Geometric spin Hall effect of spatiotemporal optical vortices

The geometric spin Hall effect of light (GSHEL), which is associated with nonzero transverse angular momentum, has been demonstrated to occur without the need for light-matter interaction and is characterized by a transverse shift. Recently, there has been a surge in research on spatiotemporal optical vortices (STOV) that carry pure transverse angular momentum. In this study, we examine the transverse shift of STOV in a tilted reference frame with respect to the optical axis. Through both theoretical analysis and numerical simulations, we establish a linear relationship between this shift and the topological charge. Our findings reveal that only "spatial STOV" exhibits a GSHEL shift, this phenomenon is contingent upon the spatial distribution of their angular momentum density. When present, the shift direction is consistently perpendicular to the angular momentum vector, and its magnitude is found to be inversely proportional to the cosine of the tilt angle. We explore a maximum shift value, which is proportional to $\sqrt{l{{x}_{0}}/{{k}_{0}}}$. These discoveries open up new avenues for the application in the realms of ultrafast optics and nanotechnology, offering a fresh perspective on the manipulation and measurement of light at the micro and nanoscales.