Boundary conditions for electric quadrupolar continua

Abstract

After defining a continuum using rigorous spatial dispersion equations for the macroscopic, fundamental Floquet modal fields of 3-D periodic metamaterial arrays, boundary conditions are derived for electric quadrupolar continua under the assumption that the electric quadrupolarization density contains no extra delta functions within the interface transition layer (in addition to the delta function that usually occurs in the divergence of the electric quadrupolarization across the interface). A physically reasonable constitutive relation, in which the electric quadrupolarization density is proportional to the spatial derivatives of the electric field, is used to examine this assumption and the applicability of the boundary conditions. It is found that, although extra delta functions can be produced by a transverse magnetic plane wave incident upon an interface, the boundary conditions remain essentially unaffected by these extra delta functions for small enough electrical separation distances between the inclusions of the arrays.