This article focuses on numerically studying the eigenstructure behavior of generalized eigenvalue problems (GEPs) arising in three dimensional (3D) source-free Maxwell's equations with magnetoelectric coupling effects which model 3D reciprocal chiral media. It is challenging 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 a matrix dimension as large as 5,308,416. The eigenstructure is heavily determined by the chirality parameter $\gamma$. We show that all the eigenvalues are real and finite for a small chirality $\gamma$. For a critical value $\gamma = \gamma^*$, the GEP has $2 \times 2$ Jordan blocks at infinity eigenvalues. Numerical results demonstrate that when $\gamma$ increases from $\gamma^*$, the $2 \times 2$ Jordan block will first split into a complex conjugate eigenpair, then rapidly collide with the real axis and bifurcate into positive (resonance) and negative eigenvalues with modulus 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.

Full spectrum of a large sparse $\top$-palindromic quadratic eigenvalue problem ($\top$-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method \etc to obtain the Wiener-Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of $\top$-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized $\top$-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G$\top$SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G$\top$SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size $4000\times 4000$ at the density functional tight binding level, corresponding to a $8\times 8\; \mbox{nm}^{2}$ cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.

Calculation of band structures of three dimensional photonic crystals amounts to solving large-scale Maxwell eigenvalue problems, which are notoriously challenging due to high multiplicity of zero eigenvalues. In this paper, we try to address this problem in such a broad context that band structures of three dimensional isotropic photonic crystals in all $14$ Bravais lattices can be efficiently computed in a unified framework. In this work, we uncover the delicate machinery behind several key results of our framework and on the basis of this new understanding we drastically simplify the derivations, proofs and arguments. Particular effort is made on reformulating the Bloch condition for all $14$ Bravais lattices in the redefined orthogonal coordinate system, and establishing eigen-decomposition of discrete partial derivative operators by identifying the hierarchical structure of the underlying normal (block) companion matrix, and reducing the eigen-decomposition of the double-curl operator to a simple factorization of a $3$-by-$3$ complex skew-symmetric matrix. With the validity of the novel nullspace free method in the broad context, we perform some calculations on one benchmark system to demonstrate the accuracy and efficiency of our algorithm to solve Maxwell eigenvalue problems.