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Temporal coupled-mode theory (CMT) is an acclaimed and widely usedtheoretical framework for modeling the continuous wave (CW) response andtemporal dynamics of any integrated or free-space photonic resonant structure.It was initially employed to understand how energy is coupled into and out of acavity and how it is exchanged between different resonant modes. In the 30years that followed its establishment, CMT has been expanded to describe abroad range of nonlinear interactions as well (self- and cross-phasemodulation, saturable absorption, frequency generation, gain, etc.). In thistutorial, we thoroughly present the basic principles and the evolution of CMTthroughout the years, showcasing its immense capabilities for the analysis anddesign of linear and nonlinear resonant photonic systems. Importantly, we focuson examples of modern, open nanophotonic resonators incorporating contemporarybulk or sheet (2D) materials that may be lossy and dispersive. For eachlinear/nonlinear effect under study we follow a meticulous, step-by-stepapproach, starting from an accurate model of the physical phenomenon andproceeding to its introduction in the CMT framework all the way to theefficient solution of the resulting system of equations. Our work highlightsthe merits of CMT as an efficient, accurate, and versatile theoretical tool. Weenvision that it can serve both as an introductory reference for any reader, aswell as a comprehensive handbook on how to incorporate a broad range of linearand nonlinear effects in the CMT framework.

Temporal coupled-mode theory in nonlinear resonant photonics: From basic  principles to contemporary systems with 2D materials, dispersion, loss, and  gain