Group velocity and nonlocal energy transport velocity in finite photonic structures

Abstract

We have derived an explicit relation between the group delay velocity v gd ω, determined by the slope of the dispersion curves, and the nonlocal energy transport velocity v E ω in a lossless, finite, one-dimensional photonic crystal. Using this relation, we find a simple link between v gd ω and the group velocity v ω g ω defined in terms of the electromagnetic dwell time. It is shown that v E ω ≤ v gd ω for any frequency and v E ω v gd ω v ω g 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.