AI 中文总结
本文研究线性周期介质中布洛赫模式问题的全场与布洛赫周期因子离散化，指出PF离散化违反协变性会导致非物理模式，强调保结构算法的重要性，其结果对光学等领域的相关计算有实际意义。
AI 中文摘要
线性周期介质中的布洛赫模式问题可分为四种主要形式。首先，布洛赫簇（布洛赫波矢与对应频率的配对）的切片方式有两种常见互补形式：频率对波矢，或反之波矢对频率。本文仅关注后者，其直接适用于损耗介质、倏逝波和复能带结构。另一个关键节点是全场（FF）与晶格周期布洛赫因子（PF）形式。二者在连续层面完全等价，但常用离散化技术可能不仅近似地、还会定性地打破这种等价性。在FF问题中，主要未知特征值是布洛赫相位因子（而非波数），仅通过对边界耦合引入；而PF形式则将布洛赫波数注入微分算子，导致体积二次 pencil。因此，本文所考虑的标准PF离散化，与对应的FF离散化相比，违反了倒格子（布里渊区偏移）协变性——正如本文的理论和数值例子所示，这可能导致不准确甚至非物理的模式。从数学角度，本文强调保结构算法的作用；结果与建议的实际重要性源于PF形式在光学、光子学等领域的广泛应用，例如光子和声结构中传播与倏逝布洛赫模式的计算、拓扑光子学、有效质量与拓扑绝缘体能带理论。
英文摘要
One can distinguish four major formulations of the Bloch-mode problem in linear periodic media. First, there are two common complementary ways of slicing the Bloch variety (Bloch wave vectors paired with the corresponding frequencies): frequency vs. wave vector or, conversely, wave vector vs. frequency. This paper deals exclusively with the latter option, directly applicable to lossy media, evanescent waves, and complex band structures. A separate crossroads is the full field (FF) vs. the lattice-periodic Bloch factor (PF) formulations. These are fully equivalent on the continuous level, but common discretization techniques may break this equivalence not just approximately but qualitatively. In the FF problem, the primary unknown eigenvalue is the Bloch phase factor (not the wavenumber), which enters only through opposite-boundary coupling. The PF formulation, on the other hand, injects the Bloch wavenumber into the differential operator and leads to a volume quadratic pencil. Consequently, standard PF discretizations of the type considered here, in contrast with the corresponding FF discretizations, violate the reciprocal-lattice (Brillouin-zone-shift) covariance -- which, as the theory and numerical examples in the paper show, may lead to inaccurate or even nonphysical modes. From the mathematical perspective, the paper highlights the role of structure-preserving algorithms. The practical importance of the results and recommendations stems from the wide adoption of PF formulations in optics, photonics, and beyond -- such as the computation of propagating and evanescent Bloch modes in photonic and acoustic structures, topological photonics, effective-mass and topological insulator band theories.
Comments9 pages, 6 figures
