TM Waves in a Dielectric Waveguide Covered by a Layer of Graphene

We investigate real surface TM waves in a homogeneous dielectric waveguide of circular cross section, a dielectric rod (DR), covered by an infinitely thin sheet of graphene. The first step is the determination and analysis of TM waves in DR without a graphene cover (a ‘‘proper’’ DR) using the specifically developed analytical-numerical methods employing the parameter dependence of the wave propagation constants in the form of implicit functions defined by dispersion equations (DEs) as solutions to the uniquely solvable Cauchy problems and inverse functions. Roots of the DE for a graphene-covered DR, i.e., the TM-wave propagation constants of this waveguide, are determined as perturbations of the DE roots for proper DR. It is demonstrated that these perturbations are stable and preserve the TE spectrum structure. The developed techniques can be generalized to determine different types of complex waves in DR covered by a graphene sheet.