An optical vortex, characterized by a helical phase structure of the form $\exp (il\theta )$, features a central phase singularity and a hollow intensity profile. The topological charge, denoted by $l$, quantifies the number of phase twists accumulated during one wavelength of propagation [1–3]. In the early 1990s, Allen et al. demonstrated that the longitudinal orbital angular momentum (OAM) of a photon is inherently carried by a spatial vortex beam, with the magnitude of the OAM being proportional to its topological charge [4,5]. Since then, optical vortex beams carrying longitudinal OAM, including Laguerre–Gaussian beams, Bessel–Gaussian beams, and others, have been extensively demonstrated and applied in a wide range of applications, such as light–matter interactions [6,7], optical imaging [8,9], quantum information [10,11], and optical communications [12,13], as well as nonlinear optics [14,15].

Recently, spatiotemporal optical vortices (STOVs) defined in the space-time domain have garnered increasing attention due to their ability to carry unique transverse OAM of light [16–19]. The development of STOVs can be traced back to early predictions [20,21] and has recently been experimentally demonstrated through both nonlinear [22] and linear methods [23,24]. The successful generation of STOV wavepackets has catalyzed further exploration into spatiotemporal topology [25,26], harmonic generations [27–29], and other forms of matter waves [30]. This progress has led to the emergence of novel spatiotemporal optical wavepackets [16,17], including spatiotemporal Bessel [31,32], crystal [33], and Laguerre and Hermite Gaussian wavepackets [34]. However, the beam radii of these conventional optical vortices—whether spatial vortex beams or STOV wavepackets—strongly depend on their topological charge number, which can hinder their applications in many cases. For instance, this dependence presents challenges in coupling STOVs into a single optical fiber for mode-division multiplexing [35,36], in the angular momentum transfer to particles for optical trapping and manipulation [37], and in the OAM-dependent divergence that accompanies the increase in the mode index for optical transmission [38].

Consequently, the concept of perfect optical vortices has been proposed to overcome the above limitations, as their beam radius is independent of the topological charge number [39,40]. This feature introduces an unprecedented paradigm for applications in ultra-secure image encryption [41], high-dimensional quantum teleportation [42], trapping particles [43], and information encoding and transmission [38]. Very recently, Ponomarenko et al. introduced the concept of perfect space-time vortices by theoretically incorporating a time lens [44]. In this approach, a Bessel-type STOV is transformed into a perfect STOV through a sequence of an ordinary lens and a time lens. However, implementing a time lens directly in the temporal domain is very challenging, as it requires precise quadratic phase modulation on the picosecond timescale [45]. Hence, the perfect STOV has never been experimentally realized and investigated.

In this work, we report the experimental generation of perfect spatiotemporal optical vortices, where the ring size in both space and time remains independent of the topological charge. We theoretically and experimentally demonstrate that the proposed perfect STOV wavepackets are the spatiotemporal Fourier transforms of polychromatic Bessel–Gaussian beams based on space-time duality. Experimental results are in excellent agreement with theoretical predictions. Furthermore, we show that the spatiotemporal collision of two perfect STOVs produces STOV lattices, where the number of sub-vortices and symbols can be freely controlled.

The perfect STOV wavepacket in the spatiotemporal domain ($X-T$) can be generated by modulating the spatial–spectral field in the $k_x-\omega$ plane. Assume the optical field in the $k_x-\omega$ domain is given by ${\psi }_0(\rho ,\varphi )$, where $(\rho ,\varphi )$ are the corresponding polar coordinates, $\rho =\sqrt{k_x^2+{\gamma }^2{\omega }^2}$, $\phi =\arctan (k_x/\gamma \omega )$, and $\gamma$ is a scaling factor with a unit of ps/mm for controlling the aspect ratio of Bessel–Gaussian mode at the $k_x-\omega$ plane. After applying a spiral phase of $e^{-il\phi }$, a two-dimensional (2D) Fourier transform yields the field in the $X-T$ domain, which can be expressed as $$\psi (r,\theta )=\mathrm{FT}\{{\psi }_0(\rho ,\phi )e^{-il\phi }\},$$(1)where $r=\sqrt{X^2+{\alpha }^2{T}^2}$, $\theta =\arctan (X/\alpha T)$, and $\alpha ={\gamma }^{-1}$ is the scaling factor in the spatiotemporal domain. FT represents the 2D Fourier transform. The spiral phase and its related optical OAM are conserved after a 2D Fourier transform from the spatial frequency–frequency domain ($k_x-\omega$ plane) to the space–time domain ($X-T$ plane) [23].

