Propagation of transverse-electric waves along a homogeneous plane dielectric waveguide is considered. The waveguide is placed between two half spaces with constant permittivities. The permittivity inside the waveguide is described by the Kerr law. The problem is to determine propagation constants of eigenmodes. It is theoretically predicted that there exists a novel type of propagation modes that does not reduce to linear modes in the limit in which the nonlinear coefficient reduces to zero. It is proved that in the presence of the Kerr effect, infinitely many new propagation constants arise. An analysis of this intriguing case is given.

The propagation of TE-polarized electromagnetic waves along a Kerr-type nonlinear dielectric, nonabsorbing, nonmagnetic, and isotropic (circular) cylindrical waveguide is investigated. For axially (azimuthal) symmetric solutions the problem is reduced to a cubic-nonlinear integral equation that is solved by iteration leading to a sequence uniformly convergent to the solution of the integral equation. The dispersion relations associated to the exact and iterate solutions, respectively, are derived and solved, subject to certain constraints. The roots of the exact dispersion relation are approximated by the roots of the dispersion relations generated by the iterate solutions. All statements of existence and convergence are based on results of a previous paper. Numerical results (concerning solutions of dispersion relations, field patterns, dependence of the propagation constant and of the cutoff radius on the nonlinearity parameter, power flow) are included.

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.

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 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.