On the Exact Maxwell evolution equation of resonator dynamics

TL;DR

This paper provides a rigorous derivation of the Maxwell evolution equation for resonator dynamics, clarifying the origin of the temporal derivative term in coupled-mode theory and addressing previous mathematical inaccuracies.

Contribution

It offers a mathematically rigorous derivation of the Maxwell evolution equation, clarifying the assumptions and origins of the temporal derivative term in classical coupled-mode theory.

Findings

The original derivation of the EME lacked mathematical rigor.

A new rigorous derivation clarifies the origin of the temporal derivative in the excitation term.

The approach explains the approximations in classical CMT.

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.