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

The scattering of electromagnetic waves by resonant systems is determined by the excitation of the quasinormal modes (QNMs), i.e. the eigenmodes, of the system. This Review addresses three fundamental concepts in relation to 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 guarantees that the scattered field can be faithfully expanded into the complete QNM basis. These concepts are not trivial for non-conservative (non-Hermitian) systems and have driven many theoretical developments since initial studies in the 70's. Yet, they are not easy to grasp from the extensive and scattered literature, especially for newcomers in the field. After recalling fundamental results obtained in initial studies on the completeness of the QNM basis for simple resonant systems, we review recent achievements and the debate on the normalization, clarify under which circumstances the QNM basis is complete, and highlight the concept of QNM regularization with complex coordinate transforms. (C) 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

Automated summary

This Review addresses the scattering of electromagnetic waves by resonant systems and focuses on representing the scattered field as a superposition of the excited quasinormal modes (QNMs). It discusses the fundamental concepts of normalization, orthogonality, and completeness, which are crucial for understanding the excitation strength and basis expansion of QNMs. The paper reviews recent achievements, debates on normalization, and introduces the concept of QNM regularization with complex coordinate transforms.