Attosecond pulses, as a milestone technique of attosecond science, have led to great advances in the research communities of ultrafast optics and strong-field physics in recent years [1,2]. Three scientists achieved the key experimental methods to produce attosecond light pulses for the study of electron dynamics in matter, thus being awarded the 2023 Nobel Prize in Physics [3]. Attosecond pulses originate from high harmonic generation (HHG) essentially, which is well understood by the semiclassical three-step model [4,5]. When atoms or molecules are irradiated by an intense laser field, the outer electron first tunnels through the distorted Coulomb potential dressed by the laser field, then is accelerated and driven back by the remaining field, and finally may recombine to the initial bound state and emit a high-energy photon simultaneously, realizing upconversion of the driving laser frequency to the extreme ultraviolet (EUV) region. Thus, attosecond pulses and HHG respectively describe the time- and frequency-domain information of this coherent emission. From the quantum perspective, the underlying mechanism can be well revealed by the Lewenstein model or solving the time-dependent Schrödinger equation (TDSE) numerically [4,6,7].

Since its advent, a lot of effort has been paid to control and shape this EUV radiation. At the beginning, researchers explored the effective methods to overcome technical hurdles from attosecond pulse trains (APTs) [8] to an isolated attosecond pulse (IAP) [9,10] and tailor its conventional properties, such as cutoff frequency [11,12], duration [13–15], brightness [16–20], harmonic odevity [21–25], and polarization [26–28]. Up to now, the highest photon energy based on HHG has reached up to 5.2 keV [12] and the shortest IAP has a duration of about 50 as [14,15].

As a type of electromagnetic wave, attosecond pulses not only carry spin angular momentum, but also possess orbital angular momentum (OAM), which is structured light fields, called optical vortex [29]. Its Poynting vector circulates around a local axis parallel to the propagation direction, forming longitudinal OAM, which has potential technological applications in optical communication, chirality recognition, and micromanipulation [30–33]. Then the manipulation of attosecond pulses entered the era of “vortex.” Attosecond optical vortices with longitudinal OAM from HHG have been reported [34,35] and gained much exciting progress in controlling the optical spin-orbit coupling [36] and generating EUV beams with time-varying OAM [37]. Just recently, free-space optical skyrmions based on HHG have been demonstrated theoretically [38].

Lately, a newly discovered spatiotemporal vortex with phase and energy circulation in a space-time plane was demonstrated experimentally [39], which was theoretically proposed in 2012 [40]. It offers a new degree of freedom, transverse OAM, and has generated important advances in optics, terahertz, and acoustics fields [41–48]. As a new form of structured light, its property of topological protection has shown promising application value in optical communication [49,50]. Meanwhile, its nonlinear effects and relativistic spatiotemporal optical vortex (STOV) HHG in plasma have recently gained increasing attention [51,52]. Significantly, Fang et al. turned their attention to HHG transverse OAMs and attempted to control them [53]. However, to the best of our knowledge, STOV attosecond pulses have not been explored yet, which would have a promising perspective for both attosecond science and extreme optics.

Here, we demonstrated a theoretical scheme for the generation of a STOV APT from atom gas pumped by the two-color femtosecond light field composed of the fundamental wave (FW) and its third harmonic (TH), with both carrying a STOV structure. Different from a single-color excitation, we found that the employment of a weak TH field with optimized phase is crucial for upconversion of STOV from FW to the high-order harmonics. It was revealed that this optimized method can effectively suppress the contribution of the electron long trajectories to the harmonics generation and result in broadband harmonics carrying the STOV feature. With a properly truncated HHG spectrum, an APT carrying spatial-temporal OAM was achieved. Further studies revealed that the intensity ratio of the two pumping fields, which may lead to ionization depletion, also plays an important role for the robust formation of the STOV APT.

