A new formulation to shape the concept of bounds in effective dielectric tensors for superlattices with two directions of periodicity

This paper introduces a new theoretical formulation based on a composition method and a statistical discretization approach by matrical block. Anisotropic properties and boundary conditions are considered, introducing the analytical bounds expressions of the effective dielectric constants in the limit of the long-wavelength regime for an idealized superlattice (SL) possessing two directions of periodicity (2D-SL). Such a SL can be described as a multilayer array of alternating cells (N x M) rectangular dielectric bars, allowing the structure to be shaped as a function of the dielectric constants of each of the anisotropic constituents. It is worth noting that in the simplified case of a 2D-SL made of only two different isotropic materials showing off the same periodicity in both directions, our general matrix formulation, due to the alternative composition laws, leads to the well-established results called, respectively, 'Wiener's and Lichtenecker's bounds' regarding the dielectric constant. This new formalism refashions the concept of bounds of effective dielectric tensors and the notion of form birefringence applied to 2D-SL, with relevant (N x M) rectangular anisotropic columns for arbitrary symmetries in the low-frequency model