The Maxwell evolution equation of electromagnetic resonators: a mathematical proof with explicit derivation

In a recent publication [1], the authors employ electromagnetic quasinormal-mode theory to establish an "exact" Maxwell evolution (EME) equation governing resonator dynamics. This resulting equation bears resemblance to the classical evolution equation proposed heuristically in the temporal coupled-mode theory (CMT), yet it also presents differences. One significant difference is that the driving force in the EME equation is proportional to the derivative of the incident field, whereas in the CMT evolution equation, it is proportional to the incident field itself. This unexpected finding was substantiated by a numerical test and a mathematical demonstration in [1]. The test remains undisputed in this study. However, the demonstration unfortunately relies on unjustified shortcuts. We promptly highlight these shortcuts while proposing an alternative demonstration that is direct and indisputable. The new demonstration provides a firm mathematical foundation for the EME equation and reinforces its credibility.