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arXiv:2401.11586 · Physical Review Letters

Entangled-Beam Reflectometry and Goos-Hänchen Shift

Q. Le Thien, R. Pynn, Gerardo Ortíz

Published Mar 5, 2025• 6 citations
arXivHTMLPDF

Abstract

We introduce the technique of entangled-beam reflectometry for extracting spatially correlated (magnetic or nonmagnetic) information from material surfaces or thin films. Our amplitude- and phase-sensitive technique exploits the coherent nature of an incoming entangled probe beam, of matter or light waves, undergoing reflection from the surface. Such reflection encodes the surface spatial structure into the probe's geometric and phase-derived Goos-Hänchen shifts, which can then be measured to unveil the structure. We investigate the way these shifts depend on the wave packet widths, and illustrate our technique in the case of in-plane periodic (non)magnetic structures by utilizing spin-path mode-entangled neutron beams.

DOI
10.1103/physrevlett.134.093802

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We introduce the technique of entangled-beam reflectometry for extracting spatially correlated (magnetic or nonmagnetic) information from material surfaces or thin films. Our amplitude- and phase-sensitive technique exploits the coherent nature of an incoming entangled probe beam, of matter or light waves, undergoing reflection from the surface. Such reflection encodes the surface spatial structure into the probe's geometric and phase-derived Goos-Hänchen shifts, which can then be measured to unveil the structure. We investigate the way these shifts depend on the wave packet widths, and illustrate our technique in the case of in-plane periodic (non)magnetic structures by utilizing spin-path mode-entangled neutron beams.
The Goos-Hänchen shifts are tiny deviations, in the incidence plane, from the laws of geometrical optics, which are observed when applying these laws to the central axes of incident and reflected light beams. The axis of the reflected beam does not generally emanate from the surface at the incidence point of the impinging beam (spatial shift) and the angle of the axis of the reflected beam does not generally match exactly the incidence angle (angular shift). In this theoretical work, we show how to exploit pre- and post-selected weak measurements of the light polarization to probe the spatial and angular Goos-Hänchen shifts occurring for paraxial Gaussian beams emitted by a surface nonlinear optical process, such as second-harmonic (SHG) of sum-frequency generation (SFG). Pre- and post-selected weak measurements, which were initially developed to probe foundational issues in quantum mechanics, proved very useful to enhance the minute shifts in linear optics that were undetectable otherwise, through a process called weak-value amplification. We further reveal the usefulness of amplified weak measurements in the context of nonlinear optics.
As a paraxial wave packet is reflected or refracted from a planar interface separating two material media, it experiences spatial and angular shifts of its center position with respect to predictions of the geometrical ray picture. These in-plane and out-of-plane beam shifts are known as Goos-Hänchen and Imbert-Fedorov shifts, respectively. We discover a universal link between the phase-space nonseparability of an incident wave packet of any degree of spatial coherence and the reflected beam shifts. We unveil coherence Goos-Hänchen and coherence Hall effects, absent in the fully coherent limit. While the former effect can trigger a pronounced enhancement of the spatial Goos-Hänchen shift, the latter enables control of the spatial Imbert-Fedorov shift, from complete cancellation at a certain incidence angle to dramatic enhancement of the shift to giant magnitudes for nearly incoherent incident wave packets. Our results are equally applicable to optical, x-ray, and neutron, as well as matter waves, and they showcase novel phenomena in wave-matter interactions.
The Goos-Hänchen effect is a spatial shift along an interface resulting from an interference effect that occurs for total internal reflection. This phenomenon was suggested by Sir Isaac Newton, but it was not until 1947 that the effect was experimentally observed by Goos and Hänchen. We provide the first direct, absolute, experimental determination of the Goos-Hänchen shift for a particle experiencing a potential well as required by quantum mechanics: namely, wave-particle duality. Here, the particle is a spin-polarized neutron reflecting from a film of magnetized material. We detect the effect through a subtle change in polarization of the neutron. Here, we demonstrate, through experiment and theory, that neutrons do exhibit the Goos-Hänchen effect and postulate that the associated time shift should also be observable.
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.
Optical beam shifts are very small and challenging phenomena to measure. Weak measurement techniques have been developed to amplify and more easily measure the shifts of spatially coherent beams. We here investigate theoretically a weak measurement technique for measuring the Goos–Hänchen shift of partially spatially coherent beams. We show that the amplification factor of the technique depends on the global coherence degree of the field and on the size of the Goos–Hänchen shift experienced by the beam. Large amplification factors can be obtained for low shifts, even for very incoherent beams. But, for higher beam shifts, such as those occurring near critical incidence, the amplification factors are modest to low for beams of low coherence.
We study the classical optics effects known as Goos–Hänchen and Imbert–Fedorov shifts, occurring when reflecting a bounded light beam from a planar surface, by using a quantum-mechanical formalism. This new approach allows us to naturally separate the spatial shift into two parts, one independent on orbital angular momentum (OAM) and the other one showing OAM-induced spatial-versus-angular shift mixing. In addition, within this quantum-mechanical-like formalism, it becomes apparent that the angular shift is proportional to the beams angular spread, namely to the variance of the transverse components of the wave vector. Moreover, we extend our treatment to the enhancement of beam shifts via weak measurements and relate our results to the recent experiments.