Abstract

A rich variety of knot-like topological structures can be realized in laboratory light beams such as lasers. The simplest description of propagating light is a map from three-dimensional space to the complex numbers, where optical vortices (phase singularities, nodal lines) are circulations of the Poynting vector energy flow.

A variety of knotted and linked vortex filaments can be embedded in propagating beams shaped by computer-controlled holograms, embedding designs inspired by algebraic geometry, such as Milnor maps. Furthermore, a huge variety of knots and links occur in random waves modelling wave chaos or optical speckle. This echoes Lord Kelvin’s vision of the periodic table of atoms as knotted vortices in the ether.

Controlling polarization adds a further layer of topology, creating three-dimensional polarization textures with spatially structured phase and polarization. Topological particle-like skyrmionic beams realise the Hopf fibration by wrapping around the optical hypersphere parametrizing the full optical state.

Moving beyond the scalar and paraxial regime, structured nonparaxial vector beams reveal various rich mathematical structures and point toward an experimentally accessible, topological pre-quantum field theory for light.