Orbital angular momentum of spatiotemporal vortices: a ray-mechanical analogy

TL;DR

The transverse orbital angular momentum (OAM) of spatiotemporal vortex pulses (STVPs) has been debated due to origin-dependent results. This work develops a simple ray-particle framework where STVPs are loops of non-interacting particles moving at speed $c$ with slightly different directions, revealing how OAM depends on the chosen reference point and the momentum-density distribution. By incorporating both uniform and non-uniform mass densities and enforcing a semiclassical vorticity quantization $rac{ ext{ abla}oldPhi}{ ext{ abla}\xi}=k_0{f u}(\xi)igrrac{ ext{ abla}{f r}}{ ext{ abla}\xi}$ with $oldsymbol{ abla}oldsymbol{ ext Phi}(2oldsymbol{ abla}
9)=2oldsymbol{ extell}$, the model reproduces prior wave-based OAM results across the $xz$ and $xt$ frameworks. The study shows how centroids (mass vs particle) and dispersion determine intrinsic versus extrinsic OAM, and it demonstrates that a non-uniform momentum density brings the mechanical results into full agreement with established wave results, including fractional STVPs when the quantization condition is not satisfied. The framework provides a transparent, coordinate-aware explanation for the previously reported discrepancies and extends to other waves with linear dispersion, highlighting the role of dispersion and centroid choice in STVP OAM. Overall, the work offers a compact, intuitive bridge between ray optics and wave theory for STVPs and their transverse OAM dynamics.

Abstract

Spatiotemporal vortex pulses (STVPs) are wavepackets that carry transverse orbital angular momentum (OAM), whose proper quantification has been the subject of recent debate. In this work, we introduce a simplified mechanical model of STVPs, consisting of a loop of non-interacting point particles traveling at a uniform constant speed but at slightly di!erent angles. We examine di!erent initial conditions for the particle loop, including configurations that are elliptic in space at a given time and configurations that are elliptic in spacetime at a fixed propagation distance. Furthermore, employing a non-uniform mass distribution allows the particle loop to mimic the STVP not only in configuration space but also in momentum space. Remarkably, when supplemented by a semiclassical vorticity quantization condition, our mechanical model exactly reproduces di!erent wave-based OAM results previously reported for paraxial STVPs.

Figures (8)

• Figure 1: Rays corresponding to trajectories of particles for modeling the evolution of STVPs vo2024closed1. In all figures, hue colors are used to identify the value of $\xi \in [0,2\pi)$ for the corresponding ray.
• Figure 2: Ray-optics modeling of $xz$ STVPs: (a) Ray bundle in space and time for $w_x=w_z$ and $\Delta_x=0.3$; (b,c) Cross-sections at $z=0$ (b), and $t=0$ (c), where the reference points $x_{\rm cent}$ and $x_{\rm min}$ are also indicated.
• Figure 3: Ray-optics modeling of $xt$ STVPs: (a) Ray bundle in space and time for $w_x=w_t$ and $\Delta_x=0.3$; (b,c) Cross-sections at $z=0$ (b), and $t=0$ (c), where the reference points $x_{\rm cent}$ and $x_{\rm min}$ are also indicated.
• Figure 4: Intensity and real part of the wavefield $\Psi(x,z,0)$, Eq. \ref{['eq:field-safe']}, for an $xz$ STVP (a) and an $xt$ STVP (b) with topological charge $\ell=3$, directional spread $\Delta_x=0.3$, and widths $w_x=w_z=w_t=3.29\lambda_0$ ($\gamma=1$), where $\lambda_0=2\pi/k_0$. The axes are in units of $\lambda_0$.
• Figure 5: Effect of the non-satisfaction of the quantization condition, $\Phi(2\pi) = 2\pi \ell$, on the wavefield intensity of an $xz$ STVP (left) and an $xt$ STVP (right), for $\Delta_x=0.3$ and $w_x=w_z=w_t=2.8\lambda_0$, which give $\Phi(2\pi)/2\pi=2.64$. The positions of the interruptions depend on the chosen region of integration, in this case $\xi\in[0,2\pi)$.