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American Journal of Physics

Electromagnetic Analog of the Quantum-Mechanical Tunnel Effect

Julius J. Hupert, Gary Ott

Published Mar 1, 1966• 17 citations
Publisher

Abstract

An electromagnetic model of a “potential barrier” may be constructed from three sectors of waveguide, two outer sectors being filled with dielectric, the short inner sector air-filled. Using the mathematical analogy between the propagation data of the De Broglie wave and the wave inside a waveguide—above and below cutoff—it is demonstrated that, for a frequency of operation between the cutoff frequencies of the air-filled and the dielectric-filled waveguide, the air-filled section represents the potential barrier region. Thus the described apparatus, the construction of which has been carried out in full detail, enables the modeling of the tunnel effect experimentally and in a quantitative fashion.

DOI
10.1119/1.1972898

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Applied Physics Letters•Feb 18, 1991•148 citations
An electromagnetic model of a “potential barrier” may be constructed from three sectors of waveguide, two outer sectors being filled with dielectric, the short inner sector air-filled. Using the mathematical analogy between the propagation data of the De Broglie wave and the wave inside a waveguide—above and below cutoff—it is demonstrated that, for a frequency of operation between the cutoff frequencies of the air-filled and the dielectric-filled waveguide, the air-filled section represents the potential barrier region. Thus the described apparatus, the construction of which has been carried out in full detail, enables the modeling of the tunnel effect experimentally and in a quantitative fashion.
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 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.
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.