Electromagnetic TM wave propagation in nonlinear multilayered waveguides. Numerical technique to obtain propagation constants

A plane multilayered waveguide structure is considered. The layers are located between two half-spaces with constant permittivities. The permittivities inside the layers can be constants or nonlinear (depending on the electric field intensity). This structure can be treated as a 1D (nonlinear) photonic crystal [1]. We consider propagation of TM polarized electromagnetic waves in such a structure. Usually, in such problems, the main goal is to obtain a dispersion equation (DE) for propagation constants (eigenvalues). For many physically interesting nonlinear permittivities it is far beyond our abilities to obtain and analyze exact DEs. A numerical approach to calculate propagation constants (eigenvalues) for (nonlinear) layered waveguide structures is suggested.