Optical vortex beams have been active research topics^[1–3]. A variety of applications have been demonstrated, such as optical communications^[4], optical spanners and tweezers^[5], and optical imaging^[6]. A typical vortex beam has a phase singularity in the transverse plane with orbital angular momentum (OAM) in the longitudinal direction^[7]. Recently, a new type of vortex referred to as a spatiotemporal optical vortex (STOV) has been demonstrated^[8,9]. The STOVs have phase singularities in the space–time domain with transverse OAM, which is a feature very distinct from traditional optical vortex beams.

In general, the STOV is not stable in propagation. In dispersive media propagation, the pulse broadening by the group velocity dispersion (GVD) will distort the wave packet severely to lose the phase singularity eventually^[9]. A more robust STOV is desired to tolerate the propagation in a lengthy dispersive medium as an optical fiber. One idea is to combine STOVs with propagation-invariant localized waves^[10] to preserve wave-packet structures against external perturbations, such as propagation in a dispersive medium. For example, a spatiotemporal Bessel structure has been explored to generate a nonspreading STOV^[11]. Another profile one can use is the Airy pulse (beam), which exhibits not only dispersion (diffraction)-free propagation^[12], but also other interesting propagation effects such as self-acceleration^[13,14] and self-healing^[15]. Due to their intriguing propagation characteristics, Airy beams have been utilized in applications such as curved plasma channel generation^[16], particle clearing^[17], optical imaging^[18], and optical trapping^[19-23]. Exploring unique propagation properties, combining Airy profiles with optical vortex beams has also been demonstrated in various previous research studies^[24–27].

In this Letter, we combined an Airy pulse and a STOV to form an Airy–STOV wave packet to manipulate the STOV in time. Due to the Airy pulse’s dispersion-free property, such an Airy–STOV wave packet maintains its temporal profile for long-distance propagation. Due to the self-healing property, it also self-heals against external perturbations. In addition, by the self-acceleration of the Airy–STOV wave packet, one can manipulate the phase singularity location in time as well.

An Airy pulse is generated by applying a third-order dispersion (TOD) in the frequency domain^[28]. Meanwhile, a spiral phase is added in the spatial-frequency–frequency domain to form a STOV. We can combine the TOD and the spiral phase in the spatial-frequency–frequency domain to form an Airy–STOV wave packet, as shown in Eq. (1), $$\tilde{E}(k_x,\omega ;z)=g(r)e^{il\theta }{e}^{\frac{j{\beta }_3{z}_0{\omega }^3}{6}}.$$(1)

The $g(r)$ is the radial Gaussian profile, $l$ is the topological charge, ${\beta }_3$ is the TOD coefficient, and therefore ${\beta }_3{z}_0$ is the total TOD, where $r=\sqrt{k_x^2+{\omega }^2}$ and $\theta =\arctan (\omega /k_x)$. The two-dimensional (2D) inverse Fourier transform gives the spatiotemporal electric field of the Airy–STOV in the space–time domain, $$E(x,t;z)=2\pi {(-i)}^{l}G(\rho )e^{il\varnothing }\mathrm{Ai}(at)\delta (x),\text{with}a={\left(\frac{2}{{\beta }_3{z}_0}\right)}^{\frac{1}{3}}.$$(2)

The resulting spatiotemporal profile is shown in Eq. (2), where $\otimes$ represents the convolution, and $\rho =\sqrt{x^2+t^2}$, $\varnothing =\arctan (t/x)$ and $G(\rho )$ is the Henkel transform of $g(r)$. As shown in Eq. (2), the resulting wave packet is the convolution of the STOV and the Airy pulse. Figure 1(a) shows the theoretical intensity profile of a typical Airy–STOV with topological charge $l=1$. The convolution of the STOV results in multiple spatiotemporal phase singularities, as shown in the theoretical phase map [Fig. 1(b)]. The phase map indicates that all singularities have the same topological charge ($l=1$ for this case).

The sign of ${\beta }_3$ determines the direction of the sidelobe tail. For ${\beta }_3>0$, the tail leads in time, as shown in Fig. 1. For ${\beta }_3>0$, the tail will lag in time. In case of a higher topological charge ($l\ge 2$), the STOV will split into multiple $l=1$ STOVs as the Airy–STOV propagates^[8,9].

