Dispersion relations and their symmetry properties of electromagnetic and elastic metamaterials in two dimensions

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 hexagonal 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. When the anisotropy is small, 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.