Yury Smirnov

Nonlinear Double-Layer Bragg Waveguide: Analytical and Numerical Approaches to Investigate Waveguiding Problem

The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function. The existence of propagating TE waves for chosen nonlinearity (the Kerr law) is proved using the contraction mapping method. Conditions under which k waves can propagate are obtained, and intervals of localization of the corresponding propagation constants are found. For numerical solution of the problem, a method based on solving an auxiliary Cauchy problem (the shooting method) is proposed. In numerical experiment, two types of nonlinearities are considered and compared: the Kerr nonlinearity and nonlinearity with saturation. New propagation regime is found.

Coupled electromagnetic TE-TM wave propagation in a layer with Kerr nonlinearity

Coupled polarized electromagnetic wave propagation in a nonlinear dielectric layer filled with lossless, nonmagnetic, and isotropic medium is considered. The layer is located between two half-spaces with constant permittivities. The permittivity in the layer is described by Kerr law. Considered coupled wave is formed of transverse electric (TE) and transverse magnetic (TM) polarized waves. This nonlinear coupled wave is called coupled TE-TM wave. The analysis is reduced to the nonlinear two-parameter eigenvalue problem. We look for coupled eigenvalues of the problem and reduce the question to the analysis of the corresponding system of two dispersion equations. Existence and uniqueness of solution to the two-parameter eigenvalue problem is proved. It is shown that a new regime for coupled TE and TM wave propagation exists in a layer with Kerr nonlinearity.

Problem of nonlinear coupled electromagnetic TE-TE wave propagation

Propagation of two TE coupled electromagnetic waves in a nonlinear plane layer is considered. Nonlinearity in the layer is described by Kerr law. It is shown that a new nonlinear propagation regime exists for a pair of TE waves. The physical problem is reduced to a nonlinear two-parameter eigenvalue problem for a system of (nonlinear) ordinary differential equations. It is proved that TE and TE waves that form a (nonlinear) coupled TE-TE wave can propagate at different frequencies ω1, ω2 with different propagation constants γ1, γ2, respectively. These frequencies can be chosen independently. The existence of a surface coupled TE-TE wave is proved. Intervals of localization of coupled eigenvalues are found.

Coupled electromagnetic transverse-electric–transverse magnetic wave propagation in a cylindrical waveguide with Kerr nonlinearity

Nonlinear coupled electromagnetic TE-TM wave propagation in a cylindrical nonlinear dielectric waveguide with circular cross section is considered. Nonlinearity inside the waveguide is described by Kerr law. Physical problem is reduced to a nonlinear two-parameter eigenvalue problem for a system of (nonlinear) ordinary differential equations. It is proved that TE and TM waves that form (nonlinear) coupled TE-TM wave can propagate at different frequencies ωE and ωM, respectively. It is shown that nonlinear coupled TE-TM wave propagates at different frequencies ωE, ωM and with different propagation constants γE, γM in the waveguide. Frequencies ωE, ωM can be chosen independently. Existence of coupled surface TE-TM waves is proved. Intervals of localization of coupled eigenvalues (γE, γM) are found.

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.

On the Problem of Electromagnetic Waves Propagating along a Nonlinear Inhomogeneous Cylindrical Waveguide

Electromagnetic TE wave propagation in an inhomogeneous nonlinear cylindrical waveguide is considered. The permittivity inside the waveguide is described by the Kerr law. Inhomogeneity of the waveguide is modeled by a nonconstant term in the Kerr law. Physical problem is reduced to a nonlinear eigenvalue problem for ordinary differential equations. Existence of propagating waves is proved with the help of fixed point theorem and contracting mapping method. For numerical solution, an iteration method is suggested and its convergence is proved. Existence of eigenvalues of the problem (propagation constants) is proved and their localization is found. Conditions of k waves existence are found.

Analysis of inverse scattering in a waveguide using the method of volume singular integral equation

We develop a method of solution to the inverse problem of reconstructing the media parameters (complex permittivity) of a body in a waveguide. The analysis is based on the use of a volume singular integral equation and its reduction to an equation which can be solved, using iterations, both numerically and analytically. This enables us to determine, for a given single-mode rectangular waveguide, the complex permittivity from the transmission coefficient. The approach also yields the proof of uniqueness of reconstruction of complex permittivity in a rectangular waveguide from the transmission characteristics. The results of test computations performed for the inclusion in the form of a parallelepiped justify the proposed approach.

Nonlinear coupled wave propagation in a n-dimensional layer

The paper focuses on the generalization of a nonlinear multi-parameter eigenvalue problem for a system of nonlinear differential equations. The problem is reduced to a system of nonlinear integral equations on a segment. The notion of eigentuple is introduced, the existence of a finite number of isolated eigentuples is proved, and their distribution is described. The corresponding linear multi-parameter eigenvalue problem is studied as well; it is proved that the linear problem has an infinite number of isolated eigentuples. Applications to nonlinear electromagnetic wave propagation theory are demonstrated.