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Journal of Physics A Mathematical and General

Electromagnetic barrier penetration: wave packets and signals

Alojz Kodre, Janez Strnad

Published Apr 1, 1975• 2 citations
Publisher

Abstract

The time delay connected with the transmission of an electromagnetic wave across a barrier is investigated. The stationary phase method implies that the effective group velocity of the transmitted wave, inside the barrier, can exceed the velocity of light in vacuo. Consequently the transmission of a signal with a sharp front is studied to determine the upper bound for the signal velocity. The approximation employed does not yield a conclusive answer in general, while for the special case of critical incidence the proof is given that the effective signal velocity does not exceed the velocity of light in vacuo.

DOI
10.1088/0305-4470/8/4/015

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The time delay connected with the transmission of an electromagnetic wave across a barrier is investigated. The stationary phase method implies that the effective group velocity of the transmitted wave, inside the barrier, can exceed the velocity of light in vacuo. Consequently the transmission of a signal with a sharp front is studied to determine the upper bound for the signal velocity. The approximation employed does not yield a conclusive answer in general, while for the special case of critical incidence the proof is given that the effective signal velocity does not exceed the velocity of light in vacuo.
In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function V(x)V(x)V(x) exceeds the value of the total energy EEE, attention has been confined to wave functions of the form $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function V(x)V(x)V(x) is taken as follows: V(x),=0forx<0andforx>a,=V0>0for0<x<a.{V(x),=0 \mathrm{for} x<0 \mathrm{and}\mathrm{for} x>a,}{={V}_{0}>0 \mathrm{for} 0<x<a.}V(x),=0forx<0andforx>a,=V0​>0for0<x<a. A wave function is set up which initially represents a wave packet moving toward the point x=0x=0x=0 from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point x=ax=ax=a at about the time at which the incident packet reaches the point x=0x=0x=0, so that there is no appreciable delay in the transmission of the packet through the barrier.
We report on microwave measurements of barrier traversal time of electromagnetic wave packets in an undersized waveguide. This time was significantly shorter than the ratio L/cL/cL/c, with LLL the barrier length and ccc the velocity of light in vacuum. Consequently the transport velocity for the wave packet inside the barrier has to be superluminal, i.e. larger than ccc.
The time history of subsequent tunneled wave packets can represent a more meaningful way to describe signal evolutions and to determine the group delay than the detection of a demodulated monopulse envelope. In the present experimental research, the delay of transmitted tunneled packets was found to be shorter than that measurable in vacuum, in good agreement with the phase time and Esposito's equation and independent of the barrier length. The wave packets did not show narrowing and "reshaping," so these phenomena did not appear to be the reason of the short delay experienced. The latter was not found to be proportional to the amplitude of the incident signal.
An interesting analytical expression is obtained for the coordinate probability distribution of a wave packet transmitted through a potential barrier. In the large-time asymptotics the distribution is given in terms of the transmission amplitude and the phase-space quasidistribution of the initial (not necessarily pure) state. The distortion of a (broad) wave packet due to the barrier penetration is determined by two timelike parameters, given by the energy derivative of the complex transmission amplitude. The resulting change of the packet shape reveals a dependence on the detector position and incorporates a possible speed-up effect and a negative dispersion.
It is shown that, in the case of the supraluminal group velocity of a wave packet in a dispersion medium, a wave packet with a smooth (analytical) envelope does propagate with a supraluminal velocity. In the case of a negative group velocity, the signal maximum does arrive at the detector earlier than at the transmitter. These facts are consistent with both the finiteness of the velocity of light in free space for information transfer (in the case of supraluminal propagation velocity) and the principle of causality (in the case of negative delay time). Basically, the effect of negative delay time may be employed for predicting an observable effect.