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

Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time

Anedio Ranfagni, Pasquale Fabeni, Gian Paolo Pazzi, Daniela Mugnai

Published Aug 1, 1993• 156 citations
Publisher

Abstract

Measures of pulse delay in microwave propagation, in open air and for short distances (not much greater than 1 m), were made by using launcher and receiver horns. When these are facing each other we observe a delay time corresponding to a speed equal to c while, if the receiver horn is shifted or tilted with respect to the launcher horn, the delay time decreases showing a superluminal behavior. In other words the modulation phase shift, interpreted as a propagation time, turns out to be surprisingly smaller than the one relative to the light speed. This effect, which disappears for longer distances, is here interpreted on the basis of the existence of a special kind of evanescent waves (leaky waves). Just the presence of evanescent waves allows one to make a comparison with the tunneling processes where superluminal transport properties have been theoretically predicted.

DOI
10.1103/physreve.48.1453

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Measures of pulse delay in microwave propagation, in open air and for short distances (not much greater than 1 m), were made by using launcher and receiver horns. When these are facing each other we observe a delay time corresponding to a speed equal to c while, if the receiver horn is shifted or tilted with respect to the launcher horn, the delay time decreases showing a superluminal behavior. In other words the modulation phase shift, interpreted as a propagation time, turns out to be surprisingly smaller than the one relative to the light speed. This effect, which disappears for longer distances, is here interpreted on the basis of the existence of a special kind of evanescent waves (leaky waves). Just the presence of evanescent waves allows one to make a comparison with the tunneling processes where superluminal transport properties have been theoretically predicted.
There have been many recent theoretical and experimental reports on the propagation of light pulses at speeds exceeding the speed of light in vacuum ccc within media with anomalous dispersion, either opaque or with gain. Superluminal propagation has also been reported within vacuum, in the case of inhomogeneous pulses. In this paper we show that the observations of superluminal and non-causal propagation of evanescent pulses under the conditions of frustrated internal reflection are only apparent, and that they can be simply explained employing an explicitly (sub)luminal causal theory. However, the usual one-dimensional approach to the analysis of pulse propagation has to be abandoned and the spatial extent of the incoming pulse along the directions normal to the propagation direction has to be accounted for to correctly interpret the propagation speed of these evanescent waves. We illustrate our theory with animations of the time development of a pulse built upon the Huygen's construction.
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.
The concept of time diffraction of evanescent electromagnetic waves is proposed. Transient propagation of electromagnetic waves is studied in two forms: propagating and evanescent waves. Differences and similarities between quantum particle tunneling and photon tunneling are clearly demonstrated. Traversal time cannot be accurately defined due to diffraction in time. Nevertheless, a delay time is defined as the difference between the time of flight and the peak time of the transient light. The delay time is found to increase linearly and slowly with the traversal distance. Superluminal tunneling is found possible only for evanescent waves, and Einstein causality is not violated.
An alternative model for near-field propagation and optical tunneling is proposed following the lines of the path-integral method developed by Feynman, and in particular by using a transition-elements analysis. Such a model was able to account for the frequency dependency of delay-time results of an experiment involving microwave propagation in the near field using two horn antennas [A. Ranfagni et al., Phys. Rev. E 66, 036111 (2002)]. Furthermore, this approach is also capable of interpreting delay-time results as a function of the barrier width in a frustrated total internal reflection experiment performed at the microwave scale and in the optical region.