On the physical significance of non-local material parameters in optical metamaterials

Abstract

When light interacts with a material made from subwavelength periodically arranged constituents, non-local effects can emerge. They occur because of either a complicated response of the constituents or possible lattice interactions. In lowest-order approximations of a general non-local response function, phenomena like an artificial magnetism and a bi-anisotropic response emerge. However, investigations beyond these lowest-order descriptions of non-local effects are needed for optical metamaterials (MMs) where a significant long-range interaction becomes evident. This highlights the need for additional material parameters to account for spatial non-locality in an effective medium description. These material parameters emerge from a Taylor expansion of the general and exact non-local response function. Even though these non-local parameters improve the effective description, their physical significance is yet to be understood. To improve the situation, we consider a conceptional MM consisting of scatterers characterized by a prescribed multipolar response arranged on a square lattice. Lorentzian polarizabilities describe the scatterers in the electric dipolar, electric quadrupolar, and magnetic dipolar terms. A slab of such a MM is homogenized while considering an increasing number of non-local terms in the constitutive relations at the effective level. We show that the effective permittivity and permeability are linked to the electric and magnetic dipole moments of the scatterers. The non-local material parameters are related to the higher-order multipolar moments and their interaction with the dipolar terms. Studying the effective material parameters with the knowledge of the induced multipolar moments in the lattice facilitates our understanding of the significance of each material parameter. Our insights aid in deciding on the order to truncate the Taylor expansion of the considered constitutive relations for a given MM.