Eugene Smolkin

Nonlinear Goubau line: analytical–numerical approaches and new propagation regimes

We consider propagation of surface TE waves in the Goubau line (GL) assuming that the dielectric cover is non-linear and inhomogeneous. The problem at hand is reduced to a non-linear integral equation with a kernel in the form of the Green function of an auxiliary boundary value problem on an interval. The existence of propagating TE waves for the chosen nonlinearity (Kerr law) is proved by the method of contraction. Conditions under which several higher-order waves can propagate are obtained, and the intervals of the corresponding propagation constants are determined. For the numerical solution, a method based on solving an auxiliary Cauchy problem (a version of the shooting method) is proposed. In numerical experiment two types of nonlinearities are considered and compared: Kerr nonlinearity and nonlinearity with saturation. New propagation modes are found.

On the problem of propagation of nonlinear coupled TE-TM waves in a double-layer nonlinear inhomogeneous cylindrical waveguide

Nonlinear coupled electromagnetic TE-TM wave propagation in a cylindrical double-layer nonlinear inhomogeneous dielectric waveguide with circular cross section is considered. Inner layer of waveguide is filled with linear medium. Nonlinearity inside the outer layer of 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. For numerical solution of the problem, a method based on solving an auxiliary Cauchy problem (the shooting method) is proposed.

Diffraction of TM polarized electromagnetic waves by a nonlinear inhomogeneous metal-dielectric waveguide

In this work, we consider the diffraction of TM waves by an open metal-dielectric waveguide, a Goubau line (GL), with a nonlinear inhomogeneous dielectric cover. Numerical experiments are carried out for the nonlinearity with saturation. The physical problem is reduced to solving a nonlinear boundary value problem for a system of ordinary differential equations. Numerical results are obtained using a modification of the shooting method which makes it possible to determine and plot the amplitude of the reflected field with respect to the amplitude of the incident field. Comparison between the nonlinear problem and the corresponding linear setting is performed.

Coupled electromagnetic wave propagation in a cylindrical dielectric waveguide filled with Kerr medium: nonlinear effects

Abstract Propagation of the coupled electromagnetic wave, which is a superposition of TE and TM waves, in a dielectric circular cylindrical waveguide filled with non-linear inhomogeneous medium is studied (if the permittivity is linear, the coupled wave does not exist). Non-linear coupled TE–TM wave is characterized by two (independent) frequencies and two (coupled) propagation constants (PCs). The physical problem is reduced to a non-linear two-parameter transmission eigenvalue problem for Maxwell’s equations. The system of dispersion equations with respect to PCs is derived and solved numerically. Two types of coupled PCs and coupled guided modes are found: non-linear solutions of the first type become solutions of the corresponding linear problems as the nonlinearity coefficient tends to zero; solutions of the second type seem to be ’purely’ non-linear as they stay away from any linear solutions as coefficient of the nonlinearity tends to zero. Coupled PCs and coupled eigenmodes are calculated and plotted.

The new type of non-polarized symmetric electromagnetic waves in planar nonlinear waveguide

ABSTRACT Paper focuses on the propagation of monochromatic nonlinear symmetric hybrid waves in a planar dielectric waveguide filled with nonlinear medium. The wave propagation problem is reduced to a transmission eigenvalue problem. Eigenvalues of the problem depend on an additional parameter and correspond to propagation constant. Using perturbation method, it is theoretically proved the existence of a finite number of isolated eigenvalues and therefore, guide waves. The found guide regime is novel in the theory of nonlinear waveguides. Numerical results are presented.

Numerical analysis of electromagnetic wave propagation in metal-dielectric waveguides filled with nonlinear medium

The propagation of monochromatic electromagnetic waves in metal-dielectric waveguides of simple geometry (circular cylindrical) filled with nonlinear inhomogeneous medium is considered. The Kerr nonlinearity is studied. A physical problem is reduced to solving a nonlinear transmission eigenvalue problem for a system of ordinary differential equations. Eigenvalues of the problem correspond to propagation constants of the waveguide. A method is proposed for finding approximate eigenvalues of the nonlinear problem based on solving an auxiliary Cauchy problem (by the shooting method). The existence of eigenvalues that correspond to a new propagation regime is predicted. A comparison with the linear case is given.

On the existence of non-polarized azimuthal-symmetric electromagnetic waves in circular metal-dielectric waveguide filled with nonlinear radially inhomogeneous medium

Abstract Propagation of monochromatic nonlinear symmetric hybrid waves in a cylindrical nonlinear inhomogeneous metal–dielectric waveguide is considered. The physical problem is reduced to solving a transmission eigenvalue problem for a system of ordinary differential equations where spectral parameters are the wave propagation constants. The setting under study is reduced to a new type of nonlinear eigenvalue problem. An analytical method for solving this problem is elaborated. For the numerical solution, a method is proposed based on solving an auxiliary Cauchy problem (a version of the shooting method). As a result of comprehensive numerical modeling, new propagation regimes are discovered.

Eigenwaves in a lossy metal-dielectric waveguide

ABSTRACT The problem of normal waves in a closed (shielded) regular waveguide of arbitrary cross-section is considered. This problem is reduced to the boundary eigenvalue problem for longitudinal components of electromagnetic field in Sobolev spaces. To find the solution, we use the variational formulation of the problem. The variational problem is reduced to study an operator-function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operator-function on the complex plane is found. We also consider properties of system of eigenvectors and associated vectors of the operator-function. Double completeness of system of eigenvectors and associated vectors with a finite defect is established.

Numerical Method for Electromagnetic Wave Propagation Problem in a Cylindrical Inhomogeneous Metal Dielectric Waveguiding Structures

The propagation of monochromatic electromagnetic waves in metal circular cylindrical dielectric waveguides filled with inhomogeneous medium is considered. The physical problem is reduced to solving a transmission eigenvalue problem for a system of ordinary differential equations. Spectral parameters of the problem are propagation constants of the waveguide. Numerical results are found with a projection method. The comparison with known exact solutions (for particular values of parameters) is made.

Numerical study of multilayered nonlinear inhomogeneous waveguides in the case of TM polarization

We consider propagation of surface TM waves in a circular dielectric waveguide filled with nonlinear (Kerr nonlinearity) multilayered inhomogeneous medium. Each layer is characterized by a specific value of the nonlinearity coefficient α. Analysis is reduced to solving a nonlinear transmission eigenvalue problem for an ordinary differential equation; eigenvalues of the problem correspond to propagation constants of the waveguide. For the numerical solution, a method is proposed based on solving an auxiliary Cauchy problem (a version of the shooting method). As a result of comprehensive numerical modeling, new propagation regimes are discovered.