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arXiv:0904.2478 · Physical Review A

Goos-Hänchen shifts in frustrated total internal reflection studied with wave-packet propagation

Xi Chen, Chun-Fang Li, Rong-Rong Wei, Yan Zhang

Published Jul 30, 2009• 27 citations
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Abstract

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.

DOI
10.1103/physreva.80.015803

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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.
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.
We study the quantum Goos-H\"anchen (GH) shift and the tunneling transmission at a curved step potential by investigating the time evolution of a wave packet. An initial wave packet is expanded in terms of the eigenmodes of a circular step potential. Its time evolution is then given by the interference of their simple eigenmode oscillations. We show that the GH shift along the step boundary can be explained by the energy-dependent phase loss upon reflection, which is defined by modifying the one-dimensional (1D) effective potential derived from the two-dimensional (2D) circular system. We also demonstrate that the tunneling transmission of the wave packet is characterized by a free-space image distant from the boundary. The tunneling transmission exhibits a rather wide angle divergence and the direction of maximum tunneling is slightly rotated from the tangent at the incident point, which is consistent with the time delay of the tunneling wave packet computed in the 1D modified effective potential.
The negative and positive Goos-H\"anchen (GH) shifts in reflected light are revisited for a partial coherent light incident on a cavity. A three-level dilute gaseous atomic medium, which follows two-photon Raman transitions, is considered in a cavity. The anomalous (negative group index) and normal (positive group index) dispersions of the intracavity medium lead to negative and positive GH shifts, respectively. The effects of beam width, spatial coherence, and mode index of partial coherent light fields are studied on the negative and positive GH shifts. It is observed that the amplitude of negative and positive GH shifts are relatively large for a small range of beam width and for a small value of spatial coherence of partial coherent light beams. The amplitude of GH shifts becomes small for large beam width and spatial coherence of incident light. Further, the distortion in the reflected light field increases when the amplitude of the GH shifts increases and vice versa.
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.
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.
A new scheme focusing on modification and control of distortion in the reflection and transmission of Goos–Hänchen (GH) shifted beams, is discussed. The absorptivity, reflectivity, and transmissivity are measured for the new atomic scheme. The GH-shifted beam is modified by control fields and incident angle. The sum of absorption, transmission, and reflection coefficients is equal to one. For positive reflection GH-shift region, there is negative transmission GH shift. The distortion in the reflected and transmitted beam is controlled with incident angle and control fields. The results discussed in this article show significant applications in micro-optics and nano-optics, acoustics, and plasma physics.