Closed-form expression for the scattering coefficients at an interface between two periodic media

We use the Bloch-mode orthogonality to derive simple closed-form expressions for the scattering coefficients at an interface between two periodic media, a computationally-challenging electromagnetic scattering problem that can be solved only with advanced numerical tools. The derivation relies on the assumptions that the interface is illuminated by the fundamental Bloch mode and that the two media have only slightly different geometrical parameters. Through comparison with fully-vectorial three-dimensional computations, the analytical expressions are shown to be highly predictive for various geometries, including dielectric waveguides and metallic metamaterials. They can thus be used with confidence for designing and engineering stacks of periodic structures.