields.