Group velocity and nonlocal energy transport velocity in finite photonic structures

Abstract

We have derived an explicit relation between the group delay velocity vgd(ω), determined by the slope of the dispersion curves, and the nonlocal energy transport velocity vE(ω) in a lossless, finite, one-dimensional photonic crystal. Using this relation, we find a simple link between vgd(ω) and the group velocity vg(ω)(ω) defined in terms of the electromagnetic dwell time. It is shown that vE(ω)≤vgd(ω) for any frequency and vE(ω)=vgd(ω)=vg(ω)(ω) at the resonance frequencies of the transmission coefficient. It was established that the band structure of a finite periodic crystal is of great importance to describe and understand the properties of these velocities. In particular, it was shown that the occurrence of superluminal group delay velocities in these structures is closely related to the existence of null gaps in their dispersion relation. Calculations performed for a half-wave-quarter-wave stack show that the energy velocity remains always subluminal.