The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities

Abstract

We show that the group delay in tunneling is not a traversal time but a lifetime of stored energy or stored probability escaping through both ends of the barrier. Because it is a lifetime associated with both forward (transmitted) and backward (reflected) fluxes, it cannot be used to define a group velocity for forward transit in cases where a wavepacket is mostly reflected. For photonic tunneling barriers the group delay is identical to the dwell time which is also a property of an entire wave function with reflected and transmitted components. Theoretical predictions and experimental reports of superluminal group velocities in barrier tunneling are re-interpreted.

Giant gain enhancement in photonic crystals with a degenerate band edge

“…Such a quantity is frequently used in the laser physics to express the resonator finesse [37,39,52] as well as to provide a good estimation of the Winger time, as explained in many different situations [51,[53][54][55][56][57]. Although lifetime and group delay are different concepts, they are intimately related.…”

“…Although lifetime and group delay are different concepts, they are intimately related. Indeed, for a uniform lossless FPC at resonance we have an exact identity c w τ τ = [55,57], which means that stored energy lifetime in such cavity identically equals to the time it takes the wave packet to traverse the whole FPC [57], but this identity is not exactly satisfied in general. The importance of c τ relies in its explicit link to the Qfactor via c Q ωτ = [39,52], whereas there exist approximations of the Q factor using the Wigner time [54].…”

Relationships among group delay, energy storage, and loss in dispersive dielectric mirrors

“…The fact that a relationship exists between the electromagnetic energy stored in the volume of a thin-film structure and its GD is known in the field of telecommunication technology [3] . This relationship is also a starting point in resolving the long-haul scientific problem of superluminal delay times of electromagnetic wave packets during transmission through highly reflective, lossless photonic bandgap structures [4] . In practical laser systems, however, one usually cannot neglect the absorption (or scattering) loss of dielectric multilayer mirrors because laser performance, such as intracavity loss, beam quality, maximum output power, laser damage threshold power, and others, strongly depends on these physical quantities.…”

“…Using the continuity equation for energy, Ref. [4] analytically showed that the weighted sum of the reflection GD (τ gr ) and the transmission GD (τ gt ) is equal to the so-called dwell time (τ d ), which is the ratio of the stored energy (U ) and the incident power (P i ): …”

“…[4], in which the transmission of a relatively narrowband light pulse through a lossless photonic barrier was investigated. Using the continuity equation for energy, Ref.…”