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Journal de Physique I

About Superluminal Propagation of an Electromagnetic Wavepacket Inside a Rectangular Waveguide

Fabio Cardone A, R. Mignani, V. S. Olkhovsky

Published Oct 1, 1997• 14 citations
Publisher

Abstract

We discuss the propagation of an electromagnetic wavepacket inside a rectangular waveguide, of the type employed in recent experiments on superluminal tunneling of electromagnetic signals. By exploiting the analogy between particle and photon tunneling, we consider both evanescent and growing waves inside the narrowed part of the waveguide. The Fourier expansion of such waves shows that the barrier behaves in a nonlocal way. Such a nonlocality is accounted for in an effective way by means of a deformation of the spacetime inside the waveguide. As a consequence, the wavepacket propagates at superluminal speed according to an effective metric tensor, built up in analogy with the Cauchy stress tensor in a deformable medium.

DOI
10.1051/jp1:1997118

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We discuss the propagation of an electromagnetic wavepacket inside a rectangular waveguide, of the type employed in recent experiments on superluminal tunneling of electromagnetic signals. By exploiting the analogy between particle and photon tunneling, we consider both evanescent and growing waves inside the narrowed part of the waveguide. The Fourier expansion of such waves shows that the barrier behaves in a nonlocal way. Such a nonlocality is accounted for in an effective way by means of a deformation of the spacetime inside the waveguide. As a consequence, the wavepacket propagates at superluminal speed according to an effective metric tensor, built up in analogy with the Cauchy stress tensor in a deformable medium.
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.
Though both theoretical and experimental investigations have revealed the superluminal behavior of evanescent electromagnetic waves, there are many disputes about the physical meaning and validity of such superluminal phenomenon, which is due to the fact that the traditional investigations are based on the theory of tunneling time and concerned with the problem of what the group velocity of evanescent waves means. In this paper, by studying the quantum probability amplitude for photons to propagate over a spacelike interval along an undersized waveguide, we present theoretical evidence for such superluminality.
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 quantum mechanical tunneling effect was theoretically described in the first half of the last century, which witnessed the rise of quantum theory and groundbreaking experimental results in nuclear physics. A similar tunneling effect also takes place in the case of EM waves. The wave account (in the case of evanescent waves), the quantum-mechanical formalism, as well as the electrodynamical one, revealed a phenomenon that is counterintuitive and possibly even inconsistent with the special relativity theory paradigm. In the quantum formalism, the time of wave propagation within a photonic barrier is expressed as an imaginary number, which can be interpreted as the zero speed of propagation. In the nineties, scientists offered further experimental proofs of the theory; in experimental systems constructed for testing EM-waves tunneling, superluminal speed of propagation has been recorded. In the article, I introduce the problem of photon tunneling in the experimental context of the possibility of nonlocal wave propagation and superluminal transmission of information.
Abstract Contrary to mechanical waves, the two‐slit interference experiment of single photons shows that the behavior of classical electromagnetic waves corresponds to the quantum mechanical one of single photons, which is also different from the quantum‐field‐theory behavior such as the creations and annihilations of photons, the vacuum fluctuations, etc. Owing to a purely quantum effect, quantum tunneling particles including tunneling photons (evanescent modes) can propagate over a spacelike interval. With this picture we conclude that the superluminality of evanescent modes is a quantum mechanical rather than a classical phenomenon.
We analyse the reshaping mechanism leading to apparently 'superluminal' advancement of a pulse traversing an undersized section of a waveguide. For frequencies below the first inelastic threshold (cut off one), there are only evanescent modes in the narrow region, and the problem becomes similar to quantum mechanical tunnelling across an effective rectangular 'barrier'. In the coordinate representation, the barrier is shown to act as an effective beamsplitter, recombining envelopes of the freely propagating pulse with various spacial shifts. Causality ensures that none of the constituent envelopes are advanced with respect to free propagation, yet the resulting pulse is advanced due to a peculiar interference effect, similar to the one responsible for 'anomalous' values which occur in Aharonov's 'weak measurements'. In the momentum space, the effect is understood as a bandwidth phenomenon, where the incident pulse probes local, rather than global, analytical properties of the transmission amplitude T(p). The advancement is achieved when T(p) mimics locally an exponential behaviour, similar to the one occurring in Berry's 'superoscillations'.