A linearly-polarized FW STOV pulse is described as [41] $$E(X,Y,Z,{\omega }_{\mathrm{FW}}t)=E_0{N}_0{(X^2/w_X^2+{\xi }^2/w_{\xi }^2)}^{|l_{\mathrm{FW}}|/2}{e}^{-(X^2/w_X^2+Y^2/w_Y^2+{\xi }^2/w_{\xi }^2)}{e}^{i({\omega }_{\mathrm{FW}}t-k_{\mathrm{FW}}Z-l{\phi }_f)}.$$(1)As shown in Fig. 1, the laser electric field polarizes along the $X$ axis and propagates along the $Z$ axis. $\xi =ct-Z$ is the traveling longitudinal distance, and $c$ is the light velocity. $E_0$ is the peak amplitude, $N_0$ is a normalization factor, ${\omega }_{\mathrm{FW}}$ is the angular frequency, and $k_{\mathrm{FW}}$ denotes the wave number. $w_{\xi }$ and $w_X$ refer to the pulse spatial widths, and ${\phi }_f=\arctan (X{w}_{\xi }/\xi w_X)$ is the azimuthal angle in the space-time plane with a topological charge $l_{\mathrm{FW}}$. Its $Y$ component is trivial with a symmetrical Gaussian distribution. Hence, we focus only on electron dynamics in the $X$–$Z$ plane at $Y=0$ below. Since the range of electron motion contributing to HHG is much smaller than the spatial size of the laser focal point and wavelengths, a discrete spatial electric dipole approximation is adopted reasonably, where atoms at different $X$ positions will experience different phase and intensity of the STOV field.

Here, a one-dimensional TDSE describes the strong laser–atom interaction, whose velocity gauge is expressed as (atomic units are used throughout unless otherwise stated) $$i\frac{\partial }{\partial t}\mathrm{\Psi }(X,x,t)=[-\frac{{\nabla }^2}{2}+V_0(x)-iA(X,t)\nabla ]\mathrm{\Psi }(X,x,t),$$(2)where $V_0(x)=-1/\sqrt{x^2+1.92}$ is a reduced potential energy for the hydrogen atom and $x$ is the electron dynamic parameter. Such a reduced model is able to capture the main picture qualitatively since only free diffusion happens in the other two dimensions. The laser field consisting of FW and TH pulses with the same spatiotemporal envelope in the local atomic coordinate system is written as $$\begin{gathered} E(X,t)=E(X,{\omega }_{\mathrm{FW}}t)+E(X,{\omega }_{\mathrm{TH}}t+\varphi )=N_0{(X^2/w_X^2+{\xi }^2/w_{\xi }^2)}^{|l_{\mathrm{FW}}|/2}{e}^{-(X^2/w_X^2+{\xi }^2/w_{\xi }^2)} \\ \times [E_0^{\mathrm{FW}}{e}^{i({\omega }_{\mathrm{FW}}t-l_{\mathrm{FW}}{\phi }_f)}+E_0^{\mathrm{TH}}{e}^{i({\omega }_{\mathrm{TH}}t-l_{\mathrm{TH}}{\phi }_f+\varphi )}], \end{gathered}$$(3)where ${\omega }_{\mathrm{TH}}=3{\omega }_{\mathrm{FW}}$, $\varphi$ is the phase difference between both pulses, and $A(X,t)=-{\int }_0^{t}E(X,t)\mathrm{d}t$ is the laser vector. To get APT, we first calculate the dipole acceleration by the Ehrenfest theorem [54]: $$a(X,t)=⟨\mathrm{\Psi }(X,x,t)|-\frac{\partial V_0(x)}{\partial x}-E(X,t)|\mathrm{\Psi }(X,x,t)⟩.$$(4)The TDSE is solved by the split-operator algorithm [55]. The initial state is numerically obtained via the imaginary time propagation [56]. A mask function ${\cos }^{1/6}$ is utilized in borders to prevent the unphysical reflections from boundaries [57].

Then the $X$-position-dependent HHG spectra are obtained by the Fourier transform of $a(X,t)$: $$\tilde{S}(X,\omega )=|\tilde{a}(X,\omega {)|}^2={\left|\int a(X,t)e^{-i\omega t}\mathrm{d}t\right|}^2.$$(5)Finally, a pulse envelope in time domain can be synthesized via $$P(X,t)={\left|{\int }_{{\omega }_1}^{{\omega }_2}\tilde{a}(X,\omega )e^{i\omega t}\mathrm{d}\omega \right|}^2,$$(6)where ${\omega }_1$ and ${\omega }_2$ define the truncated frequency range. Equation (6) includes time-space dimension and thus gives the spatial-temporal structure of HHG and APT. Here HHG emission from atoms with the same $X$ values is assumed to be perfectly in phase, and the HHG interference from different $X$-position atoms as well as the propagation effect is not considered.