To simulate the dispersion-free and the self-acceleration feature, a large amount of the GVD (quadratic phase of ${\beta }_{2}z$ as a function of the propagation distance) is added to the Airy–STOV wave packet. In the spatial-frequency–frequency domain, the Airy–STOV propagation in a dispersive medium is shown in Eq. (3), $$\tilde{E}(k_x,\omega ;z)=g{e}^{il\varnothing }{e}^{i{\beta }_{2}z{\omega }^2/2}{e}^{i{\beta }_3{z}_0{\omega }^3/6}.$$(3)

After the 2D inverse Fourier transform, the spatiotemporal Airy–STOV propagation under the GVD effect is expressed in Eq. (4), $$E(x,t;z)=2\pi {(-i)}^{l}G(\rho )e^{il\varnothing }\otimes a\exp [-i(\frac{2{\beta }_2^3{z}^3}{3{\beta }_3^2{z}_0^2}+\frac{{\beta }_{2}z}{{\beta }_3{z}_0}t\left)\right]\mathrm{Ai}\left[a(t-\frac{{\beta }_2^2{z}^2}{2{\beta }_3{z}_0})\right]\delta (x).$$(4)

According to Eq. (4), the overall wave packet can be characterized as the STOV convoluted with the temporal shifted Airy pulse. The amount of the temporal shift is proportional to the propagation distance square, which is the feature of the free acceleration. If the GVD effect (${\beta }_{2}z$) is much smaller than the TOD (${\beta }_3{z}_0$), the GVD will not significantly change the Airy–STOV structure, which is referred to as the dispersion-free propagation. To observe an obvious dispersion-free phenomenon, a sufficiently large ${\beta }_3{z}_0$ should be adopted. On the other hand, the temporal displacement is determined by $\frac{{\beta }_2^2{z}^2}{2{\beta }_3{z}_0}$, which indicates a larger amount of ${\beta }_3{z}_0$ will lead to a smaller self-acceleration.

The experimental setup to generate the Airy–STOV is very similar to the pulse-shaper setup with a 2D spatial light modulator (SLM) in Ref. [8]. The experimental setup and phase patterns on the SLM are shown in Fig. 2. The surface of the SLM can be understood as the spatial-frequency–frequency domain. To generate the Airy–STOV, a spiral phase and a cubic frequency phase are applied. Since the seed pulse is positively chirped, a quadratic frequency phase is applied to compensate for the residual chirp of the seed pulse. The spatial quadratic phase is also added to compensate for the diffraction effect of the free-space propagation.

To diagnose the Airy–STOV wave packet, the three-dimensional (3D) measurement technique in Ref. [29] is used. For the generated Airy–STOV, a short coherent reference pulse can be overlapped and scanned to retrieve the 3D profile of the Airy–STOV wave packet. The experimental iso-intensity of a typical Airy–STOV shown in Fig. 3(b) is in good agreement with the simulated profile.

We borrowed the dispersion length ($L_D$) concept of the Gaussian pulse to estimate the amount of GVD applied in the Airy–STOV propagation. To demonstrate the dispersion-free propagation, the regular STOV without the Airy temporal structure and the Airy–STOV have been compared. The numerical simulations show that regular STOV dramatically broadens in time to lose the integrity of its spatiotemporal phase singularity [Figs. 4(a) and 4(c)] when a significant GVD ($3.7L_D$) is applied. The amount of GVD is expressed in terms of the dispersion length of a Gaussian pulse with the same temporal duration of the STOV at 1/e of maximum intensity, which is 57 fs. In contrast, under the same amount of GVD propagation, the numerical simulation shows that the Airy–STOV has much less distortion while maintaining the spatiotemporal phase singularities intact [Figs. 4(b) and 4(d)].

The experimental result [Fig. 4(e)] verifies the dispersion-free propagation showing that the Airy–STOV maintains the main lobe and the phase singularity in $3.7L_D$ propagation. The experimentally measured wave packet matches very well with the simulation results.

