Phase transition from photonic crystals to all-dielectric metamaterials
We considered two different scenarios for the transition of a two-dimensional square lattice of dielectric rods from photonic crystals (PhC) to all-dielectric metamaterials (MM) . The first scenario occurs when the lattice constant a decreases in comparison with the fixed radius of rods r and dielectric permittivity ε remains constant. This transformation leads to an increase of the filling ratio r/a and it makes possible the homogenization of the periodic dielectric structure with the negative effective permeability (µ<0) at higher values of ε. The second scenario occurs when the dielectric permittivity of rods ε varies in a fixed square lattice. This transformation leads to an increase of the Mie scattering wavelength, and it can also provide the conditions for the homogenization approach with negative µ.
We show that a PhC structure transforms into a MM when the Mie gap opens up below the lowest Bragg bandgap. Our theoretical approach was confirmed by microwave experiments for a metacrystal composed of tubes filled with heated water. We also introduced the concept of a phase diagram in the permittivity-filling ratio plane. The boundaries of the MM phase have been obtained theoretically and confirmed experimentally. On the basis of the proposed approach, one can obtain different PhC–MM phase diagrams by altering the dimension, symmetry, composition, size and geometry of the structural elements within a unit cell. This analysis yields deep insight into the properties of periodic structures, and provides a useful tool for designing different classes of electromagnetic materials with variable parameters.
 M.V. Rybin, D.S. Filonov, K.B. Samusev, P.A. Belov, Yu.S. Kivshar, and M.F. Limonov. Phase diagram for the transition from photonic crystals to dielectric metamaterials. Nature Commun., 6, 10102 (2015).