Wigner-Smith Time Delay Matrix for Electromagnetics: Systems with Material Dispersion and Losses

The Wigner-Smith (WS) time delay matrix relates a system's scattering matrix to its frequency derivative and gives rise to so-called WS modes that experience well-defined group delays when interacting with the system. For systems composed of nondispersive and lossless materials, the WS time delay matrix previously was shown to consist of volume integrals of energy-like densities plus correction terms that account for the guiding, scattering, or radiating characteristics of the system. This study extends the use of the WS time delay matrix to systems composed of dispersive and lossy materials. Specifically, it shows that such systems' WS time delay matrix can be expressed by augmenting the previously derived expressions with terms that account for the dispersive and lossy nature of the system, followed by a transformation that disentangles effects of losses from time delays. Analytical and numerical examples demonstrate the new formulation once again allows for the construction of frequency stable WS modes that experience well-defined group delays upon interacting with a system.