Wigner–Smith Time-Delay Matrix for Electromagnetics: Guiding and Periodic Systems With Evanescent Modes

The Wigner–Smith (WS) time-delay matrix relates an electromagnetic system’s scattering matrix and its frequency derivative. Previous work showed that the entries of WS time-delay matrices of systems excited by propagating waves consist of volume integrals of energy-like field quantities. This article introduces a generalized WS relationship that applies to systems excited by mixtures of propagating and evanescent fields. Just like its predecessor, the generalized WS relationship allows for the identification of so-called WS modes that interact with the system with well-defined time delays. Furthermore, a technique is developed to compute the WS time-delay matrix of a composite system from the WS time-delay matrices of its subsystems. Numerical examples demonstrate the usefulness of the generalized WS method when characterizing time delays experienced by fields interacting with guiding and periodic structures that have ports supporting evanescent modes.