Vector Fresnel equations and Airy formula for one-dimensional multilayer and surface-relief gratings

A simple and intuitive formalism is presented to describe diffraction in multilayered periodic structures. A modal theory of diffraction is used to show how well-known results from scalar analysis (wave propagation in homogeneous layered media) can readily be generalized to vector problems. Specifically, the following results are derived: (1) generalized Fresnel equations appropriate for reflection and transmission from an infinitely thick grating, (2) a generalized equation for power conservation for diffraction gratings, (3) a generalized Airy formula for thin film to describe reflection and transmission of light through a lamellar grating, and (4) a matrix propagation method akin to that used to calculate reflection and transmission of multilayer thin films. Some numerical results are also presented to illustrate the applications of this research and its relationship to previous modal theories.