Transition from thin-film to bulk properties of metamaterials

Effective material parameters for metamaterials are usually extracted for a single thin layer only. We show that, in general, these parameters cannot directly be assigned to a multilayered (bulk) metamaterial. The principal issue is the presence of higher-order Bloch modes. These modes are usually negligible for a single layer but can have considerable influence on the results for a periodically stacked structure. In particular, we show examples where a stack of single-layer negative refractive index metamaterials exhibits a positive index. Furthermore, on the basis of the dispersion relation of the respective Bloch modes, we derive criteria for the validity of the effective parameter approach.