Diffraction and dispersion, which tend to distort optical wave packets during propagation, are ubiquitous phenomena. However, certain optical waves, known as localized waves, maintain their shape during propagation. Certain localized beams such as Bessel beams and Airy beams are well known for resisting diffraction^[1]. These localized beams exhibit a unique property of propagation invariance: so-called non-diffractive propagation. Another unique property is the self-healing effect, where localized beams reconstruct their spatial shape during propagation even though parts of the beam are blocked^[2]. With these properties, the localized beams have numerous potential applications, such as micromanufacturing^[3], tomography^[4], and optical communication^[5].

In contrast to two-dimensional (2D) beams, the Airy beam is a one-dimensional (1D) localized beam as the solution of a paraxial equation^[6]. As the localized beam, the Airy beam also exhibits non-diffractiveness during propagation along with the self-healing property^[7]. However, the peak of the Airy beam follows a unique parabolic trajectory during propagation, a phenomenon known as free-acceleration^[8]. With these properties, the Airy beam has found numerous applications, including nonlinear optics^[9,10], plasma physics^[11], optical communications^[12], metasurfaces^[13], and high-energy physics^[14]. There are various methods to generate Airy beams, but a common approach is to use a spatial light modulator (SLM)^[15].

The free-acceleration effect enables a unique beam manipulation. When the Airy beam profile is radially distributed (in the cylindrical coordinates), the beam freely accelerates toward the center and eventually forms a highly intense central spot. This process is called autofocusing since the focusing effect occurs solely through free-space propagation. At the same time, the high-intensity spot emerges abruptly, transitioning from nearly zero intensity to high intensity. This abruptly autofocusing effect was first reported, characterized by two-dimensional radially symmetric Airy beams converging to the center point during propagation^[16]. This abruptly autofocusing phenomenon is widely exploited in various applications^[17–20] and also observed in other beams, such as vortex beams and Pearcey beams^[21,22]. Recently, numerically simulated results of the generation and dynamics of several spatiotemporal Airy-related waves or other autofocusing waves were reported^[23–27].

In time, a pulse is broadened due to the group velocity dispersion (GVD) effect. However, since the pulse propagation is also governed by a 1D paraxial equation, an Airy pulse remains non-dispersive during propagation. In addition to its non-dispersive propagation, an Airy pulse also exhibits self-healing and self-acceleration in the presence of GVD^[28]. Based on these properties, the Airy pulse is useful for several application fields, such as nanomachining^[29] and biophotonics^[30,31]. By placing two counter-propagating Airy pulses, they accelerate toward each other in a dispersive material^[32], and eventually constructively interfere, resulting in a high-peak-power pulse^[33]. This process can be understood as the temporal autofocusing.

By combining the temporal autofocusing and the spatial autofocusing of the radial Airy beam, a three-dimensional (3D) abruptly autofocusing wave packet is predicted^[16]. In this Letter, we demonstrate this 3D abruptly autofocusing wave packet. First, two counter-propagating Airy pulses are generated using an SLM-based pulse shaper^[34]. The counter-propagating Airy pulses acquire the radially symmetric Airy beam profile using another SLM. The spatiotemporal profile of this abruptly autofocusing wave packet is measured using a 3D pulse intensity diagnostic technique^[35]. So, it has potential applications in many fields such as nanomaterial processing.

Spatiotemporal wave packets in a dispersive medium are governed by a 3D paraxial wave equation: ∂2ψ∂x2+∂2ψ∂y2+2ik∂ψ∂z−kβ22∂2ψ∂T2=0.(1)Here, k is the wave number of light and z is the propagation distance. The parameter β2 denotes a GVD coefficient, defined as β2=∂2k/∂ω2, and T=t−(z/vg) represents the local time frame, where vg is the group velocity of light. For the two counter-propagating Airy pulses, which will be referred to as dual Airy pulses, with the radially distributed Airy beam, which will be referred to as the Airy ring beam, in this Letter, the wave packet is mathematically described as $$\begin{gathered} ψ(r,T)=A0Ai(r−r0w)exp[α(r0−r)] \\ ×{Ai(T−bT0)exp[γ(T−b)] \\ +Ai(−T+bT0)exp[−γ(T+b)]}. \end{gathered}$$(2)

In Eq. (2), A0 is the amplitude of the wave packets, Ai(x) is an Airy function, and r0 is the radius of an initial Airy ring beam. Since the ring radius decides the required propagation distance for the ring to autofocus, r0 determines the focal length. r is the cylindrical radial distance where r=x2+y2. w is a scaling factor to determine the radial width of the Airy ring beam. After autofocusing, the focused spot size α denotes the decaying parameter to generate a finite-energy Airy ring beam. In the time domain, γ denotes the decaying factor for finite-energy Airy pulses, while T0 determines the temporal width of wave packets (the main lobe duration of the Airy pulse). b sets the temporal separation of the counter-propagating Airy pulses. Similar to the spatial case, temporally focused pulse duration is determined by T0, whereas the focal length is governed by b. Equation (2) represents the superposition of two time-reversed Airy pulses, each possessing the same radial Airy ring beam distribution.

