Testing the reliability of a velocity definition in dispersive medium

TL;DR

This paper presents a method to verify if a velocity definition in dispersive media reflects actual physical flow by comparing two Fourier-based calculations of velocity, revealing discrepancies in superluminal regions.

Contribution

The authors introduce a novel approach to test the physical validity of velocity definitions in dispersive media using Fourier expansions and comparison of two velocity calculations.

Findings

Significant differences (4%) in velocities in superluminal regions.

The method challenges the physical interpretation of certain velocity definitions.

Discrepancies persist even for established velocity definitions.

Abstract

We introduce a method to test if a given velocity definition corresponds to an actual physical flow, when the pulse propagation in dispersive medium is considered. i) We calculate the mean arrival time between two positions in space, using the Fourier expansion in real-\omega space. ii) We calculate the mean spatial displacement between two points in time, using the Fourier expansion in real-k space. We compare the velocities calculated in the two approaches. If the velocity definition truly corresponds to an actual flow, the two velocities must be the same. However, we show that the two velocities differ significantly (4%) in the region of superluminal propagation even for the successful definition introduced by Peatross et al. [Phys. Rev. Lett. 84, 2370 (2000)].