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The European Physical Journal D

Quantum (not frustrated) theory of the total internal reflection as the source of the Goos-Hänchen shift

Jacek Jakiel, W. Kantor

Published Oct 1, 2014• 3 citations
Publisher

Abstract

The frustrated total internal reflection theories (FTIR) from previous century are thoroughly recalculated from the, so called, monodromy operator’s point of view – a theory lunched by Born and Wolf [Principles of Optics (Pergamon Press, 1975), Chap. 1.6] and Arnold [Geometric Methods in the Theory of Ordinary Differential Equations (Springer, 1987)]. Monodromy is a theory of simultaneous solution (for both reflection and transmission amplitudes) of one dimensional Schrödinger equation (for the wavefunction and its derivative) and the Maxwell equation (for electric and magnetic fields). Introducing new quantities: the dwell distance and the phase distance, we get general Goos-Hänchen (G-H) shift formula for optical tunneling for three layer system with refraction indexes n 0, n 1, n 2. This formula reduces itself to expressions known from the scientific literature for infinite air gap (infinite width of second layer). Extension to many layers is possible.

DOI
10.1140/epjd/e2014-50156-9

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The frustrated total internal reflection theories (FTIR) from previous century are thoroughly recalculated from the, so called, monodromy operator’s point of view – a theory lunched by Born and Wolf [Principles of Optics (Pergamon Press, 1975), Chap. 1.6] and Arnold [Geometric Methods in the Theory of Ordinary Differential Equations (Springer, 1987)]. Monodromy is a theory of simultaneous solution (for both reflection and transmission amplitudes) of one dimensional Schrödinger equation (for the wavefunction and its derivative) and the Maxwell equation (for electric and magnetic fields). Introducing new quantities: the dwell distance and the phase distance, we get general Goos-Hänchen (G-H) shift formula for optical tunneling for three layer system with refraction indexes n 0, n 1, n 2. This formula reduces itself to expressions known from the scientific literature for infinite air gap (infinite width of second layer). Extension to many layers is possible.
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.
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].
Frustrated Total Internal Reflection (FTIR) has been researched since the time of Sir Issac Newton. In this paper, its application in modern optics is reviewed. A theoretical description of the phenomenon is presented beginning with Snell’s Law. The analogy between FTIR and quantum tunneling in one dimension is discussed. A quantitative and inexpensive method of demonstrating FTIR based on the attenuation of a microwave through an air gap between two wax prisms is described and a quantitative comparison of the theory with experiment is made. An extension of FTIR using microwave, as it is used in an Evanescent Wave Atomic Force Microscope (EW-AFM), is looked into and evaluated.
The Goos-H\"anchen (GH) shift is a specifical optical phenomenon that describes a shift parallel to the reflected light inside the plane of incidence, when a finite-width light undergoes total internal reflection at the interface of medium. Although the GH shift in optics has been widely observed experimentally, its generalization remains uncovered completely in relativistic quantum mechanics for the spinor structure and the possible existence of Klein's paradox. Through approximation and angular spectrum analysis, we calculate the GH shift for Dirac fermions under relativistic conditions when they are incident obliquely on a three-dimensional infinite potential barrier. Furthermore, we find that the relativistic quantum GH shift will degenerate into the result based on Schr\"odinger equation under nonrelativistic limit, while the GH shift can be negative in Dirac's framework, which is different from the nonrelativistic case.
The Goos-Hänchen (GH) shift refers to a lateral displacement (from the path expected from geometrical optics) along an interface in totally internal reflection. This phenomenon results from a coherence effect. In order to bring to light the role of coherence, the reflection of partially coherent light fields was investigated within the framework of the theory of coherence. A formal expression for the GH shifts of partially coherent light fields is obtained in terms of Mercer's expansion. It is shown that both the spatial coherence and the beam width have an important effect on the GH shift, especially near the critical angles (such as totally reflection angle). These results are important to observe the GH shifts of the beams with imperfect coherence, like x-ray and matter-wave beams.
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.