Title：Fast Algorithm for Maxwell Eigenvalue Problems Arising from 3D Photonic Crystals
Abstract：This work focuses on numerically studying the eigenstructure behavior of the generalized eigenvalue problems (GEPs) arising in the three dimensional (3D) source-free Maxwell's equations with magnetoelectric coupling effects which model 3D reciprocal chiral media. It is a challenging problem to solve such a large-scale GEP efficiently. We combine the null-space free method with the inexact shift-invert residual Arnoldi method and MINRES linear solver to solve the GEP with the matrix dimension being as large as 5,308,416. The eigenstructure behavior is heavily determined by the chirality parameter. We show that all the eigenvalues are real and finite for a small chirality. For a critical value, the GEP has 2 by 2 Jordan blocks at infinity eigenvalues. Numerical results demonstrate that when chirality parameter increases from the critical value the Jordan block will split into a complex conjugate eigenpair, and then rapidly collide on the real axis and bifurcate into a new negative eigenvalue and a new positive eigenvalue (resonance mode) smaller than the other existing positive eigenvalues. The resonance band also exhibits an anticrossing interaction. Moreover, the electric and magnetic fields of the resonance modes are localized inside the structure, with only a slight amount of field leaking into the background (dielectric) material.