Symbolic-numeric research of leaky modes in planar dielectric electromagnetic waveguide as inhomogeneous waves

In the case of guided and radiation modes of open waveguides, the Sturm-Liouville problem is formulated for self-adjoint second-order operators on the axis 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 boundary conditions for the leaky modes are not self-adjoint, so that the eigenvalues can turn out to be complex quantities. The problem of finding eigenvalues and eigenfunctions is associated with finding the complex roots of the nonlinear dispersion equation. The presence of complex eigenvalues corresponding to the leaky modes leads to an infinite increase of eigenfunctions corresponding to the leaky modes. In the present work, the leaky modes are considered as solutions of the problem for the wave equation and are analyzed as a wave process. Complex eigenvalues define the leaky modes as inhomogeneous waves. The proposed approach allows to obtain a mathematically sound representation of the leaky modes, within which one can see the region of existence of each particular leaky mode. The calculations of model structure, demonstrating the application of the described approach, are presented in the paper. © 2019 Author(s).