Surface polaritons in magnetic metamaterials from perspective of effective-medium and circuit models

Surface waves are responsible for many phenomena occurring in metamaterials and have been studied extensively. At the same time, the effects of inter-element coupling on surface electromagnetic waves (polaritons) remain poorly understood. Using two models, one relying on the effective-medium approximation and the other on equivalent circuits, we studied theoretically surface polaritons propagating along an interface between air and a magnetic metamaterial. The metamaterial comprised split rings that could be uncoupled or coupled to each other in the longitudinal or transverse directions (along or perpendicular to the propagation direction). A metamaterial without inter-element coupling supported a single polariton. When a moderate longitudinal coupling was included, it changed the wave dispersion only quantitatively, and the results of the effective-medium and the circuit models were shown to agree at low wavenumbers. However, the presence of a transverse coupling changed the polariton dispersion dramatically. The effective-medium model yielded two branches of polariton dispersion at low values of the transverse coupling. As the coupling increased, both polaritons disappeared. The validity of the effective-medium model was further tested by employing the circuit model. In this model, surface polaritons could exist in the presence of a transverse coupling only if the boundary layer of the metamaterial included additional impedances, which could become non-Foster. The results reveal that the inter-element coupling is a major mechanism affecting the properties of the polaritons. They also highlight the limitations of using bulk effective-medium parameters for interface problems in metamaterials. I. INTRODUCTION