Electromagnetic wave interaction with 3D arrays of quadrupolar inclusions

In the homogenization of metamaterials, one of the important challenges yet to be fully resolved resides in the proper modeling of the influence of higher-order multipolar terms describing the basic constituent inclusions. As an important step in this direction, we consider here a metamaterial composed of a three-dimensional cubic array of purely quadrupolar inclusions, with zero electric and magnetic dipole moments, so that the leading term in their multipole expansion is the electric quadrupole moment. Applying a first-principle homogenization approach similar to (A. Alu, Phys. Rev. B, 84, 075153, 2011) the spatially dispersive permittivity of such an array in the long-wavelength limit is derived. This tensorial permittivity is a quadratic function of the propagation constant in the medium and it is shown to depend on the propagation direction, as a symptom of strong spatial dispersion.