Generalized Mode-Matching Technique in the Theory of Guided Wave Diffraction. Part 2: Convergence of Projection Approximations”

Abstract

A rigorous justification of applicability of the truncation procedure to solution of infinite matrix equation of the mode‐matching technique still remains an open question throughout the years of its intensive use. The generalized mode‐matching technique suggested for solving the problems of mode diffraction by a step‐like discontinuity in a waveguide leads to the Fresnel formulas for matrix operators of wave reflection and transmission, rather than to standard infinite systems of linear algebraic equations. The present paper is aimed at constructing projection approximations for the mentioned operator‐based Fresnel formulas and investigating analytically the qualitative characteristics of their convergence. To that end the theory of operators in the Hilbert space is used. The unconditional strong convergence of the finitedimensional approximations of the operator‐based Fresnel formulas to the true scattering operators is proved analytically. The condition number of the truncated matrix equation is estimated. The obtained results can be used for a rigorous justification of the mode‐matching technique intended for efficient analysis of microwave devices.