Goos-H\"{a}nchen and Imbert-Fedorov Shifts of Vortex Beams at Air-Left-Handed Material Interfaces

In this paper, we present a systematic study of beam shifts and angular momenta of paraxial vortex beams at air-left-handed material (LHM) interfaces. It is shown that, compared to their counterparts at air-right-handed material (RHM) interfaces, the spatial Goos-H\"{a}nchen (GH) and Imbert-Fedorov (IF) shifts remains the same, while the angular GH and IF shifts are reversed, at air-LHM interfaces. The spatial and angular shifts of paraxial vortex beams have their respective origins in transverse angular momenta and transverse linear momenta. The spatial GH and IF shifts remain unreversed as a result of the both reversions of transverse angular momenta and z-component linear momentum, while the angular GH and IF shifts are reversed due to that the z-component linear momentum is reversed and the transverse linear momenta are unreversed, at air-LHM interfaces. In addition, we perform a quantitative analysis on spin-orbit angular momentum conversion and orbit-orbit angular momentum conversion, which further helps us understand the essence of vortex beam shifts at air-LHM interfaces and their fundamental distinctions with those at air-RHM interfaces.