Time-Varying Materials in Presence of Dispersion: Plane-Wave Propagation in a Lorentzian Medium with Temporal Discontinuity

TL;DR

This paper investigates how a monochromatic plane wave interacts with a Lorentzian medium experiencing an abrupt change in plasma frequency, revealing the need for additional boundary conditions due to dispersion effects.

Contribution

It introduces a novel analysis of plane-wave scattering at a temporal discontinuity in a dispersive Lorentzian medium, highlighting the necessity of extra boundary conditions.

Findings

Two shifted frequencies arise due to dispersion during temporal discontinuity.

Four plane waves (forward and backward) are generated instantaneously.

A transmission-line model effectively describes the dispersive time-discontinuous scenario.

Abstract

We study the problem of a temporal discontinuity in the permittivity of an unbounded medium with Lorentzian dispersion. More specifically, we tackle the situation in which a monochromatic plane wave forward-travelling in a (generally lossy) Lorentzian-like medium scatters from the temporal "half-space interface" that results from an abrupt temporal change in its plasma frequency (while keeping its resonance frequency constant). In order to achieve momentum preservation across the temporal discontinuity, we show how, unlike in the well-known problem of a nondispersive discontinuity, the second-order nature of the dielectric function now gives rise to two shifted frequencies. As a consequence, whereas in the nondispersive scenario the continuity of the electric displacement D and the magnetic induction B suffice to find the amplitude of the new forward and backward wave, we now need two extra temporal boundary conditions. That is, two forward and two backward plane waves are now instantaneously generated in response to a forward-only plane wave. We also include a transmission-line equivalent with lumped circuit elements that describes the dispersive time-discontinuous scenario under consideration.