The bi-Lebedev scheme for the Maxwell eigenvalue problem with 3D bi-anisotropic complex media

This paper focuses on studying the eigenstructure of generalized eigenvalue problems (GEPs) arising in the three-dimensional source-free Maxwell equations for bi-anisotropic complex media with a 3-by-3 permittivity tensor ε>0, a permeability tensor μ>0, and scalar magnetoelectric coupling constants ξ=ζ̄=ıγ. The bi-Lebedev scheme is appealing because it preserves the symmetry inherent to the Maxwell eigenvalue problem exactly and because full degrees of freedom of electromagnetic fields at each grid point are taken into account in the discretization. The resulting GEP has eigenvalues appearing in quadruples {±ω,±ω̄}. We consider two main scenarios, where γ<γ_∗ and γ>γ_∗ with γ_∗ as a critical value. In the former case, all the eigenvalues are real. In the latter case, the GEP has complex eigenvalues, and we particularly focus on the bifurcation of the eigenstructure of the GEPs. Numerical results demonstrate that the newborn ground state has occurred after γ=γ̃>γ_∗, and the associated eigenvector has an exotic phenomenon of localization. Moreover, the Poynting vectors of the newborn eigenvector not only are concentrated in the material but also display exciting patterns.