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.

Figure 1

(a) The EFS of a square lattice of silicone rubber cylinders embedded in an epoxy host at ${\tilde{f}}_1=0.01$. The rubber cylinder’s radius is 0.2 lattice constant. (b) $Ka$ as a function of ${\varphi }_K$. The open circles are calculated by the MST method while the solid lines are predicted by EMT. (c) and (d) are the same as (a) and (b), respectively, where the frequency is ${\tilde{f}}_2=0.03$.

Figure 2

(Color online) Open circles are angle-averaged dispersion relations calculated using MST for a square array of silicone rubber cylinders embedded in an epoxy host. The cylinder has a radius of 0.2 lattice constant. The solid and dashed curves are, respectively, the quasilongitudinal and quasitransverse branches obtained from EMT. Dimensionless frequencies, $\tilde{f}=fa/c_{t0}=\omega a/\left(2\pi c_{t0}\right)$, are used, where $c_{t0}$ is the transverse wave speed in the host.

Figure 3

(Color online) $Ka$ as a function of the radius of the cylinder $r_s$ for a square array of silicone rubber cylinders embedded in epoxy host. The frequency is fixed at $\tilde{f}=0.01$. Open circles are obtained from the angle-averaged dispersion relations calculated by MST method. Solid and dashed curves are quasilongitudinal and quasitransverse branches obtained from the EMT.

Figure 4

Similar to Fig. 1 with the lattice structure changed from a square lattice to a hexagonal lattice. The radius of the silicone rubber cylinder remains unchanged, i.e., $r_s=0.2a$. (a) and (c) EFS at $\tilde{f}=0.01$ and 0.03, respectively. The corresponding $Ka$ as a function of ${\varphi }_K$ are plotted in (c) and (d), respectively, in which the circles are calculated by the MST and the solid lines are given by the EMT.