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

TL;DR

This paper extends the Wigner-Smith time delay matrix concept to systems with both propagating and evanescent electromagnetic modes, enabling better characterization of time delays in complex guiding and periodic structures.

Contribution

It introduces a generalized WS relationship applicable to mixed propagating and evanescent fields and develops a method to compute the WS matrix of composite systems from their subsystems.

Findings

Generalized WS relationship applicable to evanescent modes.

Method to compute composite WS matrices from subsystem matrices.

Numerical validation on guiding and periodic structures.

Abstract

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 paper 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.