发表机构
Saint Louis University; The American University in Cairo（圣路易斯大学; 开罗美国大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立非厄米波导中电场双正交性与洛伦兹互易模态框架的统一算子理论，阐明左本征模与反向传播模式的关系及异常点自正交性。
AI 中文摘要
双正交性在非厄米光子系统的分析中起着核心作用，其中模态展开通常需要右本征态和左本征态。尽管这些概念在标量和耦合模描述中较为直接，但在全矢量麦克斯韦框架中的解释却不那么透明。特别是，由非共轭洛伦兹互易导出的标准电磁模态正交关系涉及电场和磁场，其形式不同于从常规伴随算子论证获得的双正交关系。在此，我们建立了这两种表述在具有增益和损耗的非厄米波导系统之间的联系。从全矢量电场波动方程出发，我们将导模问题表述为传播常数中的二次本征值问题，构造其伴随问题，并识别相应的左本征模。我们证明左本征模与反向传播电场模式的复共轭相关，并展示所得电场双正交配对等价于从非共轭洛伦兹互易获得的混合电磁场关系。该框架还阐明了模态归一化的电磁意义、其与异常点处自正交性的联系，以及在无损耗极限下恢复常规共轭模态正交性。这些结果为电磁双正交性和洛伦兹互易提供了统一的算子理论解释，并在非厄米物理的左/右本征态语言与经典电磁学的基于互易的模态框架之间建立了直接联系。
英文摘要
Biorthogonality plays a central role in the analysis of non-Hermitian photonic systems, where modal expansions generally require both right and left eigenstates. While these concepts are straightforward within scalar and coupled-mode descriptions, their interpretation in the full vectorial Maxwell framework is less transparent. In particular, the standard electromagnetic modal orthogonality relation derived from unconjugated Lorentz reciprocity involves both electric and magnetic fields and differs in form from the biorthogonality relations obtained from conventional adjoint-operator arguments. Here, we establish the connection between these two formulations for non-Hermitian waveguiding systems with gain and loss. Starting from the full vectorial electric-field wave equation, we formulate the guided-mode problem as a quadratic eigenvalue problem in the propagation constant, construct its adjoint, and identify the corresponding left eigenmodes. We show that the left eigenmode is related to the complex conjugate of the backward-propagating electric-field mode and demonstrate that the resulting electric-field biorthogonal pairing is equivalent to the mixed electric--magnetic field relation obtained from unconjugated Lorentz reciprocity. The framework also clarifies the electromagnetic meaning of modal normalization, its connection to self-orthogonality at exceptional points, and the recovery of conventional conjugated modal orthogonality in the lossless limit. These results provide a unified operator-theoretic interpretation of electromagnetic biorthogonality and Lorentz reciprocity and establish a direct connection between the left/right eigenstate language of non-Hermitian physics and the reciprocity-based modal framework of classical electromagnetism.
