Wigner–Smith Time-Delay Matrix for Electromagnetics: Theory and Phenomenology

The Wigner-Smith (WS) time-delay concepts have been used extensively in quantum mechanics to characterize delays experienced by particles interacting with a potential well. This article formally extends WS time-delay theory to Maxwell's equations and explores its potential applications in electromagnetics. The WS time-delay matrix relates a lossless and reciprocal system's scattering matrix to its frequency derivative and allows for the construction of modes that experience well-defined group delays when interacting with the system. The matrix' entries for guiding, scattering, and radiating systems are energy-like overlap integrals of the electric and/or magnetic fields that arise upon excitation of the system via its ports. The WS time-delay matrix has numerous applications in electromagnetics, including the characterization of group delays in multiport systems, the description of electromagnetic fields in terms of elementary scattering processes, and the characterization of frequency sensitivities of fields and multiport antenna impedance matrices.