Goos-Hänchen Shifts of Partially Coherent Light Fields

Abstract

We investigate the Goos-Hänchen (GH) shifts of partially coherent fields (PCFs) by using the theory of coherence. We derive a formal expression for the GH shifts of PCFs in terms of Mercer's expansion, and then clearly demonstrate the dependence of the GH shift of each mode of PCFs on spatial coherence and beam width. We discuss the effect of spatial coherence on the resultant GH shifts, especially for the cases near the critical angles, such as totally reflection angle.PACS numbers: 42.50. Ar, 42.25.Gy, 42.30.Jf Goos-Hänchen (GH) Shift refers to a tiny (lateral) displacement, from the path expected from geometrical optics, upon total reflection [1]. This effect has been extended into other fields that involve the coherent-wave phenomena, such as neutron waves [2,3], electron waves [4,5], and spin waves [6]. It was explained by Artmann [7] that the different transverse wave vectors of a light beam undergo different phase changes and sum of these waves form a reflected beam with a lateral shift. Recently, it was shown [8] that the GH shift is the sum of Renard's conventional energy flux plus a self-interference shift. The self-interference shift originates from the interference between the incident and the reflected beams. Furthermore, it was discovered that the classical Fresnel formulas for laws of refraction and reflection of light are not applicable to partially coherent light [9]. These explanations indicate that the interference or coherence of light is very important to the GH shift.In 2008, we numerically showed the effect of spatial coherence on the change of the GH shift near the critical angles [10]. Later, an experiment [11] showed the large difference between the measured GH shift of a partially coherent LED light and the theoretical result of a coherent light. However, the very recent investigations [12][13][14][15][16] have raised an important issue "whether the spatial coherence of the partially coherent fields (PCFs) influences the GH shifts?" Although the exact numerical results, calculated from our previous theory [10], are in good agreement with the experimental data [13,14,16], it is necessary to reconsider this issue thoroughly and bring to light the role of spatial coherence on the GH shift.In this Letter, we use the exact theory of coherence to investigate the GH shift of PCFs. First we derive a formal expression to calculate the GH shift of PCFs in terms of the mode expansion of PCFs. Based on this expression, we explain the physical mechanism about the dependence of the GH shift on the spatial coherence and the beam width. Finally, we suggest a proposal for showing the new effect of the spatial coherence on the practical GH shift below the critical angles.First we derive the GH shift of PCFs based on the coherence theory [17]. For the two-dimensional PCFs, one usually uses the cross-spectral density (CSD), W (x 1 , z 1 ; x 2 , z 2 , ν), to describe its propagation, where (x 1 , z 1 ) and (x 2 , z 2 ) are the coordinates of the two points in the fields, and ν is the frequency of light....

Closed-form expression for the Goos-Hänchen lateral displacement

“…The interesting result was that as the spatial coherence of the beam decreases its GH shift reduces. A formal expression for the GH shift of partially coherent beams in terms of the Mercer expansion recently appeared in the literature [25]. The partial coherence as well the axial dependence have to be included in joint analytical expression with experimental data and the GH shift can be used to determine the coherence of the incoming beam [25] as well the axial angular deviations [23].…”

“…also thanks Silvânia A. Carvalho for stimulating conversations which motivated the study of an analytical expression for the GH shift in the critical region. Finally, the authors are greatly indebted to an anonymous referee for drawing their attention to the analytical work of Horowitz and Tamir [20], the paper comparing analytical formulas with experimental data [21], and the articles showing the effect of partial coherence in the GH shift [24,25].…”

Multistage Monte Carlo simulation of jet modification in a static medium

“…(6) and (8). The only parameter in this LBT model is the strong coupling constant α s that quantifies the jet-medium interaction, which is determined by comparing model calculation to experimental data of single heavy and light flavor hadron production, single inclusive jet production and γ-jet production in heavy-ion collisions [31][32][33][34]51].…”

“…For instance, in the domain of high energy and high virtuality Q 2 ≫ √q E, one expects to apply the medium modified DokshitzerGribov-Lipatov-Altarelli-Parisi (DGLAP) evolution [22,23] based on the higher-twist formalism [24,25]. As the virtuality approaches the medium induced scale, one expects to simulate a rate equation [26][27][28] based on the BDMPS/AMY [10,11,29,30] formalisms or a derivative of the higher twist formalism [31][32][33][34].…”

Quantitative extraction of the jet transport parameter from combined data at RHIC and LHC

“…Therefore, broadening of dijet correlation in azimuthal angle is only indirectly related tô q. In a recent study within Linear Boltzmann Transport (LTB) model, the large angle tail of the dijet correlation is shown to be sensitive to the value ofq [20]. High precision data are needed for any phenomenological study.…”