Understanding Band Structures in Metamaterials Through Generalized Eigenvalue Equations

Recent research by Takuma Isobe, Tsuneya Yoshida, and Yasuhiro Hatsugai has explored the band structures of metamaterials through the lens of generalized eigenvalue equations. Their paper, titled "Band structures of generalized eigenvalue equation and conic section," was submitted to arXiv on September 2, 2024, and is accessible via the identifier arXiv:2409.01191.

The authors delve into how complex bands can emerge in systems with Hermitian matrices, providing a geometrical understanding of the transition between real and complex band structures. This transition is linked to the Lifshitz transition observed in electron systems. The study further establishes a correspondence between the real and complex bands of photonic systems and the Fermi surfaces of Dirac cones in electron systems, particularly when the permittivity (ε) and permeability (μ) are frequency-independent.

Additionally, the research highlights that exceptional points (EPs) in the photonic system are influenced by the frequency dependence of ε and μ. This insight could have significant implications for the design and application of metamaterials in various optical technologies, potentially enhancing their functionality and efficiency.

The findings are expected to advance the understanding of metamaterials and their applications in optics, paving the way for innovations in photonic devices and materials science.