Yury A. Tuchkin

C-Method: Several Aspects of Spectral Theory of Gratings

The goal of the present paper is two folded. The first, the methodological one, is the complementation of well established in diffraction theory of gratings C method with certain elements of spectral theory and the development of interactive numerical algorithm that made feed back conjunction between diffraction and spectral problems. As a natural result the second goal appeared: the appearing of such tool for numerical experiments resulted in profound qualitative and quantitative study of rather peculiar phenomena in resonant scattering from periodic surface. Special attention has been paid to the investigation of electromagnetic waves diffraction from periodic boundaries of material with single and double negative parameters.

Regularization of the Dirichlet Problem for Laplace's Equation: Surfaces of Revolution

Abstract Based on the idea of analytical regularization, a mathematically rigorous and numerically efficient method to solve the Laplace equation with a Dirichlet boundary condition on an open or closed arbitrarily shaped surface of revolution is described. To improve the convergence of the series for the single-layer density, we extracted and evaluated in an explicit form the singularity of the density at the surface edge. Numerical investigations of canonical structures, such as the open prolate spheroid and the open surface obtained by the rotation of “Pascals Limaçon” or the “Cassini Oval,” exhibit the high accuracy and wide applicability of the method.

Well-conditioned algorithm for scattering by a few eccentrically multilayered dielectric circular cylinders

The new regularization of the well-known analytical formulation of the monochromatic electromagnetic wave scattering by a few eccentrically multilayered homogenous circular cylinders is presented. It is found out that a regularization of this formulation is absolutely necessary. The two-sided regularization that we made is based on the integral formulation of the mentioned problem. The polarization of the fields are parallel to the longitudinal axes of the cylinders; thus, a two dimensional problem for each both polarizations are under consideration. The condition number of the resulting algebraic system is uniformly bounded while its truncation number increases. The numerical results are validated by existing results such as near and far fields obtained under various geometrical and electrical parameters of the scattering problem. Numerical results including the condition numbers of the regularized and nonregularized systems show that only regularized system gives numerically stable results with any desired accuracy in a wide range of frequencies from quasistatic to rather high-frequency range, limited only by the capabilities of the computer with the guarantee of the physical reliability of the solution.

Modified superformula contours optimized via genetic algorithms for fastly converging 2D solutions of EFIE

It is known that solutions of the integral equations converge at the smoothness rate of the parametrical function representing the boundary contour. Thus using an infinitely smooth parametrical representation with derivatives of all orders results into exponentially converging solutions. A version of superformula tailored for this purpose is exposed to optimization of its parameters via genetic algorithms to obtain smooth parameterization for desired boundaries in two dimensional problems. The convergence of the resulting solutions of the electric-field integral equation will be presented.

Diffraction from a Grating on a Chiral Medium: Application of Analytical Regularization Method

Theoretical results on the electromagnetic wave difiraction from a periodic strip grating placed on a chiral medium are obtained. Analytical Regularization Method based on the solution to the vector Riemann-Hilbert boundary value problem was used to get robust numerical results in the resonant domain, where direct solution methods typically fail. It was shown that in the case of normal incidence of linearly polarized wave the cross-polarized fleld appears in the re∞ected fleld. For elliptically polarized incident wave the difiraction character essentially depends on the polarization direction of the incident wave. These difiraction peculiarities are more pronounced in the resonant domain. In∞uence of the dichroism caused by chiral medium losses is thoroughly studied. The combination of a chiral medium and a grating can be efiectively used for a frequency and polarization selection and for a mode conversion.

Rigorous solution by analytical regularization method to the problem of 2D E-Polarized wave diffraction by a set of PEC surfaces

The analytical regularization method is applied to the problem of 2D E-Polarized wave diffraction by perfectly conductive surfaces consisting of a set of closed and unclosed surfaces. The electromagnetic boundary value problem is reduced to the infinite algebraic system of the second kind which in principal can be solved with any predetermined accuracy by means of truncation procedure. Numerical results, including condition number behavior, current density, and fields space distribution for obstacles are presented.

Diffraction of H-polarized Plane Wave by Cylindrical Cavities of Arbitrary Shape

The diffraction problem for an arbitrary shaped cylindrical cavity excited by a H-polarized plane wave is rigorously solved by the Method of Regularization. Along with the previously solved analogous problem for E-polarization this rigorous solution completes the construction of a reliable and highly efficient analytic-numerical technique for the analysis of diffraction problems for metallic cylinders of an arbitrary cross-section. Both problems are reduced to the numerical solution of a well-conditioned infinite system of linear algebraic equations of Fredholm type. Its numerical solution is effected by a truncation method. The computational accuracy only depends on truncation number. The effectiveness of this approach is demonstrated by examples of wave scattering problems for two-dimensional airfoils and engine intakes of various shapes. The combination of well-known approximate techniques with the developed approach has been exploited for studies of wave scattering problems for elongated cylinders of arbitrary cross-section.

A numerically stable algorithm for eccentrically metamaterial covered circular cylinders

The regularization for monochromatic TM/TE-z polarized waves scattering from multiple non intersecting circular penetrable boundaries has recently been proven to be a requisite for its stable numerical implementation for a wide scope of parameters. The validity and necessity of corresponding regularization algorithm will be demonstrated for medium parameters which are from a double negative (DNG) material media. The preliminary numerical results which already prove the properties mentioned above are given in this paper.

Modeling of waveguides of various cross section by analytical regularization method

New efficient approach for simulation of waveguides and cylindrical resonators of arbitrary profile is suggested. Numerical investigations for the abilities of algorithm and comparison with known results show high efficiency and accuracy of the method.

Super-algebraically convergent algorithm for 2D TM wave scattering from dielectric cylinders with smoothly parametrized cross section boundaries

Scattering of an incident TM wave from multiple dielectric cylinders with arbitrary cross sections is elaborated. Parametrization of each cross section is assumed to be infinitely smooth, i.e. it is infinitely differentiable according to its argument. We suggest the super-algebraically convergent algorithm to solve both electric/magnetic field integral equations (EFIE/MFIE) by proper factorization of their kernels as the infinitely smooth and singular functions in addition to extraction of its singularities involving hyper-singular ones. Solutions obtained via an entire domain Galerkin procedure are given showing physical relevance and numerical stability.