GENERALIZED MODE-MATCHING TECHNIQUE IN THE THEORY OF MODE DIFFRACTION. PART 4. RATE OF CONVERGENCE FOR PROJECTIVE APPROXIMATIONS

Abstract

In this part of the work, we continue consideration about the basics of the generalized mode-matching technique, which has recently been developed for the analysis of wave diffraction. The problem of analytical estimate of the rate of convergence of projection approximations to the operator Fresnel formulae is discussed. The unconditional strong convergence of these approximations to the true scattering operators was proved previously. For the canonical scalar problem of wave diffraction on the step discontinue in a guide a measure of inaccuracy for the approximations of scattering operators has been derived analytically. These projective approximations under consideration are the truncated Fresnel formulae for the reflection and transmission operators. It is shown that the problem can be solved by examination of strong P–convergence of projective representations of an amplitude scattering operator. An analytical estimate of the rate of convergence of approximations for the scattering operators under consideration has been obtained. The found order of approximations has been verified by numerical computation. The results obtained allow us to estimate the computational efficiency of the generalized mode-matching technique, which can be useful for numerical-analytical solution of various electromagnetic problems

Keywords

• mode-matching technique
• operator fresnel formulae
• rate of convergence