Spatiotemporal vortex beams and angular momentum

We present a space-time generalization of the known spatial (monochromatic) wave vortex beams carrying intrinsic orbital angular momentum (OAM) along the propagation direction. Generic spatiotemporal vortex beams are polychromatic and can carry intrinsic OAM at an arbitrary angle to the mean momentum. Applying either (i) a transverse wave-vector shift or (ii) a Lorentz boost to a monochromatic Bessel beam, we construct a family of either (i) time-diffracting or (ii) nondiffracting spatiotemporal Bessel beams, which are exact solutions of the Klein-Gordon wave equations. The proposed spatiotemporal OAM states are able to describe either photon or electron vortex states (both relativistic and nonrelativistic) and have potential applications in particle collisions, optics of moving media, quantum communications, and astrophysics.

Figure 1

(a) Plane-wave spectrum (4) of a monochromatic Bessel beam with $\ell =2$: a circle on the isofrequency sphere $k=\sqrt{{\omega }_0^2-m^2}$ in $\mathbf{k}$ space. Relative phases of the constituent waves form a charge-2 optical vortex encoded by colors. (b) Transverse real-space distributions of the intensity $\rho ={\left|\psi \right|}^2$, the current $\mathbf{j}=\mathrm{Im}\left({\psi }^{*}\nabla \psi \right)=\rho \nabla \Phi$, and the phase $\Phi =\arg \psi$ for the Bessel beam (5) corresponding to panel (a). The phase singularity on the beam axis represents a screw wave-front dislocation accompanied by a vortex of the current. The parameters used here are ${\omega }_0/m=2$, $k_{\perp 0}/m=1$, and $k_{z0}/m=\sqrt{2}$, and the dimensionless coordinates are $\tilde{x}=k_{\perp 0}x$ and $\tilde{y}=k_{\perp 0}y$. The normalized momentum and the OAM, Eqs. (6) and (7), integrated over the visible area, yield $\mathbf{p}=k_{z0}{\mathbf{e}}_z$ and $\mathbf{L}=\ell {\mathbf{e}}_z$.

Figure 2

(a) Plane-wave spectrum of a polychromatic Bessel beam with $\ell =2$ and spectrum (9) shifted by $\Delta k{\mathbf{e}}_x$ in $\mathbf{k}$ space. Here the semitransparent circle represents the initial monochromatic beam, and the isofrequency sphere has radius $\sqrt{k_{z0}^2+\Delta k^2}$. (b) Transverse real-space distributions of the intensity $\rho$, the current $\mathbf{j}$, and the phase $\Phi$ for the beam (a) at different times $t$. The beam moves in the positive $x$ direction and experiences time-diffraction deformations. The forklike phase patterns in the beam center represent moving edge-screw wave-front dislocations. The parameters are $k_{z0}/m=1$, $k_{\perp 0}/m=1.4$, $\Delta k/m=2$, and ${\omega }_c/m=2.45$, and we use the dimensionless variables $\tilde{x}=k_{\perp 0}x$, $\tilde{y}=k_{\perp 0}y$, and $\tilde{t}={\omega }_{c}t$. The normalized expectation values, Eqs. (6) and (7), numerically integrated over the same area around the moving beam center yield Eqs. (11) with a good accuracy: ${\mathbf{R}}_{\perp }\simeq \left(\Delta k/{\omega }_c\right)t{\mathbf{e}}_x$, $\mathbf{p}\simeq k_{z0}{\mathbf{e}}_z+\Delta k{\mathbf{e}}_x$, and $\mathbf{L}\simeq \ell {\mathbf{e}}_z\nparallel \mathbf{p}$.

Figure 3

Spatiotemporal representation of the transversely moving Bessel beam from Fig. 2. (a) Here its spectrum is plotted on the mass hyperboloid in the $\left(\omega ,k_x,k_y\right)$ space where it is seen that it also forms a vortex loop if projected onto the $\left(\omega ,k_y\right)$ plane. (b) Real-space evolution shown here in the form of the distributions in the $(t,y)$ plane, for different values of $x$ [cf. Fig. 2b]. The corresponding components of the four-current $j^{\mu }$ are $(j^0,j_y)$, where $j^0=-\mathrm{Im}\left({\psi }^{*}{\partial }_t\psi \right)$.

Figure 4

(a) Plane-wave spectrum (15) of a Lorentz-boosted Bessel beam with $\ell =2$ in $\mathbf{k}$ space. Here the semitransparent circle represents the initial monochromatic beam, whereas the isofrequency sphere has radius $\sqrt{k_{z0}^2+{\left(\gamma v{\omega }_0\right)}^2}$. (b) Transverse real-space distributions of the probability density $\rho$, the current $\mathbf{j}$, and the phase $\Phi$ for the beam (14) corresponding to panel (a), at different times $t$. The beam moves uniformly in the positive $x$ direction without diffraction. The forklike phase pattern represents a moving edge-screw wave-front dislocation. The parameters are ${\omega }_0/m=2$, ${\kappa }_0/m=1$, $\Delta k/m=2$, and $\nu =0.8$ ($\gamma =5/3$), and we use the dimensionless variables $\tilde{x}=k_{\perp 0}x$, $\tilde{y}=k_{\perp 0}y$, and $\tilde{t}={\omega }_{c}t$. The numerically calculated normalized expectation values, Eqs. (6) and (7), correspond to Eq. (18) with good accuracy.

Figure 5

Spatiotemporal representation of the transversely moving Bessel beam from Fig. 4. (a) Here its spectrum is plotted on the mass hyperboloid in the $\left(\omega ,k_x,k_y\right)$ space where it is seen that it also forms an elliptical vortex loop if projected onto the $(\omega ,k_y)$ plane. (b) The real-space evolution of the beam is shown here in the form of the density, current, and phase distributions in the $(t,y)$ plane for different values of $x$ [cf. Fig. 4b]. The corresponding components of the four-current $j^{\mu }$ are $(j^0,j_y)$, where $j^0=-\mathrm{Im}\left({\psi }^{*}{\partial }_t\psi \right)$.

Figure 6

Same evolution of the Lorentz-boosted Bessel beam (14) as in Fig. 4, but here with the scaling $\tilde{\psi }\to \sqrt{\omega }\tilde{\psi }$ of the spectral Fourier amplitudes. Thus, $\rho ={\left|\psi \right|}^2$ becomes the true relativistic probability density, and $\mathbf{j}=\mathrm{Im}\left({\psi }^{*}\nabla \psi \right)$ becomes the energy current (momentum density). This causes the relativistic Hall effect [9]: a tiny $y$ deformation and subwavelength shift (19) of the beam which is seen here owing to the nonparaxial character (small radius) of the beam. The dot indicates the center of gravity of the beam according to Eqs. (18) and (19): ${\mathbf{R}}_{\perp }\simeq \left(vt,-v\ell /2{\omega }_0\right)$, and the small $x$ deviations of the dot are due to the nonparaxial corrections.