Scalar optical hopfions

Hopfions are three-dimensional (3D) topological states discovered in field theory, magnetics, and hydrodynamics that resemble particle-like objects in physical space. Hopfions inherit the topological features of the Hopf fibration, a homotopic mapping from unit sphere in 4D space to unit sphere in 3D space. Here we design and demonstrate dynamic scalar optical hopfions in the shape of a toroidal vortex and expressed as an approximate solution to Maxwell's equations. Equiphase lines correspond to disjoint and interlinked loops forming complete ring tori in 3D space. The Hopf invariant, product of two winding numbers, is determined by the topological charge of the poloidal spatiotemporal vortices and toroidal spatial vortices in toroidal coordinates. Optical hopfions provide a photonic testbed for studying topological states and may be utilized as high-dimensional information carriers.