Sculpting light in spatial and temporal domains, commonly known as light field modulation, presents unique opportunities across optical communication^[1], bioscience^[2], materials science^[3,4], and physical science^[5]. In the spatial domain, the studies primarily concentrate on the modulation of the intensity^[6], polarization^[7], phase^[8], and coherence^[9] of optical fields capable of generating various exotic beams such as Bessel beams^[10,11], Airy beams^[6,12], vortex^[13,14], and cylindrical vector beams^[15,16]. Among these, Airy beams have attracted considerable research interest and are widely utilized in optical and electron microscopies^[17,18], self-bending surface plasmons^[19,20], and particle manipulation^[21,22] due to their distinctive self-accelerating, self-healing, and diffraction-free properties. An Airy beam consists of a central lobe accompanied by a series of side lobes, which can be generated through the introduction of a cubic phase with spatial-light modulators^[11], holographic interference technique^[23], or single diffractive elements^[24,25].

In the temporal domain, the intensity and profile of mode-locked pulses can be modulated through chirped-pulse amplification^[26] and pulse shaping techniques^[27], respectively. For example, square pulses can be generated by applying phase modulation proportional to the initial spectral intensity outside the laser cavity^[28], while pure-quartic solitons^[29] and multi-color soliton compounds^[30] are achieved through frequency phase modulation of pulses within the laser cavity. Analogous to the Airy beams, ultrashort Airy pulses are generated by applying cubic phase modulation in the frequency domain, constituting the unique dispersion-free solution in one dimension^[31,32]. Such pulses have also found important applications in supercontinuum generation^[33], soliton trapping and guiding^[34], as well as the formation of curved surface plasma waves^[20].

Recently, there has been a growing interest in Airy-related spatiotemporal wavepackets. A notable example belongs to the generation and propagation of Airy–Bessel bullets^[35], which involves a grating-telescope compensator to introduce third-order dispersion for the generation of ultrashort Airy pulses, followed by modulating the transverse modes into Bessel beams using an axicon. Then, Abdollahpour et al. achieved spatiotemporal Airy light bullets through a grating-telescope compensator and investigated their propagation characteristics in both linear and nonlinear regimes^[36]. Thereafter, Wang et al. demonstrated quantum Airy bullets by integrating temporal waveform modulation and spatial single-photon shaping^[37] in an atomic ensemble. These methods exhibit features of independent modulation in the temporal and spatial domains while requiring complex discrete spatial optical systems.

On the other hand, the combined modulation can directly generate Airy-related spatiotemporal coupled wavepackets like autofocused Airy–Gaussian vortex wavepackets^[38] and acceleration-free Airy wavepackets^[39]. Such an approach requires that the spatial frequency correlates with each wavelength of the pulse, and it involves a complex balance between the diffraction and dispersion effects^[40,41]. In this paper, we obtain spatiotemporal Airy wavepackets from a fiber laser based on a programmable pulse shaper (PPS) and a liquid crystal cubic-phase plate. By imparting the cubic phase in both the frequency domain and spatial domain, the Gaussian pulse is transformed into an Airy pulse, while the LP01 mode is shaped into an Airy beam, respectively.

The setup of the system is shown in Fig. 1(a), which is comprised of a mode-locked fiber laser, a PPS (commercial device: Santec WSS-1000), and a liquid crystal cubic-phase plate (fabricated via the photoalignment technology and a digital micro-mirror device-based micro-lithography system^[42]). The total length and net cavity dispersion of the fiber laser are 12.4 m and −0.3622ps2, respectively. Using a carbon nanotube saturable absorber (CNT-SA), sub-picosecond chirp-free pulses can be generated from the fiber laser. Subsequently, a PPS incorporating a diffraction grating, a cylindrical mirror, and a spatial light modulator adjusts the amplitude and phase of each frequency. The temporal profile of the Airy pulse is characterized using the cross-correlation technique, where the initial pulse is scanned against a modulated pulse to obtain its temporal characteristics. Subsequently, the Airy pulse is collimated and directed onto a liquid crystal plate, which introduces a cubic phase modulation to generate the desired spatiotemporal Airy wavepackets. Figure 1(b) summarizes the formation principle of spatiotemporal Airy wavepackets.

