Analitical theory Flouqet-Bloch waves for gyrotropic magnetophotonic krystals

The relevance of the problem considered in the work is determined by the widespread use of magnetophotonic crystals in various devices of the terahertz microwave and optical ranges. The key is the analytical solution of the third boundary value problem for the Hill equation with mixed Cauchy boundary conditions. This approach made it possible to explicitly find new solutions for electromagnetic fields in the crystal layers and dispersion characteristics for TE and TM waves, which is important for the development of new multifunction devices in the terahertz range.. The purpose of the work is to develop an analytical theory of Floquet-Bloch waves for gyrotropic magnetophotonic crystals with a transverse magnetic field. Materials and methods. Magnetophotonic crystals consist of gyrotropic (gyroelectric or gyromagnetic materials) two-layer structures over a period, the parameters of which vary from the magnitude of the applied magnetic field. Analytical methods for solving the Hill equation through fundamental solutions of the third boundary value problem. Results. The fundamental solutions of the Hill equation are determined in an analytical form. Analytical expressions for the dispersion characteristics of TE and TM Floquet-Bloch waves are found. The existence of bulk and surface waves in the transmission zones of a magnetophotonic crystal is established. The existence of an extraordinary surface wave with an atypical field distribution in the crystal layers for positive effective electric or magnetic permeability is shown. Conclusions. The proposed new approach for determining the solutions of the Hill equation based on the fundamental solutions of the third boundary-value problem made it possible to obtain in an analytical form the dispersion characteristics and fields of controlled gyromagnetic magnetophotonic crystals for TE and TM Floquet-Bloch waves. This will make it relatively easy to calculate various devices based on controlled Bragg structures.

INTEGRAL TRANSFORMATION FOR THE THIRD BOUNDARY-VALUE PROBLEM OF NON-STATIONARY HEAT CONDUCTIVITY WITH A CONTINUOUS SPECTRUM OF EIGENVALUES

The mathematical theory of constructing an integral transformation and the inversion formula for it for the third boundary value problem in a domain with a continuous spectrum of eigenvalues are developed. The method is based on the operational solution of the initial problem with an initial function of general form satisfying the Dirichlet condition and a homogeneous boundary condition of the third kind. On the basis of the obtained relations, a series of analytical solutions of the third boundary value problem for a parabolic equation in various equivalent functional forms is proposed. An integral representation of the analytic solutions of the third boundary-value problem is proposed for the general form of the representation of boundary-value functions in the initial formulation of the problem. The corresponding Green's function is written out.