Toroidal phase topologies within paraxial laser beams

Control of topologies in structured light fields with multi-degrees of freedom integrates fundamental optical physics and topological invariance. Beyond the simple phase vortex, three-dimensional (3D) topological singularities and related nonsingular textures have recently gained significant interest. Here, we experimentally demonstrate the creation of a family of toroidal phase topologies within paraxial laser beams. By employing single two-dimensional (2D) phase control, we generate propagating 3D topological textures, effectively embodying the topological configuration of a four-dimensional (4D) parameter space. The resulting light fields exhibit amplitude isosurfaces of toroidal vortices and hopfionic phase textures, both controlled by topological charges. The ability to prepare scalar phase textures of light offers new insights into the high-dimensional control of complex structured textures and may find significant applications in light-matter interactions, optical manipulation, and optical information encoding. Exploring non-trivial topologies and related properties has long been a fascinating and challenging task in mathematics and physics. The authors experimentally demonstrate the realization of optical toroidal vortices and hopfionic phase textures within paraxial continuous wave laser beams, which may provide new insight for topologically structured light fields.