In the spatial domain, spatially perfect vortex beams can be generated through the Fourier transform of spatial Bessel beams [46]. Motivated by this, the perfect STOV can be generated by the Bessel Gaussian mode spatial frequency–frequency domain ($k_x-\omega$ plane) through the 2D spatiotemporal Fourier transform. In the experiment, we can implement the 2D spatiotemporal Fourier transform process by a 2D 4f pulse shaper with a focusing lens. Hence, we introduce a spatial–spectral coupled polychromatic Bessel–Gaussian beam on the $k_x-\omega$ plane, which is given by [46] $$\psi (\rho ,\phi )=\exp (-\frac{{\rho }^2}{w_0^2})J_l(a\rho )e^{-il\phi },$$(2)where $w_0$ is the waist radius of the Bessel–Gaussian beam on the $k_x-\omega$ plane. $J_l$ is the first kind Bessel function with an order of $l$. $a$ is the radial wavevector of the Bessel–Gaussian beam, which determines the size of the Bessel–Gaussian beam. In the experiment, this parameter is applied to define the radius of the phase pattern in the computer-generated hologram. $e^{-il\phi }$ is the spiral phase in the spatial–spectral plane. In the experiment, the Fourier spectral field in Eq. (2) can be generated through a recently developed spatiotemporal holographic shaping method that can generate the complex amplitude field distribution with the amplitude distribution being $\exp (-\frac{{\rho }^2}{w_0^2})J_l(a\rho )$ and the phase distribution being $e^{-il\phi }$. The resulting spatiotemporal field on the $X-T$ plane can be then calculated by a 2D Fourier transform [32,46]: $$\begin{gathered} \psi (r,\theta )=\mathrm{FT}\{\psi (\rho ,\phi )\} \\ =\frac{1}{2\pi }{\int }_0^{2\pi }{\int }_0^{\infty }\exp (-\frac{{\rho }^2}{w_0^2})J_l(a\rho )e^{-il\phi }{e}^{i\rho r·\cos (\phi -\theta )}\rho \mathrm{d}\rho \mathrm{d}\varphi \\ \approx i^{l-1}\frac{w_0}{w}\exp (-\frac{{(r-r_0)}^2}{w^2})e^{-il\theta }. \end{gathered}$$(3)

The above equation represents the complex amplitude of perfect STOV with topological charge of $l$ and ring radius $r_0=af/k_0$. In the spatiotemporal domain, the perfect STOV ring thickness is $w=2f/k_0{w}_0$, where $f$ is the focal length of a focusing lens performing a spatial Fourier transform, $k_0=2\pi /\lambda$, and $\lambda$ is the wavelength. It can be seen that the ring size of a perfect STOV is independent of the topological charge and can be adjusted by changing $a$ and $f$.

Equations (1)–(3) suggest that perfect STOV can be synthesized via spatiotemporal Fourier transform by employing a spatiotemporally coupled polychromatic Bessel–Gaussian seed beam. Figures 1(a)–1(d) present the numerical simulation results for the generation of perfect STOV wavepackets at different free-space propagation distances. The simulation uses a spatial–spectral Bessel–Gaussian beam with a topological charge of $l=+5$, a spatial width of 0.7 mm, a central wavelength of 1.03 μm, and a spectral width of 10 THz. A perfect STOV wavepacket [Fig. 1(d)] is faithfully generated at a distance $L=300\mathrm{mm}$ following a 4f pulse shaper. On the other hand, in a medium with anomalous dispersion, the perfect STOV exists only within a finite propagation distance, beyond which it degrades into a Bessel–Gaussian form (see Fig. 5 in Appendix A).

