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Original method for studying the dispersion characteristics of photonic crystals – media with a dielectric constant that varies periodically in space – is considered. The method is based on the representation of the wave functions and permittivity of a periodic medium in the form of Fourier series and their subsequent substitution into the wave equation, which leads to the formulation of the dispersion equation. Using the latter, for each value of the wave vector it is possible determined a set of eigen frequencies. Each of eigen frequency forms a separate dispersion curve as a continuous function of the wave number. The Fourier expansion coefficients of the permittivity, which depend on the vectors of the reciprocal lattice of the photonic crystal, are determined on the basis of data on the geometric characteristics of the elements that form the crystal, their electrophysical properties and the density of the crystal. The solution of the dispersion equation found makes it possible to obtain complete information about the number of modes propagating in a periodic structure at different frequencies, and about the possibility of forming band gaps, i.e. frequency ranges within which wave propagation through a photonic crystal is impossible. The focus of this work is on the application of this method to the analysis of the dispersion properties of metallic photonic crystals. The difficulties that arise in this case due to the presence of intrinsic dispersion properties of the metals that form the elements of the crystal are overcome by an analytical description of their permittivity based on the model of free electrons. As a result, a dispersion equation is formulated, the numerical solution of which is easily algorithmized. That makes possible to determine the dispersion characteristics of metallic photonic crystals with arbitrary parameters. Obtained by this method the results of calculation of dispersion diagrams, which characterize two-dimensional metal photonic crystals, are compared with experimental data and numerical results obtained using the method of self-consistent equations. Their good agreement is demonstrated.

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Original methods for determining the effective electrophysical parameters of two-dimensional metallic photonic crystals, applicable in a wide frequency range from radio to visible ranges are researched. In contrast to the currently widespread analytical approaches to describing such parameters, the subject of this article is the methods that can be used both for the rigorous numerical analysis and for the direct practical application. It is shown that in the radio and infrared ranges in the first allowed zone of photonic crystals the determination of their effective dielectric constant can be carried out either on the basis of studying the intrinsic resonance properties of spatially limited structures or by studying the processes of reflection and refraction at their boundaries. In the visible range metallic photonic crystals exhibit properties that are largely similar to those of solid metals. However, photonic crystals have significantly lower heat losses and a positive effective dielectric constant <1. ​​This makes them promising for creating various devices for converting optical radiation. Determination of the effective electrophysical parameters at these frequencies is possible on the basis of a direct comparison of the amplitude-phase distributions of the field in photonic crystals and continuous media.