The rigorous solution of the scattering problem for a finite cone embedded in a dielectric sphere surrounded by the dielectric medium

Abstract

Wave scattering from a finite hollow cone with perfectly conducting boundaries embedded in a dielectric sphere is considered. The structure is excited axially symmetrically by the radial electric dipole. The scattering problem is formulated in the spherical coordinate system as the boundary value problem for the Helmholtz equation. The diffracted field is given by expansion in the series of eigenfunctions. Owing to the enforcement of the conditions of continuity together with the orthogonality properties of the Legendre functions the diffraction problem is reduced to infinite system of linear algebraic equations (ISLAE) of the first kind. The usage of the analytical regularization approach transforms the ISLAE of the first kind to the second one and allows one to justify the truncation method for obtaining the numerical solution in the required class of sequences. This system is proved to be regularized by a pair of operators, which consist of the convolution type operator and the corresponding inverse one. The inverse operator is found analytically using the factorization technique. The numerical examples are presented. The static and the low-frequency approximations as well as the transition to the limiting case when the cone degenerates into the disc are considered.This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Guest Editorial: Method of analytical regularisation for new frontiers of applied electromagnetics

“…A subject of relevance to radioscience, near-field optics and nanotechnologies is considered in the paper 'The rigorous solution of the scattering problem for the finite cone embedded in the dielectric sphere surrounded by the dielectric medium' by Kuryliak [5]. Here, a first-kind matrix equation, obtained by means of the mode-matching technique and the orthogonality properties of the Legendre functions, is analytically regularised by a pair of operators consisting of the convolution type operator and the corresponding inverse one expressed in closed form.…”

Analysis of the Scattering from a Two Stacked Thin Resistive Disks Resonator by Means of the Helmholtz–Galerkin Regularizing Technique

“…Usually, it can be a suitable asymptotic part (e.g., the static part or the high-frequency part, etc.) or a canonical-shape part, which can be inverted by means of functional techniques such as Wiener-Hopf, Cauchy, Titchmarsh, Abel, Riemann-Hilbert Problem, and the separation of variables techniques [12][13][14][15][16][17][18][19][20]. Following this line of reasoning, the resulting integral equation is of the Fredholm second-kind; hence, the Fredholm theory [21], generalized by Steinberg for operators [22], can be applied.…”