Nature of “Superluminal" Barrier Tunneling

We show that the distortionless tunneling of electromagnetic pulses through a barrier is a quasistatic process in which the slowly varying envelope of the incident pulse modulates the amplitude of a standing wave. For pulses longer than the barrier width, the barrier acts as a lumped element with respect to the pulse envelope. The envelopes of the transmitted and reflected fields can adiabatically follow the incident pulse with only a small delay that originates from energy storage. The theory presented here provides a physical explanation of the tunneling process and resolves the mystery of apparent superluminality.

Figure 1

The electric and magnetic fields in the barrier over a few cycles of the optical carrier wave. Here ${\beta }_{0}L=20\pi$ and $\kappa L=4$. The envelopes are in steady state.

Figure 2

(a) Incident (dotted), transmitted (solid line), and reference (dashed) pulses for the barrier of strength $\kappa L=4$. The input narrowband pulse has normalized width ${\tau }_p=3$ and a peak at $t_0=5{\tau }_p$. The tunneled pulse is undistorted, has a peak intensity of $1.4\times {10}^{-3}$, and is delayed by $t_d^{\prime }=0.25$. (b) Snapshots of the spatial distribution of energy density in the tunneling pulse. These are taken at time instants from $t^{\prime }=13$ to $t^{\prime }=17$ in steps of 0.25 around the peak of the incident pulse. Note that the entire distribution moves up and down in phase.

Figure 3

(a) Incident (dotted), transmitted (solid line), and reference pulse (dashed) for a short pulse of width ${\tau }_p=0.1$. (b) Snapshots of spatial distribution of energy density in the “tunneling” pulse of width ${\tau }_p=0.1$.