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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Group delay, stored energy, and the tunneling of evanescent electromagnetic waves

Herbert G. Winful

Published Jul 28, 2003• 78 citations
Publisher

Abstract

A general relation between group delay and stored electric and magnetic energies is presented for two-port networks. It generalizes the results of Dicke to situations where electric and magnetic stored energies differ. The general result is applied to tunneling evanescent waves in cutoff waveguides. It is shown explicitly that the group delay is equal to the dwell time plus a self-interference delay which is proportional to the net reactive stored energy. The Hartman effect, the saturation of group delay with length in cutoff waveguides, is explained on the basis of saturation of stored energy with guide length. It is pointed out that the anomalously short delays observed in tunneling experiments are not propagation delays and should not be associated with superluminal velocities. A strictly luminal energy velocity is derived and a method is suggested for the measurement of dwell time and energy velocity.

DOI
10.1103/physreve.68.016615

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A general relation between group delay and stored electric and magnetic energies is presented for two-port networks. It generalizes the results of Dicke to situations where electric and magnetic stored energies differ. The general result is applied to tunneling evanescent waves in cutoff waveguides. It is shown explicitly that the group delay is equal to the dwell time plus a self-interference delay which is proportional to the net reactive stored energy. The Hartman effect, the saturation of group delay with length in cutoff waveguides, is explained on the basis of saturation of stored energy with guide length. It is pointed out that the anomalously short delays observed in tunneling experiments are not propagation delays and should not be associated with superluminal velocities. A strictly luminal energy velocity is derived and a method is suggested for the measurement of dwell time and energy velocity.
We show that the anomalously short delay times observed in barrier tunneling have their origin in energy storage and its subsequent release. The observed group delay is proportional to the energy stored. This delay is not a propagation delay and should not be linked to a velocity since evanescent waves do not propagate. The "Hartman effect", in which the group delay becomes independent of thickness for opaque barriers, is shown to be a consequence of the saturation of stored energy with barrier length.
We show that the group delay in tunnelling 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 tunnelling barriers the group delay is identical to the dwell time which is also a property of an entire wavefunction with reflected and transmitted components. Theoretical predictions and experimental reports of superluminal group velocities in barrier tunnelling are re-interpreted.
The group delay in frustrated total internal reflection (FTIR) is shown to equal the dwell time plus a self-interference delay. The self-interference delay originates from the energy stored in the overlap between incident and reflected waves above the interface. The group delay in FTIR for finite beams is thus shown to be the lifetime of stored energy in the gap between the prisms and in the partial standing wave above the gap. As in other manifestations of tunneling the saturation of group delay with barrier length (Hartman effect) is explained by the saturation of stored energy in the region defined. No superluminal propagation is implied.
Tunneling without distortion is not a propagation phenomenon but a quasistatic process in which the slowly varying envelope of the input pulse modulates the amplitude of a standing wave in the barrier. Because the envelope has a bandwidth that is small compared to the cutoff frequency of the barrier it does not propagate; it merely stands and waves. The envelope of the reflected and transmitted fields can adiabatically follow the envelope of the input pulse with only a small delay proportional to the energy stored in the barrier. The Hartman effect, in which the delay becomes independent of barrier length, is explained on the basis of saturation of stored energy with increasing length. Because the delay is not a propagation delay, it should not be linked to a velocity of propagation. True evanescent waves have zero speed and not infinite velocity as often asserted.
The analogy between quantum tunneling of particles and evanescent electromagnetic waves can be used to study particle tunneling. Possibilities for the measurement of the time delay resulting from the transmission of waves through an evanescent region, say with lower dielectric constant, are discussed. In contrast to particle tunneling, an electromagnetic pulse can consist of many photons and can be probed in a noninvasive way. We emphasize the delay of the centroid of an electromagnetic pulse transmitted below cutoff through a portion of a waveguide with the same cross section as the adjacent propagating guides. The boundary conditions at the interface between the propagating and the evanescent region lead to the same transmission and reflection coefficients as for a square-barrier tunneling problem. For a pulse restricted to a narrow frequency range, the time delay depends only on the frequency derivative of the phase shift associated with transmission. The delay of a centroid is just that; there is no deeper physical sense which links the incoming centroid to the outgoing centroid. For sufficiently long evanescent regions, the delay is independent of thickness.