Projection approximations to the matrix scattering operators and relative convergence phenomenon

Abstract

A matrix‐operator model of the mode‐matching technique is examined for a scalar problem of mode diffraction on abrupt discontinuity in the waveguide. The convergence of approximations,which are found via a truncation procedure, to the matrix operators of mode reflection and transmission is studied analytically. Various estimates for these projection approximations are obtained. On the basis of these results, the unconditional and relative convergence phenomenaare analyzed. The impact of the “Mittra rule” upon the rate of convergence is validatedanalytically. The uniform‐convergence condition for scattering operatorapproximations is stated. Properties of the condition number for the truncated matrix‐operator equation arediscussed. The results thus obtained may well be used to substantiatea highly practically class of mathematical models related to the mode‐matching technique