Maxwell's equations with arbitrary self-action nonlinearity in a waveguiding theory: Guided modes and asymptotic of eigenvalues

Abstract

The paper treats a nonlinear eigenvalue problem that describes propagation of a transverse-magnetic wave in a plane dielectric waveguide having perfectly conducted walls at both sides. The dielectric's permittivity is characterised by an arbitrary monotonically increasing self-action nonlinearity. The full set of guided modes is described by eigenvalues of the corresponding (nonlinear) Maxwell operator with appropriate boundary conditions. We give a comprehensive analysis of this problem and develop an original approach to study its solvability and properties of solutions. Several results about existence of the eigenvalues are proved, their distribution and asymptotic are found; zeros of the eigenfunctions and their location are also determined; criterion of periodicity for the eigenfunctions is found, comparison theorem is derived, etc. It is shown that unbounded nonlinearities lead to appearance of nonlinearised solutions. This results in the existence of a novel guided regime that cannot be described within the framework of perturbation of linear guided wave regimes.