IMPROVED FORMULATION OF SCATTERING MATRICES FOR SEMI-ANALYTICAL METHODS THAT IS CONSISTENT WITH CONVENTION

Abstract

The literature describing scattering matrices for semi- analytical methods almost exclusively contains ine-cient formulations and formulations that deviate from long-standing convention in terms of how the scattering parameters are deflned. This paper presents a novel and highly improved formulation of scattering matrices that is consistent with convention, more e-cient to implement, and more versatile than what has been otherwise presented in the literature. Semi-analytical methods represent a device as a stack of layers that are uniform in the longitudinal direction. Scattering matrices are calculated for each layer and are combined into a single overall scattering matrix that describes propagation through the entire device. Free space gaps with zero thickness are inserted between the layers and the scattering matrices are made to relate flelds which exist outside of the layers, but directly on their boundaries. This framework produces symmetric scattering matrices so only two parameters need to be calculated and stored instead of four. It also enables the scattering matrices to be arbitrarily interchanged and reused to describe longitudinally periodic devices more e-ciently. Numerical results are presented that show speed and e-ciency can be increased by more than an order of magnitude using the improved formulation.