Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings

TLDR

The study systematically presents two recursive, numerically stable matrix algorithms—the S‑matrix and R‑matrix—for modeling layered diffraction gratings, independent of grating models, geometries, and mountings. Multiple implementation variants are described, along with their physical interpretations, numerical stability, and efficiency characteristics. Unconditional numerical stability is achieved by avoiding exponentially growing functions at each recursion step, and although the S‑matrix algorithm is generally more efficient than the R‑matrix algorithm, exceptional cases are noted.

Abstract

Two recursive and numerically stable matrix algorithms for modeling layered diffraction gratings, the S-matrix algorithm and the R-matrix algorithm, are systematically presented in a form that is independent of the underlying grating models, geometries, and mountings. Many implementation variants of the algorithms are also presented. Their physical interpretations are given, and their numerical stabilities and efficiencies are discussed in detail. The single most important criterion for achieving unconditional numerical stability with both algorithms is to avoid the exponentially growing functions in every step of the matrix recursion. From the viewpoint of numerical efficiency, the S-matrix algorithm is generally preferred to the R-matrix algorithm, but exceptional cases are noted.