Previous research has demonstrated the generation of spatiotemporal Laguerre–Gaussian and spatiotemporal Bessel–Gaussian wavepackets, where the spatial and temporal diameters are dependent on the topological charge [31,32,34,47]. As shown in Fig. 1(e), numerical simulations reveal that the spatial and temporal widths of spatiotemporal Laguerre–Gaussian and spatiotemporal Bessel–Gaussian wavepackets increase with increasing topological charge. In contrast, the spatial and temporal diameters of perfect STOV wavepackets remain constant, regardless of the topological charge. This is one of the main features and advantages of perfect STOVs.

Figure 2 exhibits the experimental setup for perfect STOV generation and characterization. The mode-locked input laser, with a spectral bandwidth of approximately 20 nm centered at 1030 nm, is split into a probe pulse and an object pulse. The probe pulse passes through a pulse compressor consisting of a pair of parallel gratings and a right-angle prism and is then de-chirped to a Fourier-transform-limited pulse. The object pulse is directed into a folded 2D ultrafast pulse shaper that consists of a reflective grating, a cylindrical lens, and a phase-only reflective SLM (Holoeye GAEA-2, $3840\times 2160\mathrm{pixels}$ with a pitch of 3.74 μm). The programmable SLM is positioned in the spatial–spectral plane and imprints an intricate phase pattern, as shown in the bottom right corner. This pattern encodes the complex amplitude of the polychrome Bessel–Gaussian beam described in Eq. (2), which has a form of [33,34] $$\psi (x,\omega )=\mathrm{mod}(\mathrm{Arg}({\psi }_{\mathrm{BG}})+gx·\text{asinc}(1-|{\psi }_{\mathrm{BG}}|)+\pi ·\text{asinc}(|{\psi }_{\mathrm{BG}}|),2\pi ),$$(4)where ${\psi }_{\mathrm{BG}}$ is the complex amplitude of the Bessel–Gaussian beam and $g$ denotes the frequency of a linear phase ramp, the depth of which is contingent upon the modulus of the on-demand mode. Furthermore, to suppress the residual modulation, the phase-only SLM must be calibrated to a linear $2\pi$ phase response over all 256 gray levels at wavelength 1030 nm. To produce high quality perfect STOV wavepackets, the pulsed beam’s spatial–spectral bandwidths should entirely cover the Bessel–Gaussian pattern loaded on the SLM. Hence, the spatial–spectral bandwidths of input pulsed beam and the spatial resolution of the SLM (or phase hologram) jointly impose a limitation on the generation of ultrahigh-order perfect STOV wavepackets. This limitation could be mitigated by utilizing customized metasurface devices and lasers with larger spatial and spectral width [34].

The modulated beam is reflected and recombined, then it is focused by a spherical lens ($f=1.2\mathrm{m}$), and the perfect STOV wavepackets are obtained at the focal plane where a CCD camera is positioned. The spatiotemporal wavepacket is synthesized in the far field via a time-delayed spatial propagation after the grating. To characterize the wavepacket, the probe and generated wavepacket are recombined at the CCD camera with a small tilt angle. Scanning the probe pulse’s time delay enables the reconstruction of the object wavepacket’s spatiotemporal profile at a certain temporal slice from the delay-dependent fringes. Finally, we can reconstruct the three-dimensional spatiotemporal wavepacket by stitching all temporal slices [34].

Figure 3(a) presents the intensity iso-surface and sliced phase patterns of experimentally generated perfect STOV wavepackets with typical topological charges of $l=+1,+5$, and $+10$, respectively. The size of the perfect STOV remains constant in space–time coordinates. To quantitatively assess the properties of perfect STOVs, Fig. 3(c) plots the variation of spatial ($w_x$, circle marker) and temporal ($w_t$, square marker) diameters as a function of the topological charge. The experimental data (circles and squares) align well with the theoretical predictions (solid curves), confirming that the spatial and temporal diameters of perfect STOVs are indeed independent of the topological charge. This experimental verification solidifies the characteristics of perfect STOV wavepackets (see Fig. 6 in Appendix B).

