Biorthogonality of Guided Modes in Non-Hermitian Electromagnetism
Abstract
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.
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BibTeX
@misc{abdrabou2026biorthogonality,
title = {{Biorthogonality of Guided Modes in Non-Hermitian Electromagnetism}},
author = {Amgad Abdrabou and M. A. Swillam and Şahin K. Özdemir and R. El-Ganainy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28153},
eprint = {2609.28153},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28153},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Biorthogonality of Guided Modes in Non-Hermitian Electromagnetism
AU - Amgad Abdrabou
AU - M. A. Swillam
AU - Şahin K. Özdemir
AU - R. El-Ganainy
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28153
UR - https://arxiv.org/abs/2609.28153
AB - 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.
ER -