Orbital angular momentum of optical, acoustic, and quantum-mechanical spatiotemporal vortex pulses

Motivated by recent progress in the generation of optical spatiotemporal vortex pulses (STVPs), there is a theoretical discussion about the transverse orbital angular momentum (OAM) carried by such pulses. Two recent works [K. Y. Bliokh, Phys. Rev. Lett. 126, 243601 (2021)] and [S. W. Hancock et al., Phys. Rev. Lett. 127, 193901 (2021)] claimed the OAM values which differ by a factor of 2 for circular STVPs. Here we resolve this controversy by showing that the result by Hancock et al. is correct for the total OAM, while the result by Bliokh describes the suitably defined intrinsic part of the OAM. The other, extrinsic part of the OAM originates from the fact that plane waves of the same amplitude but different frequencies in the pulse spectrum contain different densities of photons, which induces a transverse vortex-dependent shift of the photon centroid even in a STVP with symmetric energy-density distribution. We describe similar peculiarities of acoustic and quantum-relativistic (Klein-Gordon and arbitrary-spin) STVPs. In all cases, only the intrinsic OAM keeps a universal form independent of the details of the problem and similar to the OAM of monochromatic vortex beams.