The simulation parameters are ${\lambda }_{\mathrm{FW}}=1600\mathrm{nm}$, $I_{\mathrm{FW}}=1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, ${\lambda }_{\mathrm{TH}}=533.3\mathrm{nm}$, and $I_{\mathrm{TH}}$ as well as the phase difference $\varphi$ between both being variable. The topological charge is fixed as $l_{\mathrm{FW}}=1$ and $l_{\mathrm{TH}}=3$, respectively. The time and spatial steps are $\mathrm{\Delta }t=0.1$ a.u. and $\mathrm{\Delta }x=0.2$ a.u., respectively. The simulation box for the range of electronic motion is $x=[-\mathrm{1000,}1000]$ a.u., which is big enough to preserve the full HHG information. The transverse range of the laser focal point is $X=[-12{\lambda }_{\mathrm{FW}},12{\lambda }_{\mathrm{FW}}]$ with 401 data points. The convergence has been tested by using a larger transverse box for the laser beam and finer spatial and time grids for the electron dynamics while almost identical harmonic spectra are obtained.

To achieve a STOV APT, we need to produce HHG spectra extended to the EUV region with well-defined transverse OAMs. The first idea is adopting the FW STOV pulse with ${\lambda }_{\mathrm{FW}}=1600\mathrm{nm}$ as a pumping laser. The result is shown in Fig. 2(a1). At first glance, for any $X$ position, odd and even harmonics are not well separated, which is very different from the common HHG spectra. In addition, there is no position-dependent frequency chirp for each harmonic. For example, the 35th harmonic pulse amplitude and phase distributions in the space-time plane are presented in Figs. 2(a2) and 2(a3). There is an indistinct intensity distribution and an irregular phase variation. Thus, harmonics in these HHG spectra do not possess STOV structure. We have systematically tested different pump laser wavelengths at various laser intensities ranging from $5\times {10}^{13}$ to $1.25\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and always cannot find the possibility for generation of broadband harmonics carrying a clear STOV structure in the plateau region. We therefore concluded that harmonics pumped with the fundamental field carrying STOV cannot possess STOV structure.

Inspired by the previous works on harmonic control with multiple color laser fields [18,53], we then added a TH laser pulse with $I_{\mathrm{TH}}=9\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ into the fundamental pump field. The new HHG spectra are shown in Figs. 2(b1) and 2(c1), corresponding to the relative phase difference $\varphi =0$ and $1.1\pi$, respectively. As one can see, Fig. 2(b1) is similar to Fig. 2(a1), and no STOV harmonics are produced. In contrast, in Fig. 2(c1) each harmonic shows an unambiguous space-dependent frequency chirp. Furthermore, this chirp scales linearly with the order of the harmonics, and the number of interference dark fringes in each harmonic is equal to its order. Let us take the 35th harmonic as an example. Its intensity profile and phase distribution in the spatial-temporal domain are presented in Figs. 2(c2) and 2(c3). Its intensity shows a clear doughnutlike structure, while its phase manifests as a phase winding of $35\times 2\pi$. We have confirmed that the same analysis holds for each harmonic in Fig. 2(c1). Therefore, it can be concluded that every harmonic in Fig. 2(c1) is a STOV light field with a topological charge $l_n=nl$, where $n$ is the harmonic order and $l$ is the topological charge of the FW laser. We speculated that the microscopic mechanism responsible for the structures in Fig. 2(c) is that the frequency and $X$ position are highly related in the nonlinear interaction between the STOV laser fields and atoms, making the FW transverse OAM $l$ transfer into the $n$th-order high harmonic ones $l_n$ by $n$ times. In other words, since the $n$th-order harmonic corresponds to the $n$th-order nonlinear effect, the complex exponent $e^{i({\omega }_{\mathrm{FW}}t-l_{\mathrm{FW}}{\varphi }_f)}$ recording the local phase information of the FW STOV field is transmitted to the $n$th-order harmonic after the $n$th-power execution, namely, $e^{in({\omega }_{\mathrm{FW}}t-l_{\mathrm{FW}}{\varphi }_f)}$.

