J. Eur. Opt. Society-Rapid Publ. 2025, 21, 5 Ó The Author( s), published by EDP Sciences, 2025 https:// doi. org / 10.1051 / jeos / 2024048 Available online at: https:// jeos. edpsciences. org
Journal of the European Optical Society-Rapid Publications
EOSAM 2024 Guest editors: Luca De Stefano and Raffaele Velotta
RESEARCH ARTICLE
On the exact Maxwell evolution equation of resonator dynamics
Tong Wu 1, Rachid Zarouf 2, 3, and Philippe Lalanne 1,* 1 Laboratoire Photonique, Numérique et Nanosciences( LP2N), IOGS – Université de Bordeaux-CNRS, 33400 Talence cedex, France 2 Aix-Marseille Université, Laboratoire ADEF, Campus Universitaire de Saint-Jérôme, 52 Avenue Escadrille Normandie Niémen,
13013 Marseille, France 3 CPT, Aix-Marseille Université, Université de Toulon, 13288 Marseille, France
Received 29 October 2024 / Accepted 10 December 2024
Abstract. In a recent publication [ Opt. Express 32, 20904( 2024)], the accuracy of the main evolution equation that governs resonator dynamics in the coupled-mode theory( CMT) was questioned. The study concluded that the driving force is proportional to the temporal derivative of the excitation field rather than the excitation field itself. This conclusion was reached with a derivation of an“ exact” Maxwell evolution( EME) equation obtained directly from Maxwell’ s equations, which was further supported by extensive numerical tests. Hereafter, we argue that the original derivation lacks mathematical rigor. We present a direct and rigorous derivation that establishes a solid mathematical foundation for the EME equation. This new approach clarifies the origin of the temporal derivative in the excitation term of CMT and elucidates the approximations present in the classical CMT evolution equation through a straightforward argument.
Keywords: Coupled-mode theory, Electromagnetic resonance, Quasinormal mode, Resonant scattering.
1 Introduction
Many concepts in physics are based on normal modes of conservative systems, such as molecular orbitals and excitation energies. In these self-adjoint problems, the response of the system can be represented as a sum over the complete set of normal modes.
However, when energy dissipation occurs through processes like absorption or leakage in open systems, the system ceases to be conservative. The corresponding operator becomes non-self-adjoint and its spectrum becomes continuous. The spectrum also includes an infinite set of quasinormal modes( QNMs) characterized by complex eigenfrequencies( or eigenenergies), which produce characteristic damped oscillations in the system’ s temporal response. A significant challenge across various disciplines [ 1 – 4 ] focuses on using QNM expansions to represent this response, akin to the methods applied to closed systems with normal modes.
In electromagnetism, this challenge was effectively addressed long ago with the temporal coupled-mode theory( CMT) [ 5, 6 ]. CMT offers a computationally effective, systematic, and intuitive framework for characterizing interactions between different modes within resonators and predicting their temporal dynamics. As a result, it
* Corresponding author: philippe. lalanne @ institutoptique. fr has become an essential tool for designing and optimizing optical resonators, facilitating the development of innovative photonic technologies in both linear [ 3, 4, 7 ] and nonlinearoptics [ 8 ].
Typically, CMT assumes the presence of m modes and n ports. In this context, a mode corresponds to a QNM, while a port represents a propagating channel where the QNMs may decay. It is perhaps worth noting that our understanding of ports may not fully reflect reality, as resonators are open systems that dissipate energy across a continuum of“ ports”. A typical scenario involves a resonator on a layered substrate, illuminated by a plane wave( one of the system’ s many ports). This wave scatters into a continuum of ports composed of all the plane waves in the substrate or superstate, as well as any potential guided modes of the layered substrate [ 9 ].
The CMT equations consist of two primary equations [ 5 – 8 ]. The first, known as the evolution equation, describes how the amplitudes of the modes evolve in response to incoming waves from the ports. This equation is crucial for understanding the dynamics of the resonator. The second equation coherently combines resonant-assisted and background pathways to predict the fields that couple out into the ports, enabling the calculation of reflection and transmission coefficients for the various ports.
Despite its extensive application and many successes, CMT remains a phenomenological theory for Fano- Feshbach resonances [ 10 ]; all coefficients describing the
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