Spatiotemporal Optical Vortices

We present the first experimental evidence, supported by theory and simulation, of spatiotemporal optical vortices (STOVs). A STOV is an optical vortex with phase and energy circulation in a spatiotemporal plane. Depending on the sign of the material dispersion, the local electromagnetic energy flow is saddle or spiral about the STOV. STOVs are a fundamental element of the nonlinear collapse and subsequent propagation of short optical pulses in material media, and conserve topological charge, constraining their birth, evolution, and annihilation. We measure a self-generated STOV consisting of a ring-shaped null in the electromagnetic field about which the phase is spiral, forming a dynamic torus that is concentric with and tracks the propagating pulse. Our results, here obtained for optical pulse collapse and filamentation in air, are generalizable to a broad class of nonlinearly propagating waves.

Popular Summary

When an intense laser pulse propagates through a medium such as air or glass, it can perturb the local atoms or molecules in such a way that they act to focus the pulse. The pulse is then said to “self-focus.” Above a certain laser power threshold, the self-focusing process runs away, and the pulse attempts to collapse to a singularity. However, short of reaching the singularity, the pulse collapse self-terminates owing to multiple mechanisms brought on by the high intensity itself. Here, we show, both experimentally and theoretically, that self-focusing collapse arrest is always accompanied by the self-generation of spatiotemporal optical vortices, which are line- or loop-shaped structures that direct the circulation of electromagnetic power in a spatiotemporal plane defined by the local axis of the vortex. One form of spatiotemporal optical vortex measured in the experiments is a toroidal volume of circulating electromagnetic power surrounding the propagating laser pulse, evoking an electromagnetic “smoke ring.”

More generally, it appears that spatiotemporal optical vortices can be a fundamental element of all laser pulses, irrespective of their intensity. While we have demonstrated their spontaneous generation as integral to the nonlinear self-focusing collapse of intense pulses, spatiotemporal optical vortices can also be imprinted on much less intense, linearly propagating pulses using optical elements that impose spatiotemporal or spatiospectral phase shifts.

Our discovery of spatiotemporal optical vortices is accordingly expected to pave the way for a wide range of future studies and applications of ultrafast laser pulses.

Figure 1

(a) Phase and intensity projections of simulated pulse, in local coordinates, showing STOV generation. Propagation medium: atmospheric pressure air. Black dashed lines encircle vortices $v_1$, $v_2$, and $v_3$, and arrows point in the direction of increasing phase. Our phase winding convention considers a $(\xi -{\xi }_0)+i(r-r_0)$ winding about a null at (${\xi }_0$, $r_0$) to denote a $+1$ STOV, giving the $v_1$ STOV a $+1$ charge. The white dots in the intensity projection are centered on the locations of vortices $v_1$, $v_2$, and $v_3$. (b) Simulation of an air filament crossing the air-helium boundary for the conditions of (a), showing the beam fluence and plasma density. Nonlinear propagation terminates as the beam transitions from air to helium, whereupon the beam and a reference pulse (not shown; see Appendix pp1) are directed to an interferometer.

Figure 2

Toy model showing birth of a vortex pair via spatiotemporal phase front shear. The white curve and arrow depicts the axial (temporal) intensity profile and propagation direction, while the “core” and “periphery” labels denote the spatial intensity step. The $z=0$ panel shows the initial condition where phases are aligned, the $z=z_v$ panel shows the birth of the null (vortices overlap), and the $z=2z_v$ panel shows continued shear carrying the vortices apart, with the $+1$ vortex moving to the temporal front and the $-1$ vortex moving backward. Our phase winding convention considers a $(\xi -{\xi }_0)+i(x-x_0)$ winding about a null at (${\xi }_0$, $x_0$) to denote a $+1$ STOV. In the third panel, red and black arrows indicate the direction of increasing phase. The two arrows connect the same lines of constant phase; the arrows show that the phase difference between head and tail is ill defined due to the vorticity of the $-1$ STOV.