Why does the two-color STOV field scheme with a definite phase difference $\varphi =1.1\pi$ in Fig. 2(c) generate EUV high harmonics with clear STOV structures? To answer this question, we resort to the Gabor transform to trace the HHG bursts in the temporal domain [58]. Here we examine the representative space position $X=0$ case, and the results are shown in Fig. 3, where Figs. 3(a), 3(b), and 3(c), respectively, correspond to Figs. 2(a1), 2(b1), and 2(c1). As expected, the phase singularity of the driving field leads to a middle valley at $t=0$ in three panels. We first concentrate on Figs. 3(a) and 3(b). During per half cycle of the FW field, the long and short electron trajectories make nearly equivalent contributions to the HHG. Besides these common long and short trajectories, under careful observation, there are complex bursts in the lower energy region except for the first half cycle in Figs. 3(a) and 3(b), which complicates the time and frequency structure of HHG and leads to almost continuous HHG spectra in Figs. 2(a1) and 2(b1). In addition, there noticeably exists strong low-order HHG burst around $t=0$, where the laser field is almost zero. In contrast, Fig. 3(c) shows remarkably that the bursts induced by long trajectories almost disappear and only short trajectories dominate. Meanwhile, the complex lower energy bursts in Figs. 3(a) and 3(b) also vanish in Fig. 3(c). So, the $X$-dependent HHG spectra are discrete and each harmonic shows a certain STOV structure. Based on the above results, we speculate that the multiple scattering of the long electron trajectories and its suppression may be a critical factor of the formation of harmonics with clear STOV structure.

To further reveal the physical origin behind Fig. 3, we trace electron classical trajectories via Newton’s equation in adiabatic approximation. The results are shown in Fig. 4, and other simulation details are included in Appendix A. The black dashed line describes the synthesized laser electric field. The curve color denotes the number of electron returns, and the thickness of the curve corresponds to the ionization rate with logarithmic scale. Both Figs. 4(a) and 4(b) show that the thicker lines possess more returns, which implies that the electron long trajectories born around the laser field peak contribute to the multiple scatterings responsible for low-energy structure and smear the HHG bursts in Figs. 3(a) and 3(b), leading to complex HHG spectra in Figs. 2(a1) and 2(b1). However, Fig. 4(c) shows that the additional weak TH field with a proper phase makes the combined field split into two tips. This synthesized field can restrain long trajectories with multiple scatterings and is conducive to the generation of EUV STOV, which can be understood well based on the three-step model. Ionization is solely dependent on the absolute value of the electric field, and the phase information of the STOV field is primarily imprinted in electron trajectories. The HHG plateau results from the contribution of not only first-returning trajectories but also higher-order scattering ones. The phase the latter accumulates does not synchronize with the former, thus resulting in the continuous HHG spectra. The earliest HHG bursts exclusively originate from the first return, while subsequent bursts are contributed considerably by multiple scattering channels with disordered phases. This explains why Figs. 2(a3) and 2(b3) gradually display irregular phase structures from left to right. Therefore, in order to obtain a clear STOV structure, it is essential to suppress the contribution of multiple scatterings.

Since the relevant HHG spectra have been obtained, an APT can be retrieved by an inverse Fourier transform, as shown in Fig. 5. The spatiotemporal intensity distributions of APT in Figs. 5(a), 5(b), and 5(c), respectively, correspond to Figs. 2(a1), 2(b1), and 2(c1). The truncated frequency range in three panels is from $11{\omega }_{\mathrm{FW}}$ to $39{\omega }_{\mathrm{FW}}$. As expected, there are no intensity zero points in central position in Figs. 5(a) and 5(b) since their HHG spectra have no STOV structure. In other words, spatiotemporal vortex APT cannot be produced in Figs. 5(a) and 5(b). Importantly, Fig. 5(c) gives a perfect hollow pulse train with an average topological charge $l=19.5$ at the spatiotemporal plane calculated by [53,59,60] $$l=\frac{2\iint \mathrm{Im}(ct{E}_X^{*}\partial E_X/\partial X)\mathrm{d}t\mathrm{d}X}{\iint E_X^{*}{E}_X\mathrm{d}t\mathrm{d}X},$$(7)where $E_X$ is the electric field intensity of the synthetic field. APT subpulse intensity distribution is extracted in Fig. 5(d). Its full width at half maximum (FWHM) indeed has reached attosecond scale, about 613 as.

