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Physical Review A

Photonic tunneling time in frustrated total internal reflection

A. A. Stahlhofen

Published Jun 16, 2000• 41 citations
Publisher

Abstract

Frustrated total internal reflection is considered as the classical analog of quantum-mechanical tunneling. Here the tunneling time in frustrated total internal reflection is discussed. It is shown that the phase time allows for vanishing and negative barrier traversal times. The tunneling time is shown to be dominated by a time scale based on the Goos-H\"anchen shift. The role of further nonspecular deformations of the transmitted beam is addressed. The resulting tunneling times are compared with experimental data.

DOI
10.1103/physreva.62.012112

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Frustrated total internal reflection is considered as the classical analog of quantum-mechanical tunneling. Here the tunneling time in frustrated total internal reflection is discussed. It is shown that the phase time allows for vanishing and negative barrier traversal times. The tunneling time is shown to be dominated by a time scale based on the Goos-H"anchen shift. The role of further nonspecular deformations of the transmitted beam is addressed. The resulting tunneling times are compared with experimental data.
We investigate experimentally the optical tunneling times associated with frustrated total internal reflection of a light beam. Using as physical clocks the lateral shifts and angular deviations suffered by the transmitted and reflected beams, we measure both components of a complex tunneling time: the phase time and a semiclassical time. The phase time is shown to imply superluminal velocities and to depend strongly on boundary conditions not linked to the tunneling process. By contrast, the semiclassical time yields subluminal velocities and is related solely to tunneling.
An exact analogy between wave mechanics in quantum theory and the scalar wave treatment of optics emerges from the marriage of Newtonian formulation of geometrical optics [1] and the ``formal quantum theory of light rays'' [2]. Here the incidence of a ray of light on the interface between two media is treated as the incidence of a wavefunction on a potential barrier. This leads to the coefficient of reflection identical to Fresnel's formula for s-polarised light [3], although there is no concept of the polarisation of light in this model. In the present work, we apply this model to the total internal reflection of light and evanescent waves. We also deduce a one-to-one correspondence between the transmission coefficient in wave mechanics and frustrated total internal reflection for all angles of incidence. Further, we demonstrate that this model can also be used to derive the Goos-Hänchen shift for s-polarised light. This work augments the discussion on these topics found in the standard texts in optics [4].
Abstract In this article, we work with an optical-mechanical analogy model that exploits the analogy between wave mechanics in quantum theory and the scalar wave treatment of optics. We discuss the application of this model to the evanescent waves and the frustrated total internal reflection of light. Further, we demonstrate that this model can also be used to derive the Goos–Hänchen shift for s -polarised light. This work provides the advanced undergraduate students and instructors of optics and quantum mechanics courses with a platform, where they can apply the methods of the quantum theory to optics, which is not discussed in depth in the standard texts.
We have investigated the Goos-H\"anchen (GH) shifts in frustrated total internal reflection (FTIR) studied with wave-packet propagation. In the first-order approximation of the transmission coefficient, the GH shift is exactly the expression given by a stationary phase method, thus saturating an asymptotic constant in two different ways depending on the angle of incidence. Taking the second-order approximation into account, the GH shift does not saturate with increasing gap width when the small beam size is used. The GH shift increases by decreasing the beamwidth at the small incidence angles, while for the large incidence angles it reveals a strong decrease by decreasing the beamwidth. These phenomena offer the better understanding of the GH shift and tunneling delay time in FTIR.
Abstract Tunneling is an outstanding physical process. The observation that particles surmount a high barrier in spite of the fact that they don't have the necessary energy can not be explained by classical physics. However, this so called tunneling became allowed by quantum mechanics. Experimental tunneling studies with different photonic barriers from microwave frequencies up to ultraviolet frequencies point towards a universal tunneling time [1, 2]. Tunneling can be described by virtual, that is, by unobservable particles [3]. In the case of tunneling there is a virtual particle between the real incident and the real transmitted particles. Tunneling modes are solutions of the Schrödinger and of the Helmholtz equations. The most prominent example of the occurrence of tunneling modes in optics is frustrated total internal reflection (FTIR) at double prisms. In 1949 Sommerfeld [4] pointed out that this optical phenomenon represents the analogy of quantum mechanical particle tunneling. Recent experimental and theoretical data confirmed the conjecture that the tunneling process is characterized by a universal tunneling time presumably independent of the kind of field. The observed tunneling processes proceeded at a time of the order of magnitude of the reciprocal frequency of the wave packets.