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

TL;DR

This paper extends the Wigner-Smith time delay matrix framework to electromagnetic systems with dispersive and lossy materials, enabling accurate analysis of mode delays in more realistic scenarios.

Contribution

It introduces a new formulation of the WS time delay matrix that accounts for material dispersion and losses, enhancing its applicability to practical electromagnetic systems.

Findings

The extended WS matrix accurately characterizes mode delays in dispersive, lossy systems.

Analytical and numerical examples validate the new formulation.

Constructs frequency-stable WS modes with well-defined group delays.

Abstract

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.