Revisiting vortex generation in the spin-orbit interactions of refraction and focusing of light

The generation of vortices in the focusing of circularly polarized light beams results from the spin-orbit interaction (SOI), which is very similar to the generation of vortices when a light beam normally passes through an optically thin slab. However, its physical origin and the intrinsic relationship between both systems are still not well elucidated. Here, we establish a unified full-wave theory to revisit this problem. It is revealed that the origin of vortex generation in the transverse fields of the focusing system is the momentum-dependent Pancharatnam-Berry (PB) phase, which shares the same physics as the vortex generation in the slab case. In particular, we find that, under normal incidence, the beam transmitted through an optically thin slab has a longitudinal field and also exhibits a vortex with a topological charge of +/- 1 owing to the Berry-phase mechanism, akin to the focusing case. These findings not only unify the SOI-induced vortex generation in the optically thin slab and focusing systems within a single framework, uncovering the physical origin behind the SOI-induced vortex generation in optics, but also may shed light on understanding the SOIs in other fields.