Figure 3

Simulations of beam phase (top) and intensity (bottom) at the axial or temporal slice ${\xi }_v$, where the STOV pair first appears. From left to right, the pulse is advancing along $z$, with the input shown at $z=0$, a collapsing beam at $z=156\mathrm{cm}$, ionization onset at 160 cm (not shown), just before the vortices spawn at 166 cm, just after the vortices spawn at 167 cm, and an image where the vortex pair has moved out of the $\xi ={\xi }_v$ plane. The linear intensity images are overlaid with centered lineouts of ${\log }_{10}(I/I_0)$, where $I_0={10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. Experimental parameters are used as code inputs: $w_0=1.3\mathrm{mm}$, pulse energy 2.8 mJ ($P/P_{\mathrm{cr}}=6.4$), pulse FWHM 45 fs.

Figure 4

Beam on-axis phase shift (with respect to flat-phase reference arm) as a function of pulse power at $z_h=150\mathrm{cm}$. The phase jumps abruptly by $\sim 2\pi$ at $P/P_{\mathrm{cr}}\sim 5$, providing a clear signature of the transition from the precollapse to the postcollapse beam. The red points are averages over 2600 shots (blue points) in 150 energy bins.

Figure 5

Top row: Retrieved spatial phase (radians) and intensity (arbitrary units) images at $z=165\mathrm{cm}$, $P/P_{\mathrm{cr}}=4.4$ for (i) a precollapse beam and (ii) and (iii) beams where a vortex ring is on either side of the reference central wavelength of 800 nm. The bottom row shows that as the maximal phase gradient in the images increases, the maximal phase shift saturates at $\pi$ for all cases of $P/P_{\mathrm{cr}}$ leading to beam collapse.

Figure 6

Demonstration of energy flow about a $+1$ “$R$-vortex” STOV. White arrows correspond to size and direction of $\mathbf{j}$ in Eq. (3). There are three distinct regimes: the left-hand panel shows the saddle regime, which exists in regularly dispersive media (${\beta }_2>0$), the middle panel shows the degenerate regime, which can exist in regular or anomalously dispersive media and has a dominant axis for energy flow, and the right-hand panel shows the spiral regime, which exists in anomalously dispersive media (${\beta }_2<0$).

Figure 7

Experimental setup for measuring the in-flight intensity and spatial phase profiles of collapsing and filamenting femtosecond pulses over a $\sim 2\mathrm{m}$ range.

Figure 8

(a) Line emission (in arbitrary units) of the neutral helium $3{}^3\mathrm{D}–2{}^3\mathrm{P}$, $\lambda =587.6\mathrm{nm}$ transition induced by a tightly focused 800 nm pulse as a function of helium cell position. (b) Simulated filament intensity and (c) spatial phase at 800 nm (black) just before the $\sim 4\text{−}\mathrm{mm}$ air-helium transition, as well as reconstructed intensity and phase (red). The reconstruction is performed by propagating the solution 4 mm through the transition region followed by 50 cm of helium. The simulated pulse is then backpropagated through 50.4 cm of vacuum to simulate reconstruction from imaging optics.

Figure 9

Spectral phase (radians, top) and spectral intensity (arbitrary units, bottom) taken from simulation at $z=180\mathrm{cm}$ as the vortex core in the spatiospectral domain crosses 2.35 PHz ($\lambda =800\mathrm{nm}$). Simulation parameters are the same as Fig. 3 of the main text: $w_0=1.3\mathrm{mm}$, pulse energy 2.8 mJ ($P/P_{\text{cr}}=6.4$), pulse FWHM 45 fs. From left to right, the top row (bottom row) shows the full spatial-spectral phase (intensity), as well as spatial phase (intensity) slices spectrally fore and aft of 800 nm. The dashed black line in the top left-hand panel intersects the vortex core.

Figure 10

Comparison of the spatial-spectral phase of the beam at 800 nm to the weighted spatial phase measured about 800 nm using the same simulation parameters considered in Fig. 3: $w_0=1.3\mathrm{mm}$, pulse energy 2.8 mJ ($P/P_{\mathrm{cr}}=6.4$), pulse FWHM 45 fs. Flipping of the spatial phase occurs at $z=179\mathrm{cm}$ at 800 nm and $z=185\mathrm{cm}$ for the weighted spatial phase.