Study Of The Transverse And Lateral Shifts Of Vortex Beam In Left-handed Materials

The vortex beam has a spiral wavefront, resulting in an orbital angular momentum lˉh perphoton along the propagation axis. This unique and novel degree of angular momentum hasfacilitated several applications, such as optical spanners, high data rate and multi-channel com-munications, and high-resolution imaging. On the other hand, the linear and angular momentain left-handed materials (LHMs) has been a hot topic since the emergence of LHMs. In view ofthese two issues, we believe studying the Goos-Ha¨nchen (GH) and Imbert-Fedorov (IF) shiftsof vortex beam at air-LHM interface would deepen our understanding of the essence of mo-menta in LHMs and may even usher in new applications associated with these tiny shifts, likeultra-sensitive metrological technique. Motivated by the theoretical significance and potentialapplication of this topic, the following researches are conducted:First, an angular spectrum method is adopted to derive the electric fields of the reflect-ed and transmitted vortex beams at air-LHM interfaces. Except for some special cases, forinstance, in the vicinity of critical angle of total internal reflection (TIR), the expressions ofelectric fields generally hold true. The longitudinal fields are included as well, which take ona polarization-sensitive vortex structure, providing a new perspective on the mechanism of IFshifts other than spin-orbit conversion. Based on the transmission model, a rigorous transmittedbeam profile in position space is formulated for the first time. The cross-polarization effect ofthe reflected vortex beam is also investigated. The research indicates that when the incidentbeam is p-polarized, the reflected beam has more than one singularity.Second, we utilize the operator method to calculate the spatial and angular shifts, whichis significantly different from the conventional calculation method and saves us considerabletime and effort. The GH and IF shifts in TIR and partial transmission are demonstrated andcontrasted with air–right-handed material (air-RHM) interfaces. It is shown that, comparedto their counterparts at air-RHM interfaces, the spatial GH and IF shifts of both reflected andtransmitted beam remain the same at air-LHM interfaces. The angular GH and IF shifts oftransmitted beam, however, are reversed at air-LHM interfaces. The impacts of the incidentangle and the refractive index on the beam shifts are analyzed as well. The study indicates thatfor a given refractive index, we can maximize, minimize and even eliminate the GH and IFshifts by choosing a suitable incident angle. The GH and IF shifts can be markedly suppressedby raising the absolute value of the refractive index.Third, the GH and IF shifts in TIR are investigated. It is shown that although there are spa-tial GH and IF shifts, the angular GH and IF shifts do not exist in TIR. At air-LHM interfaces, the phases of the reflected coefficients are reversed, and thus the spatial IF shift is reversed foran obliquely polarized incident beam. The orbital Hall effect of light (OHEL) in TIR is inves-tigated as well. Two novel splits are proposed. The first split is the orbital split in the incidentplane, which is proportional to the topological charge of the vortex beam. The second split isthe angular split perpendicular to the incident beam. This angular split increases proportionallywith the absolute value of the vortex charge rising.Fourth, we present a systematic study of the linear and angular momenta of transmittedand reflected vortex beam at air-LHM interfaces. Our study indicates that the spatial and angu-lar shifts of paraxial vortex beams have their respective origins in transverse angular momentaand transverse linear momenta. The spatial GH and IF shifts remain unreversed as a result ofboth reversions of transverse angular momenta and z-component linear momentum, while theangular GH and IF shifts are reversed because the z-component linear momentum is reversedand the transverse linear momenta are unreversed at air-LHM interfaces. In addition, we perfor-m a quantitative analysis on spin-orbit angular momentum conversion and orbit-orbit angularmomentum conversion, which further helps us understand the essence of vortex beam shifts atair-LHM interfaces and their fundamental distinctions from those at air-RHM interfaces.