Dispersion Relations and Their Symmetry Properties for Electromagnetic and Elastic Metamaterials in Two Dimensions

TL;DR

This paper analytically derives dispersion relations for 2D electromagnetic and elastic metamaterials using multiple-scattering theory, highlighting the influence of lattice structure on isotropy and anisotropy, and supporting effective medium theories.

Contribution

It provides explicit analytical expressions for dispersion relations in 2D metamaterials, clarifies the role of lattice symmetry, and extends effective medium theories to anisotropic elastic metamaterials.

Findings

Dispersion relations depend on lattice structure and can be isotropic or anisotropic.

Analytical results support existing isotropic effective medium theories.

Numerical verification confirms properties of anisotropic elastic metamaterials.

Abstract

In the framework of multiple-scattering theory, we show that the dispersion relations of certain electromagnetic (EM) and elastic metamaterials can be obtained analytically in the long-wavelength limit. Specific examples are given to the two-dimensional metamaterials with cylindrical inclusions arranged in square and triangular lattices. The role played by the lattice structure in determining whether a dispersion relation is isotropic or not is shown explicitly. Different lattice dependences between EM and elastic metamaterials are also shown. In the case of isotropic dispersions, our results coincide with those of isotropic effective medium theories obtained previously for EM and elastic metamaterials, respectively, and, therefore, provide a more fundamental support to those theories. In the case of elastic metamaterials with anisotropic dispersions, our analytical results can provide an anisotropic effective medium theory in the form of Christoffel's equation. In this case, the isotropic effective medium theory can describe accurately the angle-averaged dispersion relations. The properties of anisotropic dispersions are discussed and verified by numerical calculations of a realistic elastic metamaterial.