Beyond local effective material properties for metamaterials

To discuss the properties of metamaterials on physical grounds and to consider them in applications, effective material parameters are usually introduced and assigned to a given metamaterial. In most cases, only weak spatial dispersion is considered. It allows to assign local material properties, e.g., a permittivity and a permeability. However, this turned out to be insufficient. To solve this problem, we study here the effective properties of metamaterials with constitutive relations beyond a local response and take strong spatial dispersion into account. This research requires two contributions. First, bulk properties in terms of eigenmodes need to be studied. We particularly investigate the isofrequency surfaces of their dispersion relation are investigated and compared to those of an actual metamaterial. The significant improvement to effectively describe it provides evidence for the necessity to use nonlocal material laws in the effective description of metamaterials. Second, to be able to capitalize on such constitutive relations, also interface conditions need to be known. They are derived in this contribution for our form of the nonlocality using a generalized (weak) formulation of Maxwell's equations. Based on such interface conditions, Fresnel expressions are obtained that predict the amplitude of the reflected and transmitted plane wave upon illuminating a slab of such a nonlocal metamaterial. This all together offers the necessary means for the in-depth analysis of metamaterials characterized by strong spatial dispersion. The general formulation we choose here renders our approach applicable to a wide class of metamaterials.