A new analytical and numerical solution of the electrodynamic waveguide problem for leaky modes of a planar dielectric symmetric waveguide is proposed. The conditions of leaky modes, corresponding to the Gamow-Siegert model, were used as asymptotic boundary conditions. The resulting initial-boundary problem allows the separation of variables. The emerging problem of the eigen-modes of open three-layer waveguides is formulated as the Sturm-Liouville problem with the corresponding boundary and asymptotic conditions. In the case of guided and radiation modes, the Sturm-Liouville problem is self-adjoint and the corresponding eigenvalues are real quantities for dielectric media. The search for eigenvalues and eigenfunctions corresponding to the leaky modes involves a number of difficulties: the problem for leaky modes is not self-adjoint, so the eigenvalues are complex quantities. The problem of finding eigenvalues and eigenfunctions is associated with finding the complex roots of the nonlinear dispersion equation. To solve this problem, we used the method of minimizing the zero order. An analysis of the calculated distributions of the electric field strength of the first three leaky modes is given, showing the possibilities and advantages of our approach to the study of leaky modes.
Asymptotic properties of eigenvalues and eigenfunctions of a Sturm-Liouville problem with discontinuous weight function