Structured Optical Toroidal Vortices with Rotational Symmetry

Toroidal vortices, as intriguing topological structures, play a fundamental role across a wide range of physical fields. In this study, we theoretically propose a family of structured optical toroidal vortices as generalized forms of toroidal vortices in paraxial continuous wave beams. These structured optical toroidal vortices exhibit unique rotational symmetry while preserving the topological properties of standard toroidal vortices. The three-dimensional topological structures demonstrate l-fold rotational symmetry, which is closely related to the topological charges. Structured toroidal vortices introduce additional topological invariants within the toroidal light field. These topological light fields hold significant potential applications in the synthesis of complex topological structure and optical information encoding.