Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings

A formulation of the modal method for multi-element dielectric lamellar diffraction gratings is presented. It combines the conventional semi-analytic Kronig–Penny formulation with a Fresnel scattering matrix approach to the solution of the diffraction problem. The theory is intuitive, applicable to complex geometries, and provides insight into diffraction grating physics. With the Fresnel matrix extensions the method potentially has greater generality than other formulations, and is applicable to non-trivial waveguide problems, such as characterizing the coupling between 1D photonic crystal waveguides. A numerically stable way of using the modal method for multilayer stacks of lamellar gratings is described.