Normalization, orthogonality and completeness of quasinormal modes of open systems: the case of electromagnetism

TL;DR

This paper reviews the fundamental concepts of normalization, orthogonality, and completeness of quasinormal modes in electromagnetic open systems, clarifying recent developments and debates to aid understanding of their role in scattering analysis.

Contribution

It provides a comprehensive review of recent advances and clarifies key concepts related to QNM normalization, orthogonality, and completeness in non-Hermitian electromagnetic systems.

Findings

Clarification of conditions for QNM basis completeness

Discussion of QNM normalization and orthogonality in non-Hermitian systems

Introduction of QNM regularization with complex coordinate transform

Abstract

The scattering of electromagnetic waves by resonant systems is determined by the excitation of quasinormal modes (QNMs), i.e., the eigenmodes of the system. This Review addresses three fundamental concepts in relation with the representation of the scattered field as a superposition of the excited QNMs: normalization, orthogonality, and completeness. Orthogonality and normalization enable a straightforward assessment of the QNM excitation strength for any incident wave. Completeness guaranties that the scattered field can be faithfully expanded into the QNM basis. These concepts are non-trivial for non-conservative (non-Hermitian) systems and have driven many theoretical developments since initial studies in the 70's. Yet, owing to recent achievements, they are not easily grasped from the extensive and scattered literature, especially for newcomers in the field. After recalling fundamental results obtained in early studies on the completeness of the QNM basis for simple resonant systems, we review recent achievements and debates on the normalization, clarify under which circumstances the QNM basis is complete, and highlight the concept of QNM regularization with complex coordinate transform.