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arXiv:1705.05914 · Physical Review A

Oscillatory behavior of light in the composite Goos-Hänchen shift

Manoel P. Araújo, Stefano De Leo, Gabriel G. Maia

Published May 12, 2017• 32 citations
arXivHTMLPDF

Abstract

For incidence in the critical region, the propagation of Gaussian lasers through dielectric blocks is characterized by the joint action of angular deviations and lateral displacements. This mixed effect, known as the composite Goos-H\"anchen shift, produces a lateral displacement that is dependent on the axial coordinate, recently confirmed by a weak measurement experiment. We discuss under which conditions this axial lateral displacement, which only exists for the composite Goos-H\"anchen shift, presents an oscillatory behavior. This oscillation phenomenon shows a peculiar behavior of light for critical incidence and, if experimentally tested, could stimulate further theoretical studies and lead to interesting optical applications.

DOI
10.1103/physreva.95.053836

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For incidence in the critical region, the propagation of Gaussian lasers through dielectric blocks is characterized by the joint action of angular deviations and lateral displacements. This mixed effect, known as the composite Goos-H"anchen shift, produces a lateral displacement that is dependent on the axial coordinate, recently confirmed by a weak measurement experiment. We discuss under which conditions this axial lateral displacement, which only exists for the composite Goos-H"anchen shift, presents an oscillatory behavior. This oscillation phenomenon shows a peculiar behavior of light for critical incidence and, if experimentally tested, could stimulate further theoretical studies and lead to interesting optical applications.
In this work, we summarize the current state of understanding of lateral displacement and angular deviations of an optical beam propagating through dielectric blocks. In part I, the analytical formulas, found for critical incidence, are compared with numerical calculations and, when possible, extended from Gaussian to more general angular distributions. Angular deviations are discussed both for the critical and for the Brewster incidence. The combined effect, known as the composite Goos–Hänchen shift, shows an interesting axial dependence and, under particular circumstances, the intriguing phenomenon of the light oscillation. In part II, the weak measurement technique is discussed in detail and compared with direct optical measurements.
The interference between optical beams of different polarizations plays a fundamental role in reproducing the optical analog of the electron spin weak measurement. The extraordinary point in optical weak measurements is represented by the possibility to estimate with great accuracy the Goos-Haenchen (GH) shift by measuring the distance between the peak of the outgoing beams for two opposite rotation angles of the polarizers located before and after the dielectric block. Starting from the numerical calculation of the GH shift, which clearly shows a frequency crossover for incidence near to the critical angle, we present a detailed study of the interference between s and p polarized waves in the critical region. This allows to determine in which conditions it is possible to avoid axial deformations and reproduce the GH curves. In view of a possible experimental implementation, we give the expected weak measurement curves for Gaussian lasers of different beam waist sizes propagating through borosilicate (BK7) and fused silica dielectric blocks.
In conventional optics for light waves, the pioneering work by Goos-Hänchen in 1947 on the lateral shift (or displacement) of a light beam along the surface of a dielectric boundary under the condition of total reflection has stimulated a large number of studies. The key physics behind the Goos-Hänchen shift is the nature of wave interference. From the perspective of wave optics, the incident beam of a finite transverse width can be viewed as composed of plane wave components, each of which has a slightly different transverse wavevector. Each wave component, after the total internal reflection, undergoes a different phase shift, and the superposition of all the reflected wave components gives rise to the lateral shift of the intensity peak in the reflected beam.
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.