Electromagnetic Eigenvalue Problem for a Graphene Strip Placed in the Center of a Circular Dielectric Rod: Hypersingular Integral Equations and Symmetry Classes

We study the electromagnetic eigenvalue problem, adapted to the analysis of the lasing threshold conditions, for the TE-polarized modes of a circular rod made of the gain material, with a flat graphene strip inside. Thanks to the twofold symmetry, the modes split to four independent classes. In each symmetry class, the eigenvalue problem is reduced to a pair of the hypersingular integral equations (IE), serving as a basis for further numerical analysis.