Generalized Mode-Matching Technique in the Theory of Guided Wave Diffraction. Part 3: Wave Scattering by Resonant Discontinuities”

Abstract

The problem of justification of the correctness of the matrix‐operator models of the modematching technique as applied to the problems of resonant wave scattering by waveguide discontinuities has remained of great importance throughout the years of the intensive use of the method. Another unsolved problem is substantiation of using the truncation procedure to solving the obtained infinite matrix equations. The present paper is aimed at proving rigorously correctness of the mathematical model in the form of the operator‐based Fresnel formulas for the specified class of mode diffraction, constructing projection approximations for the sought‐for scattering operators and justifying their convergence. To that end a generalized mode‐matching technique is used. The “generalized operator‐based Fresnel formulas” are derived for the scattering operator matrices. The universality of the constructed operator model in the form of the Cayley transform is proven. It is shown that domain of correctness of this model is completely determined by the established operator properties of the generalized scattering matrix. The unconditional convergence of the projection approximations to the exact solution is proved analytically. The mode‐matching technique which is widely used for solving scalar problems of waveguide mode diffraction possesses a matrix‐operator nature and an adequate to this nature mathematical apparatus, specifically, the theory of operators in the Hilbert space. The suggested generalization of the mode‐matching technique can be used for rigorous analysis of microwave devices.