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

Abstract

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.

Tailoring Iridescent Visual Appearance with Disordered Resonant Metasurfaces

“…Theoretically, an infinite number of QNMs should be retained in the expansion. 36 However, only a few modes (four QNMs are sufficient hereafter) are necessary to quantitatively match the experimental data. In our view, it is crucial for inverse design to have the kind of advanced conceptualization provided by the modal expansion in which a highly multidimensional function, the form factor in this case, is represented by only a few physical quantities.…”

“…Theoretically, an infinite number of QNMs should be retained in the expansion . However, only a few modes (four QNMs are sufficient hereafter) are necessary to quantitatively match the experimental data.…”

A Theoretical Perspective on Molecular Polaritonics

“…Several methods to achieve such a few-mode quantized description have been developed in the past few years. One is based on quasinormal modes, which are eigenmodes of the Maxwell equations including losses with complex frequencies. , They can be used to expand the electric field solutions based on a master equation approach or explicitly quantized such that the EM field is represented in terms of discrete bosonic modes. , In this approach, the discrete modes are defined as superpositions of the bosonic field operators f̂ λ ( r , ω) of macroscopic QED with coefficients determined by the quasinormal modes obtained from classical EM calculations, and the resulting modes are orthonormalized to obtain approximate discrete lossy modes.…”

Microwave Vortex Transceiver System with Continuous Tunability Using Identical Plasmonic Resonators

“…The orthogonality of the two dipoles can be verified in the sense that [ 47 ] $$\[ \begin{array}{*{20}{c}}{\mathop \smallint \limits_{ - \infty }^{ + \infty } \mathop \smallint \limits_{ - \infty }^{ + \infty } {{{\bf E}}_1}\;\cdot\;{{{\bf E}}_2}dxdy = 0}\end{array} \]$$and $$\[ \begin{array}{*{20}{c}}{\mathop \smallint \limits_{ - \infty }^{ + \infty } \mathop \smallint \limits_{ - \infty }^{ + \infty } {{{\bf H}}_1}\;\cdot\;{{{\bf H}}_2}dxdy = 0}\end{array} \]$$in which E 1 and E 2 are their vectorial electric fields and H 1 and H 2 are their vectorial magnetic fields. The integral is normalized as: $$\[{{\mathop \smallint \limits_{ - \infty }^{ + \infty } \mathop \smallint \limits_{ - \infty }^{ + \infty } {{{\bf E}}_1}\;\cdot\;{{{\bf E}}_2}dxdy} \mathord{\left/ {\vphantom {{\mathop \smallint \limits_{ - \infty }^{ + \infty } \mathop \smallint \limits_{ - \infty }^{ + \infty } {{{\bf E}}_1}\;\cdot\;{{{\bf E}}_2}dxdy} {\mathop \smallint \limits_{ - \infty }^{ + \infty } \mathop \smallint \limits_{ - \infty }^{ + \infty } {{{\bf E}}_1}\;\cdot\;{{{\bf E}}_1}dxdy}}} \right.$$…”