Symmetry-compatible angular momentum conservation relation in plasmonic vortex lenses with rotational symmetries

Plasmonic vortex lenses (PVLs), producing vortex modes, known as plasmonic vortices (PVs), in the process of plasmonic spin-orbit coupling, provide a promising platform for the realization of many optical vortex-based applications. Very recently, it has been reported that a single PVL can generate multiple PVs. This work exploits the representation theory of finite groups, reveals the symmetry origin of the generated PVs, and derives a new conservation relation based on symmetry principles. Specifically, the symmetry principles divide the near field of the PVL into regions, designate integers, which are the topological charges, to the regions, and, particularly, give an upper bound to the topological charge of the PV at the center of the PVL. Further application of the symmetry principles to the spin-orbit coupling process leads to a new conservation relation. Based on this relation, a two-step procedure is suggested to link the angular momentum of the incident field with the one of the generated PVs through the symmetries of the PVL. This theory is well demonstrated by numerical calculations. This work provides an alternative but essential symmetry perspective on the dynamics of spin-orbit coupling in PVLs, forms a strong complement for the physical investigations performed before, and therefore lays down a solid foundation for flexibly manipulating the PVs for emerging vortex-based nanophotonic applications.