Spatiotemporal Vortex Pulses: Angular Momenta and Spin-Orbit Interaction

Recently, spatiotemporal optical vortex pulses carrying a purely transverse intrinsic orbital angular momentum were generated experimentally [Optica 6, 1547 (2019); Nat. Photonics 14, 350 (2020)]. However, an accurate theoretical analysis of such states and their angular-momentum properties remains elusive. Here, we provide such analysis, including scalar and vector spatiotemporal Bessel-type solutions as well as description of their propagational, polarization, and angular-momentum properties. Most importantly, we calculate both local densities and integral values of the spin and orbital angular momenta, and predict observable spin-orbit interaction phenomena related to the coupling between the transverse spin and orbital angular momentum. Our analysis is readily extended to spatiotemporal vortex pulses of other natures (e.g., acoustic).

Figure 1

The plane-wave spectra (left) and phase-intensity distributions of real-space wave functions $\psi (\mathbf{r},t)$ (right) for (a) the monochromatic Bessel beam with $\ell =2$ and (b) spatiotemporal Bessel pulse with $\ell =2$, Eqs. (1) and (2). In real-space distributions, the brightness is proportional to the intensity $|\psi {|}^2$, while the color indicates the phase $\text{A}\mathrm{rg}(\psi )$.

Figure 2

Temporal diffraction of the spatiotemporal Bessel pulse (1) with $\ell =3$ and $\mathrm{\Delta }k/k_0=0.3$. The characteristic timescale $c{t}_R=k_0/\mathrm{\Delta }{k}^2$ is analogous to the Rayleigh range of spatially diffracting beams.

Figure 3

Electric fields of plane waves in the spectra of Bessel-type vector STVPs (left) and the corresponding real-space intensity distributions $|\mathbf{E}(\mathbf{r},t){|}^2$ (right) for (a) the out-of-plane polarization and (b) the in-plane polarization. The parameters are $\ell =1$ and $\mathrm{\Delta }k=0.7$ for better visibility of nonzero intensity (5) in the center of the pulse in (b).

Figure 4

Spatial distributions of (a) the normalized transverse spin angular momentum density ${\omega }_0{S}_y/I$ and (b) the orbital angular momentum density (with the subtracted propagational term) ${\omega }_0{L}_y+k_{0}xI$ in the in-plane-polarized Bessel STVP, Eqs. (8) and (5). The parameters are $\ell =2$ and $\mathrm{\Delta }k/k_0=0.5$.

Figure 5

The plane-wave spectrum (left) and the phase-intensity distribution of the real-space wave function $\psi (\mathbf{r},t)$ (right) for an elliptical Bessel STVP with $\ell =2$ and ratio of principal axes $\gamma =2$. The normalized integral OAM of this pulse is ${\omega }_0⟨L_y⟩/⟨I⟩=2.5$, Eq. (10).