The conservation and nonlinear conversion of spatiotemporal topological charges of light

The orbital angular momentum (OAM) of light is a type of angular momentum (longitudinal OAM), associated with wavefront or phase vortices in the electromagnetic field and it has been demonstrated very useful in applications such as optical tweezer, super-resolution microscopy, telecommunication, scatterometry, and quantum information. On the other hand, light carrying spatiotemporal orbital angular momentum (ST-OAM) is a recently discovered type of structured and localized electromagnetic field. This field carries characteristic space-time spiral phase structure and transverse intrinsic OAM. ST-OAM of light was theoretically proposed 15 years ago and finally realized experimentally over the past 5 years. In this work, we present the generation and characterization of the second-harmonic of ST-OAM pulses. We uncovered the conservation of transverse OAM in a second-harmonic generation process for the first time. We found that the space-time topological charge of the fundamental field is doubled along with the optical frequency. Our experiment thus suggests a general ST-OAM nonlinear scaling rule— analogous to that in conventional OAM of light. Furthermore, we observe that the topology of a second-harmonic ST-OAM pulse can be modified by complex spatiotemporal astigmatism when using a thick nonlinear crystal. After analyzing canonical momentum density and energy density flux of fundamental and second-harmonic ST-OAM pulses, we identified the causes of spatiotemporal astigmatism in a dispersive medium that gives rise to multiple phase singularities separated in space and time. Our study opens a new route for nonlinear conversion and scaling of light carrying ST-OAM with the potential for driving other secondary ST-OAM sources from electromagnetic waves, mechanical or acoustic waves, to matter waves such as electrons and neutrons.