Testing the reliability of a velocity definition in a dispersive medium

Abstract

We introduce a method to test if a given velocity definition corresponds to an actual physical flow in a dispersive medium. We utilize the equivalence of the pulse dynamics in the real-ω and real-k Fourier expansion approaches as a test tool. To demonstrate our method, we take the definition introduced by Peatross et al. [Phys. Rev. Lett. 84, 2370] and calculate the velocity in two different ways. We calculate (i) the mean arrival time between two positions in space, using the real-ω Fourier expansion for the fields and (ii) 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 (3%) in the region of superluminal propagation even for the successful definition of Peatross et al.