We obtain representations of scalar and dyadic Green's functions for one-, two-, and threedimensional periodic photonic crystals. On this basis, volumetric, surface, and mixed surfacevolumetric integral and integro-differential equations are deduced. Numerical results of solution of the obtained equations for some types of structures are presented.
Simple nonintersecting thin metallic wire photonic crystals are studied using the integral equation method based on the Green's function for periodically arranged sources. A wavenumber and wavevector dependent effective permittivity tensor is derived upon homogenization, and the band diagram and the per mittivity for the lower dispersion branch are calculated.
Manmade media (MMMs) consisting of uniaxial photonic crystals with inserts of layers (films) or cylinders embedded in a periodic way into a dielectric substrate with dielectric permeability (DP) are considered. Approximate model-based and accurate electrodynamic methods for describing such MMMs, which are referred to in the case of metal (conductive) or ferrite (metaatom) inserts as a ‘hyperbolic metamaterial’ (HMM), are analyzed. Homogenization methods, the role of dissipation, spatial dispersion (SD), and slow plasmon-polaritons are reviewed. The feasibility of obtaining the hyperbolic dispersion law in a macroscopic description of DP of inserts using the Drude–Lorentz model is studied. In the general case with dissipation and SD, the surface of the Fresnel-equation isofrequencies is shown to differ from a rotation hyperboloid and to be bounded. The ambiguity of a description based on effective material parameters, the effect of dissipation and SD on hyperbolicity, currently observable and possible physical phenomena, and HMM applications are discussed.
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