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 a light-matter interaction and is characterized by a transverse shift. Recently, there has been a surge in research on spatiotemporal optical vortices (STOVs) that carry transverse angular momentum. In this study, we examine the transverse shift of a STOV in a tilted reference frame with respect to the propagation axis. Through both theoretical analysis and numerical simulations, we find only ``spatial symmetrical STOVs'' exhibit the GSHEL which can be divided into two types based on the direction of the vortices. Our findings reveal that this phenomenon is contingent upon the spatial distribution of their angular momentum density. In addition, we explore a maximum shift value, which is proportional to $\sqrt{l{x}_{0}/{k}_{0}}$ due to the uncertainty principle. These discoveries open up new avenues for applications in the realms of ultrafast optics and nanotechnology, offering a fresh perspective on the manipulation and measurement of light at the micro- and nanoscales.