Passive mode-locking is achieved in the fiber laser at a pump power of 120 mW, giving birth to a train of pulses centered at 1558.04 nm with a 3-dB bandwidth of 3.82 nm [Fig. 2(a)]. The pulse width is given as 945 fs from the retrieved pulse profile [Fig. 2(b)]. The corresponding time-bandwidth product is 0.446, indicating the near-chirp-free property of the pulse. The average output power of the pulse is 4 mW, corresponding to the pulse energy of 0.24 nJ at the fundamental repetition rate of 16.424 MHz.

Airy pulses can be achieved by introducing a cubic phase to a Gaussian pulse in the frequency domain^[12,43], which is mathematically expressed as Uout(ω)=Uin(ω)exp[iφ(ω)]. Here, Uin(ω) represents the complex amplitude of the Gaussian pulse, and φ(ω)=A(ω−ω0)3 is the frequency phase. ω0 denotes the central angular frequency, and A represents the cubic phase coefficient.

As depicted in Figs. 2(c) and 2(d), for a positive cubic phase coefficient of 12.67ps3, the Airy pulse exhibits a series of smaller peaks preceding the main lobe with an approximate duration of 4.3 ps. Conversely, by reversing the sign of the cubic phase, an Airy pulse can be generated in which the main lobe precedes the oscillatory trail [Fig. 2(d)]. The emergence of this phenomenon can be attributed to the quadratic chirp of the pulse, which gives rise to temporal interference between longer and shorter wavelengths. Attributing the small residual second-order dispersion of the PPS and fiber system, the minimum value of oscillatory trails cannot approach zero.

The simulation results coincide with the experimental observations, in which the ideal Airy pulses are formed by applying the same cubic phase used in the experiment [Figs. 2(e) and 2(f)]. By applying a quadratic phase φ2(ω)=B(ω−ω0)2 to the Airy pulses, their oscillatory trails slightly deviate from the ideal cases [see insets of Figs. 2(e) and 2(f), where B=−20ps2]. Thus, we confirm that the residual second-order dispersion decreases the modulation depth of the oscillating trails in the Airy pulse.

The characteristics of Airy pulses can be adjusted by changing the coefficient of the cubic phase. As illustrated in Fig. 3(a), the modulation depth of the oscillatory trails increases with the cubic frequency phase, which results from the suppression of residual second-order dispersion. The pulse exhibits a similar evolution behavior for the negative cubic phase coefficients, except for the orientation of the Airy pulses [Fig. 3(b)]. Furthermore, the separation between the main lobe and the secondary lobe is proportional to the cubic phase coefficient, enlarging from 4.1 to 7.1 ps [Figs. 3(c) and 3(d)].

In the spatial domain, the output Airy pulse operates in the fundamental mode (LP01 mode), exhibiting an intensity distribution that closely resembles that of a Gaussian beam. It can be converted into Airy beams by imparting a cubic phase Φ(kx,ky)=exp[iβ(kx3+ky3)], where kx and ky represent spatial wavenumbers, and β characterizes the coefficient of the cubic phase.

After collimating the output laser, the wrapped cubic phase pattern of the liquid crystal plate is precisely aligned with the dimensions of the laser spot by adjusting the displacement stage that holds the liquid crystal plate. Figure 4(a) shows the micrograph image of the liquid crystal cubic-phase plate, with phase modulation ranging from −20π to +20π and size 1.2mm×1.2mm. By modulating the polarization state prior to the fiber collimator, a variety of Airy beams can be generated and captured using a CCD.

