Fresnel equations and transmission line analogues for diffraction gratings

A simple and intuitive formalism is presented to describe diffraction in multi-layered periodic structures. We use the well known results from scalar analysis (wave propagation in homogeneous layered media) and show that they can be generalized rather readily to vector problems such as diffraction analysis. Specifically, we derive: (1) generalized Fresnel equations appropriate for reflection and transmission from an infinitely thick grating, (2) a generalized Airy formula for thin-film to describe reflection and transmission of light through a lamellar grating and (3) a matrix propagation method akin to that used for multi-layer thin film analysis. The results developed here complement the recent work on R-matrix and S-matrix propagation algorithms that have been used in connection with modal and differential grating theories. These algorithms have proven to be numerically stable for calculating diffraction efficiencies from deep groove gratings. The formalism developed here expands upon the earlier literature by providing important details that are hitherto unavailable