Completeness and divergence-free behavior of the quasi-normal modes using causality principle

Abstract

A fundamental feature of the quasi-normal modes (QNMs), which describe light interaction with open (leaky) systems like nanoparticles, lies in the question of the completeness of the QNMs representation and in the divergence of their field profile due to their leaky behavior and complex eigenfrequency. In this article, the QNMs expansion is obtained by taking into consideration the frequency dispersion and the causality principle. The derivation based on the complex analysis ensures the completeness of the QNMs expansion and prevents from any divergence of the field profile. The general derivation is tested in the case of a one-dimensional open resonator made of a homogeneous absorptive medium with frequency dispersion given by the Lorentz model. For a harmonic excitation, the result of the QNMs expansion perfectly matches the exact formula for the field distribution outside as well as inside the resonator.

Light Interaction with Photonic and Plasmonic Resonances

“…No rigorous mathematical proof exists, but the conjecture is supported by many numerical examples . In a very recent work published during the proofreading of the present review, the possibility to ensure completeness outside the resonator volume with expansions based on regularized QNMs has been demonstrated mathematically for 1D open resonators and confirmed by numerical simulations. This appears to be a very important theoretical finding that may represent a new departure running counter the longstanding belief that QNM expansions are complete only inside resonators.…”

Transformation optics for plasmonics: from metasurfaces to excitonic strong coupling

“…is not yet a consensus about the precise definition of these EM modes, their associated eigenfunctions and eigenvalues. As a consequence, various terms and definitions, such as resonant states [103], generalized normal [104] or quasinormal [105][106][107] modes, have been coined lately to refer to them. Indeed, the conception of a theoretical framework allowing for a general Green's function decomposition would mean a significant advance in multiple areas.…”

Quasinormal modes expansions for nanoresonators made of absorbing dielectric materials: study of the role of static modes

“…Modes are the only ingredients of these approaches, which are based on the assumption that the QNMs supported by the resonator forms a complete set. Note that the completeness has been demonstrated for simple geometries such as slabs and spheres 7,8,14,17 but not for more complex systems. Within these formalisms, if a non-resonant contribution to the scattered field shall exist, it should somehow be included in the modal expansion.…”

“…We refer to them in the following as quasinormal modes, abbreviated in QNMs, or simply modes. Owing to the non-Hermitian character of the system and the spectral dispersion of the constituent materials, the development of QNM theories for optical resonators is a non-trivial task, which, after initial works in the 90's 7,8 , has recently received much attention 6,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] . In simple words, QNM theories translate a physical observation into mathematical terms by describing the system response with a modal expansion, or decomposition, which is nothing more than a sum of resonant terms.…”

“…These questions have been explicitly or implicitly addressed by recent works on QNM theory, but no clear answers emerge. On the one hand, most approaches leading to a modal theory do not introduce an explicit non-resonant term and express the scattered field or the Green tensor as a sum of modal contributions 7,8,[10][11][12][13][14][15][16][17][18]23,24 . Modes are the only ingredients of these approaches, which are based on the assumption that the QNMs supported by the resonator forms a complete set.…”