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We discuss a quartic eigenvalue problem arising in the context of an opticalwaveguiding problem involving atomically thick 2D materials. The waveguideconfiguration we consider consists of a gradient-index (spatially dependent)dielectric equipped with conducting interior interfaces. This leads to aquartic eigenvalue problem with mixed transverse electric and transversemagnetic modes, and strongly coupled electric and magnetic fields. We derive aweak formulation of the quartic eigenvalue problem and introduce a numericalsolver based on a quadratification approach in which the quartic eigenvalueproblem is transformed to a spectrally equivalent companion problem. We verifyour numerical framework against analytical solutions for prototypicalgeometries. As a practical example, we demonstrate how an improved qualityfactor (defined by the ratio of the real and the imaginary part of the computedeigenvalues) can be obtained for a family of gradient-index host materials withinternal conducting interfaces. We outline how this result lays the groundworkfor solving related shape optimization problems.

Nonlinear eigenvalue problems for coupled Helmholtz equations modeling gradient-index graphene waveguides