Scalar optical hopfions

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