Centroid shifts of spatiotemporal vortex pulses with arbitrarily oriented orbital angular momentum at planar reflection and refraction

The vortex structure of wave packets on the transversal or spatiotemporal plane can significantly enhance their shifts during the reflection and refraction. These shifts are classified into six types: Goos-H\"anchen shift within the plane of incidence, Imbert-Fedorov shift perpendicular to the plane of incidence, longitudinal shift, together with three counterparts of angular shifts. Here, we analytically derive the expressions for these shifts for three-dimensional spatiotemporal vortex wave packets that carry orbital angular momentum of arbitrary orientation. It is found that the influence of the optical configurations (e.g., incident angle, refractive index, etc.) on the shifts can be described by a deformation tensor, while that of the wave-packet structure is an isotropic one. Interestingly, the spatial and angular shifts induced by the topological charge can be attributed to two topologically distinct tangent vector fields on the spheres, respectively. Also, the conservation laws of transverse linear momentum and vertical angular momentum are revisited. Our studies may find potential in spatiotemporal pulse shaping and shed light on the recent controversy regarding the definition of intrinsic orbital angular momentum for a spatiotemporal vortex pulse.