D. V. Valovik

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.

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.

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.

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.

Guided Electromagnetic Waves Propagating in a Two-Layer Cylindrical Dielectric Waveguide with Inhomogeneous Nonlinear Permittivity

The paper focuses on the problem of monochromatic electromagnetic TM wave propagation in a two-layer circular cylindrical dielectric waveguide. The space outside the waveguide is filled with isotropic medium having constant permittivity. The inner core of the waveguide is filled with isotropic medium having constant permittivity; the cladding of the core is filled with isotropic inhomogeneous nonlinear permittivity (the nonlinear term is expressed by Kerr law). Existence of guided modes which depend harmonically on z (the waveguide axis coincides with z-axis) is proved and their localization is found. Numerical results including different type of nonlinearities are presented. A comparison with the linear case is given. The existence of a new propagation regime is predicted.

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.

On the problem of nonlinear coupled electromagnetic transverse-electric–transverse magnetic wave propagation

Coupled electromagnetic TE and TM wave propagation in a nonlinear plane layer is considered. Nonlinearity inside the layer 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. These frequencies can be chosen independently. Existence of coupled surface TE and TM waves is proved. Intervals of localization of coupled eigenvalues are found.

Nonlinear Effects of Electromagnetic TM Wave Propagation in Anisotropic Layer with Kerr Nonlinearity

The problem of electromagnetic TM wave propagation through a layer with Kerr nonlinearity is considered. The layer is located between two half-spaces with constant permittivities. This electromagnetic problem is reduced to the nonlinear boundary eigenvalue problem for ordinary differential equations. It is necessary to find eigenvalues of the problem (propagation constants of an electromagnetic wave). The dispersion equation (DE) for the eigenvalues is derived. The DE is applied to nonlinear metamaterial as well. Comparison with a linear case is also made. In the nonlinear problem there are new eigenvalues and new eigenwaves. Numerical results are presented.