Bifurcation analysis of the eigenstructure of the discrete single-curl operator in three-dimensional Maxwell’s equations with Pasteur media

Abstract

Abstract This paper focuses on studying the bifurcation analysis of the eigenstructure of the $\gamma $-parameterized generalized eigenvalue problem ($\gamma $-GEP) arising in three-dimensional source-free Maxwell’s equations with Pasteur media, where $\gamma $ is the magnetoelectric chirality parameter. The weakly coupled case, namely, $\gamma < \gamma _{\ast } \equiv $ critical value, the $\gamma $-GEP is positive definite, has been well studied by Chern et al. (2015, Singular value decompositions for single-curl operators in three-dimensional Maxwell’s equations for complex media. SIAM J. Matrix Anal. Appl., 36, 203–224). For the strongly coupled case, namely, $\gamma> \gamma _{\ast }$, the $\gamma $-GEP is no longer positive definite, introducing a totally different and complicated structure. For the critical strongly coupled case, numerical computations for electromagnetic fields have been presented by Huang et al. (2018, Solving three-dimensional Maxwell eigenvalue problem with fourteen Bravais lattices. Technical Report). In this paper, we build several theoretical results on the eigenstructure behaviour of the $\gamma $-GEPs. We prove that the $\gamma $-GEP is regular for any $\gamma> 0$, and the $\gamma $-GEP has $2 \times 2$ Jordan blocks of infinite eigenvalues at the critical value $\gamma _{\ast }$. Then we show that the $2 \times 2$ Jordan block will split into a complex conjugate eigenvalue pair that rapidly goes down and up and then collides at some real point near the origin. Next it will bifurcate into two real eigenvalues, with one moving towards the left and the other to the right along the real axis as $\gamma $ increases. A newly formed state whose energy is smaller than the ground state can be created as $\gamma $ is larger than the critical value. This stunning feature of the physical phenomenon would be very helpful in practical applications. Therefore, the purpose of this paper is to clarify the corresponding theoretical eigenstructure of the three-dimensional Maxwell equations with Pasteur media.