Corrections to the classical continuity boundary conditions at the interface of a composite medium

Using the multiple-scales homogenization method, we derive generalized sheet transition conditions (GSTCs) for electromagnetic fields at the interface between two media, one of which is free-space and the other a certain type of composite material. The parameters in these new boundary conditions are interpreted as effective electric and magnetic surface susceptibilities, which themselves are related to the geometry of the scatterers that constitute the composite. We show that the effective tangential E and H fields are not continuous across the interface except in the limit when the lattice constant (the spacing between the scatterers—atoms, molecules or inclusions in the case of a composite material) of the composite medium is very small compared to a wavelength. We derive first-order corrections to the classical continuity conditions. For naturally occurring materials whose lattice constants are on an atomic scale, these effects are shown to be negligible for waves at optical frequencies or lower. However, once the lattice constant becomes a significant fraction of a wavelength (which is the case for many artificial dielectrics and metamaterials), the corrections can be important. In previous work we have alluded to the fact that such a GSTC is needed to correctly account for the surface effects when extracting the effective material properties of a metamaterial. The results of this current paper justify the assumptions made in that previous work. In general, these GSTCs will result in corrections to the classical Fresnel reflection and transmission coefficients (which are themselves merely zeroth-order approximations to the actual reflection and transmission coefficients), and in a separate publication we will use these GSTCs to address this issue.