Transition between a photonic crystal and a metamaterial with electric response in dielectric structures

The feasibility of dielectric structures possessing metamaterial properties owing to electric Mie resonances is theoretically demonstrated. The structures are formed by homogeneous dielectric cylinders occupying the sites of a simple hexagonal or square lattice. The (normalized cylinder radius, dielectric constant) phase diagram is constructed and demonstrates a region where the effective dielectric constant $\varepsilon_{\text{eff}}$ of the metamaterial is close to zero. The simulation of the propagation of a Gaussian beam through a prism made of this metamaterial shows that the vanishingly small value of $\varepsilon_{\text{eff}}$ is insensitive to the orientation of the crystal axes.