We demonstrate that although the well-known analogy between the time-independent solutions for two-dimensional tunneling (e.g., frustrated total internal reflection) and tunneling through a one-dimensional potential barrier cannot, in general, be extended to the time domain, there are certain limits in which the delay times for the two problems obey a simple relationship. In particular, when an effective mass is chosen such that
mc2=\ensuremath{\Elzxh}\ensuremath{\omega}, the ``classical'' traversal times for allowed transmission become identical for a photon of energy \ensuremath{\Elzxh}\ensuremath{\omega} traversing an air gap between regions of index n and for a particle of mass m traversing the analogous square barrier of height
V0 in one dimension. The quantum-mechanical group delays are also identical, given this effective mass, both for E\ensuremath{\approxeq}
V0 (\ensuremath{\theta}\ensuremath{\approxeq}${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$) and for E\ensuremath{\gg}
V0 (\ensuremath{\theta}\ensuremath{\ll}${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$). (For a smoothly varying potential or index of refraction, the agreement persists for all values of E where the WKB approximation applies.) The same relation serves to equate the quantum-mechanical ``dwell'' times for any values of E and
V0. On the other hand, in the ``deep tunneling'' limit, E\ensuremath{\ll}
V0 (\ensuremath{\theta}\ensuremath{\approxeq}\ensuremath{\pi}/2), one must choose
mc2=
n2\ensuremath{\Elzxh}\ensuremath{\omega} in order to make the group delays equal for the two problems. These equivalences simplify certain calculations, and the two-dimensional analogy may also be useful for geometrically visualizing the tunneling process and the anomalously small group delays known to occur in the opaque limit. We also demonstrate that the equality of the group delays for transmission and reflection for lossless barriers follows from a simple intuitive argument based on time-reversal invariance, and discuss the extension of the result to the case of lossy barriers.
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.
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.