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Physical Review A

Time delay of evanescent electromagnetic waves and the analogy to particle tunneling

Th. Martin, Rolf W. Landauer

Published Feb 1, 1992• 213 citations

Abstract

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.

DOI: 10.1103/physreva.45.2611

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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 tunneling of particles is in direct analogy with the transmission of evanescent electromagnetic waveguide modes as has been shown quite recently. We compare experimental data of an electromagnetic wave packet traversing an evanescent waveguide region with theoretical values derived for particle tunneling through a rectangular potential barrier. The transmission time was deduced by transformation of the experimental frequency data to the time domain. The data are in agreement and reveal superluminal wave-packet velocities for opaque evanescent regions.

Expressions are derived for the electromagnetic fields of guided waves which are analogous to the quantum-mechanical equations representing barrier tunneling. This analogy is achieved by comparing the propagation constant of the Schrödinger time-independent wave equation with that of the electromagnetic wave equation in waveguide and by comparing the de Broglie wavelength of a particle with the wavelength of the waves that propagate the energy. This results in an expression relating the form of an arbitrary one-dimensional energy barrier to the physical dimensions of a section of waveguide. The analogy is tested by the propagation of energy in the TE10 mode at both the 3- and 6-cm bands for the cases of rectangular and hyperbolic barriers. Although evanescent modes are present at the discontinuous regions, the analog for the rectangular barrier, which is considered to be the worse case, is verified when an effective barrier length leff of about 1.2l is used. This experimental verification demonstrates the possibility of waveguide simulation of quantum-mechanical energy barriers and the practicality of utilizing an electromagnetic analog for demonstrating the tunneling phenomenon and provides a method for measurement of the transmission coefficient through an arbitrarily shaped barrier.

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.

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.

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.

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.