The self-acceleration effect allows us to control the temporal location of the spatiotemporal vortex. According to Eq. (4), the temporal shift is proportional to the square of the propagation distance, which appears as a uniform acceleration motion. We compared the simulation and experimental results again in a strong GVD propagation. The simulated intensity profiles show the temporal shift of zero-intensity holes [Figs. 5(a1)–5(g1)]. Along with zero-intensity holes, the phase singularities are also temporally shifted [Figs. 5(a2)–5(g2)]. The simulation indicates that we can manipulate the temporal location of spatiotemporal singularities. In the experiment, we observed very similar temporal shifts of Airy–STOV in a dispersive propagation [Figs. 5(a3)–5(g3)]. The TOD of $0.0005{\mathrm{ps}}^3$ is applied on the SLM, while the GVD is adjusted in the range of $0.009{\mathrm{ps}}^2$ to $-0.009{\mathrm{ps}}^2$.

Figure 6 shows the temporal shift of the phase singularity in the main lobe of the Airy–STOV due to the free acceleration. The simulation and experiment have a similar temporal shift of the phase singularity, which verifies the free-acceleration feature.

It is noteworthy that asymmetric intensity distribution occurs around the main lobe phase singularity in the dispersive propagation (Fig. 5). For a normal GVD ($\mathrm{GVD}>0$), in the main lobe, the energy flows from the upper to the lower part around the singularity and vice versa for the anomalous dispersion. Such an intensity shift is the consequence of the unbalanced dispersion and diffraction effect^[8,9]. This result indicates that one can control the spatial intensity distribution with a spectral phase (GVD) revealing the spatiotemporally coupled aspect of the Airy–STOV.

Owing to the self-healing feature of the Airy wave pulses, the Airy–STOV also can recover its temporal structure in a dispersive propagation. Figure 7 shows the simulation result and the experimental result of the Airy–STOV self-healing. Figures 7(a) and 7(b) show the initial Airy–STOV with strong intensity around the main lobe singularity. In the Airy pulse, the center frequency is concentrated in the main lobe while the tail contains the red and blue parts of the spectrum. To induce a significant distortion in the main lobe of the Airy–STOV, the center frequency has been filtered out^[30]. Figures 7(c) and 7(d) show the simulated and experimental profiles with a strong perturbation in the main lobe after the spectral filtering.

In the experiment, the disturbed wave packet has propagated in a 4-inch (1 inch = 2.54 cm) SF11 glass, which provides the GVD of $0.0129{\mathrm{ps}}^2$. After the propagation in a highly dispersive glass, the main lobe of the Airy–STOV heals to gain the power around the main lobe singularity. The self-healed main lobe has an asymmetry intensity distribution due to the unbalanced dispersion and diffraction. However, it is clear that the Airy–STOV recovers the energy around the main lobe phase singularity due to the self-healing effect [Figs. 7(e) and 7(f)]. Quantitatively, we can calculate the ratio of the OAM of the main lobe to the entire wave packet to analyze the process of the self-healing. The initial transverse OAM of the main lobe to the entire Airy–STOV’s OAM is $\sim 30\%$. After filtering out the center frequency, the main lobe only contains $\sim 8\%$ of the total transverse OAM. After the self-healing in a dispersive medium, the main lobe gains OAM back to $\sim 22\%$ of the total transverse OAM. Interestingly, this observation indicates that the OAM can flow temporally within the wave packet during the dispersive propagation.

We demonstrated theoretically and experimentally that phase singularities in the Airy–STOV wave packet have properties of being dispersion-free, self-accelerating, and self-healing, inherited from the Airy pulse propagation. These effects can benefit future applications of STOVs. Due to the dispersion-free feature, it is believed that the Airy–STOVs can maintain the vortex structure even in a lengthy dispersive medium. Owing to the self-healing, Airy–STOVs are likely to be more robust against perturbations. The self-acceleration can be used to control the temporal locations of spatiotemporal singularities. Further, we can extend this concept to a family of wave packets such as Airy–Airy–Airy–STOVs and Airy–Bessel–STOVs, to enhance the maneuverability of the phase singularities.