To evaluate the performance of perfect STOVs, we conducted an experimental comparison with single-ring spatiotemporal Laguerre–Gaussian beams [34] of $p=0$. Using identical beam parameters and topological charge numbers ($l=+1,+5,+10$), we generated spatiotemporal Laguerre–Gaussian wavepackets with $p=0$. The spatial and temporal diameters of the spatiotemporal Laguerre–Gaussian wavepackets increase significantly with larger topological charge, as shown in Fig. 3(b). As seen in Fig. 3(d), there is excellent agreement between the experimental data (represented by circles and squares) and the theoretical results (solid curves). These results demonstrate that both the spatial and temporal diameters increase with increasing topological charges (see Fig. 7 in Appendix B).

The properties of the perfect STOV wavepackets are further verified by comparing with those of spatiotemporal Laguerre–Gaussian wavepackets. In addition to their unique characteristics of constant sizes, perfect STOVs could offer additional degrees of freedom in controlling their spatial and temporal dimensions. By adjusting the parameter $a$ in Eq. (2), we can freely regulate both the spatial and temporal diameters of perfect STOV wavepackets. To illustrate the advantages of “perfect” in the spatiotemporal domain, we examined two cases, following the method outlined in Ref. [47], where two perfect STOVs with applied linear phase coefficients $k_t=-1.5\mathrm{ps}$ and $k_t=+1.5\mathrm{ps}$ move in opposite directions along the time axis.

In the first case, as shown in Figs. 4(a1) and 4(a2), the initial two perfect STOV wavepackets separated in time are generated first by applying linear phases with opposite slopes to perfect STOV wavepackets with the same spatial and temporal diameters ($a_1=a_2=5{\mathrm{mm}}^{-1}$) and $l_1=+4$ and $l_2=-2$. Then, we accelerate and delay the two wavepackets to 0 ps, respectively, in time, so that the two wavepackets collide in the spatiotemporal domain, forming a petal-like interference pattern, as shown in Fig. 4(b1), and the number of petals is $N=|l_2-l_1|$. In the second case, Figs. 4(d) and 4(f) show two perfect STOV wavepackets of different spatial and temporal diameters ($a_3=4{\mathrm{mm}}^{-1}$ and $a_4=6{\mathrm{mm}}^{-1}$) and $l_3=+4$ and $l_4=-2$. When these two perfect STOV wavepackets collide completely [Fig. 4(e1)], it is clear that there are some dark dots at the contact point of the two perfect STOV wavepackets. As shown in Fig. 4(e2), the phase pattern of the superposition of the two perfect STOV wavepackets shows that a spatiotemporal optical vortex with $l=-1$ exists at each dark dot. The number of spatiotemporal phase singularity is $N=|l_4-l_3|$. Furthermore, their signs are easily determined by the sign of $N=l_4-l_3$. Therefore, the collision of these two perfect STOV wavepackets with different spatial and temporal diameters leads to the generation of an STOV lattice with controllable numbers and signs of vortices. Therefore, the perfect property and radial degrees of freedom of perfect STOV wavepackets can be used to flexibly generate richer and more complex spatiotemporal structure light fields.

Compared with other methods using different parameters for controlling the radius of the vortex ring [48] or using the combination of conical and helical phase for generating perfect STOVs [49], our method has the advantage of rigorously satisfying the spatiotemporal Fourier transform conditions both mathematically and physically while having a much simpler and straightforward experimental configuration. This is achieved by the spatiotemporal holographic shaping method to modulate the complex spatiotemporal optical field [33].

In conclusion, a perfect spatiotemporal optical vortex with spatiotemporal sizes independent of the topological charges is generated experimentally by the spatiotemporal Fourier transform of the polychrome Bessel–Gaussian beam. The properties of the perfect STOVs are verified through comparison with spatiotemporal Laguerre–Gaussian wavepackets. Furthermore, we show the spatiotemporal collision of two perfect STOV wavepackets by introducing a linear phase in the time dimension, where the spatiotemporal collision of two perfect STOVs with different diameters leads to the phenomenon of spatiotemporal vortex reconnection, resulting in STOV lattices consisting of spatiotemporal sub-vortices with a controllable number and sign. These perfect spatiotemporal optical vortices may be used to facilitate optical communications, particle trapping, and tweezing among other potential applications.