Average and instantaneous velocities of energy of evanescent modes

TL;DR

This paper clarifies that while instantaneous energy velocities of evanescent modes can appear superluminal, their average energy velocity remains subluminal, aligning with causality principles in quantum field theory.

Contribution

It demonstrates that the average energy velocity of evanescent modes in a waveguide is always less than or equal to the speed of light, resolving apparent superluminal propagation claims.

Findings

Average energy velocity is ≤ speed of light in vacuum.

Instantaneous energy velocity can be superluminal.

Superluminal instantaneous velocities do not violate causality.

Abstract

Many theoretical and experimental investigations have presented a conclusion that evanescent electromagnetic modes can superluminally propagate. However, in this paper, we show that the average energy velocity of evanescent modes inside a cut-off waveguide is always less than or equal to the velocity of light in vacuum, while the instantaneous energy velocity can be superluminal, which does not violate causality according to quantum field theory: the fact that a particle can propagate over a space-like interval does preserve causality provided that here a measurement performed at one point cannot affect another measurement at a point separated from the first with a space-like interval.