Formation and Controlling of Optical Hopfions in High Harmonic Generation.

Toroidal vortex is a kind of novel and exotic structured light with potential applications in photonic topology and quantum information. Here, we present the study on high harmonic generation from the interaction of the intense toroidal vortices with atoms. We show that the spatial distribution of harmonic spectra reveal unique structures, which are kinds of high-order topological toroidal vortex because of the rotating of the driving fields with transverse orbital angular momentum. Then we show that, by synthesizing the toroidal vortices and optical vortices with longitudinal orbital angular momentum, it is able to generate the optical hopfions in the extreme ultraviolet range, which exhibit a completely different topological property compared with both the driving fields. Harnessing the spin angular momentum selection rules, we further show that one can control the topological texture and channel each harmonic into a single mode of controllable Hopf invariant.