Yu. G. Smirnov

Propagation of TM waves in a Kerr nonlinear layer

TM electromagnetic waves propagating through a nonlinear homogeneous isotropic unmagnetized dielectric layer located between two homogeneous isotropic half-spaces are studied. The nonlinearity in the layer obeys the Kerr law. The problem is reduced to a system of nonlinear ordinary differential equations. A dispersion relation for the propagation constants is derived. The results are compared with those in the case of a linear layer.

Nonlinear effects in the problem of propagation of TM electromagnetic waves in a Kerr nonlinear layer

The dispersion equation for the problem of propagation of TM-polarized electromagnetic waves in a Kerr nonlinear layer is presented. The propagation constants are calculated. Qualitative differences of the dispersion curves in the nonlinear and linear cases are demonstrated.

Logarithmic Integral Equations in Electromagnetics

Elements of the theory of integral operators: integral operators with purely logarithmic kernel integral operators in Holder spaces logarithmic integral operators and Chebyshev polynomials integral operators defined on a set of intervals integral operators with fixed logarithmic singularities elements of spectral theory abstract pole pencils logarithmic integral operators in Sobolev spaces integral operators with kernels represented by series methods of small parameter approximate inversion approximate semi-inversion. Generalized potentials with logarithmic kernels: generalized potentials Greens potentials examples for canonical domains half-plane rectangle circle exterior of a circle ring. Summation operators: matrix representation Galerkin methods and basis of Chebyshev polynomials summation operators in the spaces of sequences matrix representation of logarithmic integral operators. Boundary value problems: formulation of the problem uniqueness and existence theorems canonical problems - diffraction by strips and slots diffraction by a slot diffraction by a strip diffraction by a screen with a rectangular slotted cavity scattering by a circular slotted cylinder eigenoscillations of open and closed slot resonators closed rectangular slot resonator open rectangular slot resonator slotted resonator with circular cross section the integral and summation equations for the strip problems summation equations in the problem on eigenfrequencies.

Permittivity Reconstruction of Layered Dielectrics in a Rectangular Waveguide from the Transmission Coefficients at Different Frequencies

Determination of electromagnetic parameters of dielectric bodies of complicated structure is an urgent problem. However, as a rule, these parameters cannot be directly measured (because of composite character of the material and small size of samples), which leads to the necessity of applying methods of mathematical modeling and numerical solution of the corresponding forward and inverse electromagnetic problems. It is especially important to develop the solution techniques when the inverse problem for bodies of complicated shape is considered in the resonance frequency range. In this paper we develop a method of solution to the inverse problem of reconstructing (complex) permittivity of layered dielectrics in the form of diaphragms in a waveguide of rectangular cross section from the transmission coefficients measured at different frequencies. The method enables in particular obtaining solutions in a closed form in the case of one-sectional diaphragm. In the case of an n-sectional diaphragm we solve the inverse problem using numerical solution of a nonlinear equation system of n complex variables. Solvability and uniqueness of the system are studied and convergence of the method is discussed. Numerical results of calculating (complex) permittivity of the layers are presented. The case of metamaterials is also considered. The results of solution to the inverse problem can be applied in nanotechnology, optics, and design of microwave devices.

Calculation of the Propagation Constants of TM Electromagnetic Waves in a Nonlinear Layer

Propagation of TM electromagnetic waves through a nonlinear homogeneous isotropic nonmagnetic dielectric layer is considered. The layer is located between two homogeneous isotropic half-spaces. The dispersion equation for the propagation constants of the waves in the layer and the first approximation for these constants are presented. The propagation constants for the cases of a linear medium and a nonlinear medium in the layer are compared. The linear and nonlinear cases and the first approximation are analyzed. Calculation results are presented.

On the infinitely many nonperturbative solutions in a transmission eigenvalue problem for Maxwell’s equations with cubic nonlinearity

The paper focuses on a transmission eigenvalue problem for Maxwell’s equations with cubic nonlinearity that describes the propagation of transverse magnetic waves along the boundaries of a dielectric layer filled with nonlinear (Kerr) medium. Using an original approach, it is proved that even for small values of the nonlinearity coefficient, the nonlinear problem has infinitely many nonperturbative solutions (eigenvalues and eigenwaves), whereas the corresponding linear problem always has a finite number of solutions. This fact implies the theoretical existence of a novel type of eigenwaves that do not reduce to the linear ones in the limit in which the nonlinear coefficient reduces to zero. Asymptotic distribution of the eigenvalues is found, periodicity of the eigenfunctions is proved, the exact formula for the period is found, and the zeros of the eigenfunctions are determined.

Pseudodifferential operator method in a problem on the diffraction of an electromagnetic wave on a dielectric body

We study the problem on the diffraction of electromagnetic waves on a solid body in free space. To analyze the integro-differential equations describing this phenomenon, we use the theory of pseudodifferential operators. We evaluate the asymptotic expansion of the symbol and prove the ellipticity and Fredholmness with index zero of the problem operator.

Calculation of the Propagation Constants and Fields of Polarized Electromagnetic TM Waves in a Nonlinear Anisotropic Layer

Propagation of polarized electromagnetic TM waves through a nonlinear homogeneous nonmagnetic dielectric layer located between two homogeneous isotropic half-spaces is considered. The dispersion equation is derived for the propagation constants (eigenvalues) of waves in the layer. The propagation constants for nonlinear and linear media in the layer are compared. The propagation constants and fields (eigenfunctions) are calculated.

A Subhierarchical Parallel Computational Algorithm for Solving Problems of Diffraction by Plane Screens

A subhierarchical method is proposed for solving problems of diffraction by thin plane perfectly conducting screens of an arbitrary shape. The Rao-Wilton-Glisson method is used for numerical solution of the problem. A parallel algorithm is applied to calculate matrix elements and right-hand sides. Screen currents are calculated on a computational cluster.

Nonlinear transmission eigenvalue problem describing TE wave propagation in two-layered cylindrical dielectric waveguides

The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with a Kerr nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of a Green’s function. The existence of propagating TE waves is proved using the contraction mapping method. For the numerical solution of the problem, two methods are proposed: an iterative algorithm (whose convergence is proved) and a method based on solving an auxiliary Cauchy problem (the shooting method). The existence of roots of the dispersion equation (propagation constants of the waveguide) is proved. Conditions under which k waves can propagate are obtained, and regions of localization of the corresponding propagation constants are found.