Figures 4(b)–4(d) illustrate the distinct morphologies of Airy beams generated under various laser polarization states. As shown in Fig. 4(b), a pair of symmetric dual Airy beams is generated when the incident Airy pulse is linearly polarized. For the left- or right-circular polarization state, LP01 modes are converted into single Airy beams facing opposite directions with the circular polarization state orthogonal to that of the incident ones [Figs. 4(c) and 4(d)]. This phenomenon arises from geometric phase modulation, which describes the relationship between the phase change and the polarization conversion when light passes through anisotropic media like liquid crystals^[44]. Jones calculus can be used to more intuitively illustrate the principle of geometric phase modulation^[45]. Briefly speaking, the magnitude of the geometric phase is proportional to the orientation angle of the liquid crystal optical axis and has a polarization-dependent sign. Herein, the optical axis orientation of the liquid crystal plate follows the cubic phase distribution, which can transfer a Gaussian beam into an Airy beam after Fourier transformation is implemented by a lens^[12]. Therefore, for left-circularly polarized incident light, an Airy beam with a right-circular polarization state can be generated after being modulated by the liquid crystal cubic-phase plate. For the right-circularly polarized incident light, a left-circularly polarized Airy beam will be produced. For linearly polarized light that can be decomposed into left- and right-circular polarizations, dual Airy beams can thus be obtained.

Although the morphologies of Airy beams vary under different polarization states, their spatial positions remain invariant. The only change observed is a slight variation in the generated geometric sizes. Both the dual- and single-mode beams exhibit a prominent main lobe accompanied by a set of two-dimensional side beamlets, whose intensity decays exponentially, perfectly aligning with the intensity distribution of the Airy beam. Thus, the temporal and spatial properties of Airy wavepackets can be precisely controlled by the PPS and the liquid crystal cubic-phase plate, respectively. Furthermore, the typical propagation distances of the Airy wavepacket in both the temporal and spatial domains are on the order of meters, allowing for the preservation of its shape during long-distance transmission.

We further investigate the self-healing characteristic of the spatiotemporal Airy wavepackets. During the transformation from Gaussian pulses to Airy pulses, the cubic phase modulation introduces delays in both the low- and high-frequency components of the pulse. As a result, the central region of the spectrum is associated with the main lobe of the Airy pulse [Fig. 5(a)]^[35]. By selectively filtering out the central region of the spectrum, it is possible to attenuate the primary temporal lobe, leading to what is commonly referred to as the “injured” Airy pulse [Fig. 5(b)]. After propagating through a 10 m single-mode fiber, the primary lobe of the “perturbed” Airy pulse undergoes significant reconstruction, with its energy primarily sourced from the oscillatory tails [Fig. 5(c)].

In the spatial domain, directly obstructing the main lobe can readily damage the integrity of the Airy beam [Fig. 5(d)]. The self-healing phenomenon of the Airy beam can be observed through its transmission in free space [Fig. 5(e)]. As the CCD is gradually moved back, we observe that the side lobe energy of the injured Airy beam gradually transfers to the main lobe. After propagating 30 cm, the main lobe demonstrates nearly complete restoration [Fig. 5(f)], thereby illustrating its intrinsic self-healing property. The self-healing property of Airy wavepackets originates from their unique energy flow distribution, which facilitates the redistribution of energy from the side lobes to reconstruct the primary lobe with partial damage. Consequently, although the experimentally generated Airy beams possess finite energy, they still exhibit essential properties of ideal Airy beams. The self-healing characteristic makes Airy wavepackets have a strong resistance to interference or turbulence in complex media.

In addition, we emphasize that the dynamic manipulation capabilities of spatiotemporal Airy wavepackets in both temporal and spatial dimensions significantly enhance focusing precision while minimizing sample damage. This makes them especially suitable for imaging dynamic specimens^[18,46] and super-resolution microscopy^[47–49]. Moreover, the self-healing properties of these spatiotemporal Airy wavepackets enable their applications in high-speed and long-distance optical data transmission^[50], thereby substantially improving the efficiency and security of multiphoton communication systems^[51].

In this paper, we successfully generated self-healing spatiotemporal Airy wavepackets using a PPS and a liquid crystal cubic-phase plate. In the temporal domain, a mode-locked pulse can be transformed into an Airy pulse by applying cubic phase modulation to its spectrum. Subsequently, the LP01 mode is converted into an Airy beam via propagation through a cubic phase plate, thereby forming the spatiotemporal Airy wavepackets. The self-healing properties of these wavepackets were experimentally verified by propagating their partially obstructed counterparts through single-mode fibers and free space. Our research demonstrates a straightforward, efficient, and stable method for generating spatiotemporal Airy wavepackets, providing a robust framework for producing other spatiotemporal optical fields.