Since the $X>0$ part of the driving STOV pulse with $l_{\mathrm{FW}}=1$ has one more optical cycle than the $X<0$ part, APT shown in Fig. 5(c) exhibits a fork-shaped structure with a fork number $N=2$, where the pulse number in $X>0$ is two more than in $X<0$. The unique structure is also attributed to the interference of different harmonics with a specific topological charge interval $\mathrm{\Delta }l$ in time and space (for more explanations, see Appendix B). The aforementioned results manifest that our two-color strategy is an effective means of generating spatiotemporal vortex APT.

We continue to explore spatially-resolved HHG spectrum by increasing the TH intensity. Figures 6(a1) and 6(a2) give the typical 35th harmonic HHG spectra with two laser intensities $I_{\mathrm{TH}}=9\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ and $I_{\mathrm{TH}}=4.9\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. Comparing the two panels, we find that a stronger TH laser significantly smears the fine interference patterns and thus destroys its due topological charge. As is known, the interference fringes are determined by the phase difference and amplitude ratio of subwaves. Since the phase difference between the FW and TH lasers remains unchanged in Figs. 6(a1) and 6(a2), the interference amplitude ratio, i.e., ionization probability, should undergo variation remarkably. So we trace the time-dependent ionization probability for Figs. 6(a1) and 6(a2), as shown in Fig. 6(b). The inset in Fig. 6(b) illustrates a STOV pulse traveling from left to right divided into two segments (Part A and Part B) in the time domain due to a phase singularity. When considering a ratio of $I_{\mathrm{TH}}/I_{\mathrm{FW}}=0.09$, the ion yield for both Part A and Part B is comparable, about 0.2. However, for the $I_{\mathrm{TH}}/I_{\mathrm{FW}}=0.49$ case, the ion generation probability for Part A is four times larger than that for Part B, resulting in a significant decrease in contrast of the interference fringes.

In order to confirm our deduction further, we calculate the HHG spectrum based on the quantum-orbit model in the strong-field approximation framework [53] (for more details, see Appendix C), as shown in Figs. 6(c1) and 6(c2). As one sees, Figs. 6(a1) and 6(a2) are well reproduced in Figs. 6(c1) and 6(c2), respectively. The above results suggest that intensity is a key parameter in generating STOV APT and ionization depletion may hinder HHG STOV formation because the target system cannot experience the intact STOV information of the driving laser.

We theoretically and numerically show that a spatiotemporal vortex APT can be produced from atom gas excited by a two-color femtosecond laser pulse with STOV. Our analyses revealed that the synthesized pump laser field can be optimized by the relative phase and intensity ratio, so that FW STOV information can be robustly transferred to each of the high-order harmonics by suppressing electron long trajectories, achieving an APT carrying an averaged topological charge of 19.5, with each sub-cycle optical burst 613 as.

The transverse OAM of light opens up an entirely new dimension, empowering us to shape light pulses spatiotemporally [61]. With the rapid development of generating intense STOV laser pulses and characterizing STOV structure [62], our scheme will be realized experimentally in the coming future. Analogous to the streaking method [63], it is well expected that superposing the STOV APT onto an intense long-wavelength Gaussian or STOV pulse will help us gain a profound understanding of the physical mechanisms underlying the interaction between matter and the transverse OAM of light [64,65]. Last but not least, our research sheds light on how STOV pulses influence electron trajectories and light emission during HHG processes, laying the groundwork for the generation of IAP with transverse OAM and bridging the two communities of STOV and attosecond science.

Acknowledgment. Authors are grateful to Qiwen Zhan for the helpful and enlightening discussions.