Yury Shestopalov

DIFFRACTION OF ELECTROMAGNETIC WAVES BY A LAYER FILLED WITH A KERR-TYPE NONLINEAR MEDIUM

The diffraction of a plane wave by a transversely inhomogeneous isotropic nonmagnetic linearly polarized dielectric layer filled with a Kerr-type nonlinear medium is considered. The analytical and numerical solution techniques are developed. The diffraction problem is reduced to a singular boundary value problem for a semilinear second-order ordinary differential equation with a cubic nonlinearity and then to a cubic-nonlinear integral equation (IE) of the second kind and to a system of nonlinear operator equations of the second kind solved using iterations. Sufficient conditions of the unique solvability are obtained using the contraction principle.

Investigation of Electromagnetic Diffraction by a Dielectric Body in a Waveguide using the Method of Volume Singular Integral Equation

The problem of diffraction of an electromagnetic field by a locally nonhomogeneous body in a perfectly conducting waveguide of rectangular cross section is considered. This problem is reduced to solving a volume singular integral equation (VSIE). The examination of this equation is based on the analysis of the corresponding boundary value problem (BVP) for the system of Maxwells equations and the equivalence of this BVP and VSIE. The existence and uniqueness for VSIE in the space of square-integrable functions are proved. A numerical Galerkin method for the solution of VSIE is proposed, and its convergence is proved.

Eigenwaves in waveguides with dielectric inclusions: spectrum

Abstract We consider fundamental issues of the mathematical theory of the wave propagation in waveguides with inclusions. Analysis is performed in terms of a boundary eigenvalue problem for the Maxwell equations which is reduced to an eigenvalue problem for an operator pencil. We formulate the definition of eigenwaves and associated waves using the system of eigenvectors and associated vectors of the pencil and prove that the spectrum of normal waves forms a non-empty set of isolated points localized in a strip with at most finitely many real points.

Determination of permittivity of an inhomogeneous dielectric body in a waveguide

The determination of permittivity of an inhomogeneous dielectric body located in a rectangular waveguide is considered. An iteration method for the numerical solution of the problem is proposed. Convergence of the method is proved. Numerical results for the determination of permittivity of a dielectric body are presented.

Eigenwaves in waveguides with dielectric inclusions: completeness

We formulate the definition of eigenwaves and associated waves in a nonhomogeneously filled waveguide using the system of eigenvectors and associated vectors of a pencil and prove its double completeness with a finite defect or without a defect. Then, we prove the completeness of the system of transversal components of eigenwaves and associated waves as well as the ‘mnimality’ of this system and show that this system is generally not a Schauder basis. This work is a continuation of the paper Eigenwaves in waveguides with dielectric inclusions: spectrum. Appl. Anal. 2013. doi:10.1080/00036811.2013.778980 by Y. Smirnov and Y. Shestopalov. Therefore, we omit the problem statements and all necessary basic definitions given in the previous paper.

Inverse scattering in guides

We present statements and a method of solution to the inverse scattering problem of reconstructing permittivity of a dielectric inclusion in a 2D or 3D waveguide from thetransmission characteristic ...

Estimation of solutions of Helmholtz problems with uncertain data

The creation and justification of the methods for minimax estimation of parameters of the external boundary value problems for the Helmholtz equation in unbounded domains are considered. When observations are distributed in subdomains, the determination of minimax estimates is reduced to the solution of integro-differential equations in bounded domains. When observations are distributed on a system of surfaces the problem is reduced to solving integral equations on an unclosed bounded surface which is a union of the boundary of the domain and this system of surfaces. Minimax estimation of the solutions to the boundary value problems from point observations is also studied.

Stationary iteration methods for solving 3D electromagnetic scattering problems

Generalized Chebyshev iteration (GCI) applied for solving linear equations with nonselfadjoint operators is considered. Sufficient conditions providing the convergence of iterations imposed on the domain of localization of the spectrum on the complex plane are obtained. A minimax problem for the determination of optimal complex iteration parameters is formulated. An algorithm of finding an optimal iteration parameter in the case of arbitrary location of the operator spectrum on the complex plane is constructed for the generalized simple iteration method. The results are applied to numerical solution of volume singular integral equations (VSIEs) associated with the problems of the mathematical theory of wave diffraction by 3D dielectric bodies. In particular, the domain of the spectrum location is described explicitly for low-frequency scattering problems and in the general case. The obtained results are discussed and recommendations concerning their applications are given.

Diffraction by a Kerr-type Nonlinear Dielectric Layer

The difiraction of a plane wave by a transversely inhomogeneous isotropic nonmag- netic linearly polarized dielectric layer fllled with a Kerr-type nonlinear medium is considered. The difiraction problem is reduced to a cubic-nonlinear integral equation (IE) of the second kind and to a system of nonlinear operator equations of the second kind solved using iterations. Su-cient conditions of the IE unique solvability are obtained using the contraction principle.

Numerical simulation of a nonuniform dielectric inclusion in a waveguide aimed at reconstructing its permittivity

We consider numerical determination of the dielectric media parameters of inclusions in a waveguide of rectangular cross-section from the transmission coefficient. We develop and apply computer codes implementing the FDTD algorithm for nonstationary Maxwells equations with perfectly matched layer (PML) absorbing boundary conditions using the Berenger layout and estimate parameter ranges providing the prescribed accuracy for solving forward and inverse scattering problems for waveguides with inclusions.