Research Article

On the exact Maxwell evolution equation of resonator dynamics

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

^* Corresponding author: philippe.lalanne@institutoptique.fr

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.