A numerical simulation has been performed to demonstrate the 3D autofocusing effect of the suggested optical wave packet. Figure 1 visualizes the initial iso-intensity profile and the internal intensity distribution of the wave packet. As the wave packet propagates through a dispersive medium, the radial Airy ring converges to the center, while the Airy pulses merge in the middle as shown in Fig. 2, which clearly illustrates the 3D abruptly autofocusing effect.

An initial pulse is emitted from a mode-locked ytterbium (Yb) fiber oscillator. The oscillator has a Gaussian beam output with a mode field diameter (MFD) of approximately 1 mm. The laser spectral bandwidth is ∼80nm at 1030 nm, which corresponds to a Fourier transform-limited pulse duration of approximately 54 fs. We describe the experimental method to generate the wave packets in Fig. 3. The initial pulsed beam is split into two paths. The object beam is modulated using an SLM-based pulse shaper^[34]. To convert the initial pulsed Gaussian beam into two counter-propagating Airy pulses, we utilize the method described in Ref. [33]. In detail, linear phases with an opposite slope (i.e., triangular spectral phase) are imposed on the first SLM (SLM 1), resulting in two temporally separated pulses. Subsequently, a cubic spectral phase is applied to each of these pulses, transforming them into Airy pulses. By controlling the sign of the cubic spectral phase, the direction of the tail for each Airy pulse can be adjusted to form dual Airy pulses that face each other. Such a phase pattern is described as Eq. (3) and shown in Fig. 4(a): $$\begin{gathered} φSLM1=a0(ξ+ξ0)−a0(ξ−ξ0) \\ −a1(ξ+ξ1)3+a1(ξ−ξ1)3. \end{gathered}$$(3)

In the pulse shaping system, the input pulse is Fourier-transformed from the time domain to the frequency domain on the SLM 1. Then, we can modulate the spectral phase by adjusting the phase on the SLM 1 according to Eq. (3), where ξ represents a horizontal coordinate on the SLM 1. The first and second terms in Eq. (3) are responsible for converting the initial pulse into two separate pulses. The third and fourth terms represent the phase modulation to shape each pulse into an Airy pulse.

Following pulse shaping, the beam is expanded three times using a telescope. The second SLM (SLM 2) works as a beam shaper, modifying the Fourier spatial frequency components according to the Fourier transform in the spatial domain. The phase applied on the SLM 2 is given by Eq. (4) [shown in Fig. 4(b)]: φSLM2=−b0(ρ−ρ1)−b1(ρ−ρ2)3.(4)

In Eq. (4), ρ=ξ2+η2 is a radial component in polar coordinates, where η is the vertical coordinate on the SLM 2. The first phase term transforms the input beam into a ring-shaped beam, while the second converts it into an Airy ring beam. As the Fourier transform is performed by a Fourier lens, the Airy ring beam is generated.

Applying a conic spatial phase, similar to the pulse shaping mechanism, generates the ring beam. Introducing a cubic phase in the radial direction induces a radial Airy profile onto the spatial ring beam. As a result, an Airy ring beam-dual Airy pulse wave packet is formed. Theoretically, the transformed beam shape can be expressed as the Hankel transform of a Gaussian beam, as shown in Eq. (5): G(r)=∫0∞ρg(ρ)J0(2πρr)exp[iφSLM2(ρ)]dρ.(5)

Here, g(ρ) represents the initial beam profile, assumed to be Gaussian, and J0 denotes the Bessel function of the first kind of order zero. For a small beam size, the J0(x) can be approximated using the Taylor expansion as J0(x)≈1−14x2≈cos(x). Hence, the Hankel transform can be approximated as a cosine Fourier transform of a radial Hermite-Gaussian function. The small beam approximation holds when ρ≪2fλr. In this expression, ρ is the beam radius on the SLM, r is the beam size after the Fourier lens, f is the focal length of the lens, and λ is the wavelength. In the range of (ρ>0), the Hermite-Gaussian function closely resembles a shifted Gaussian function. As a result, the transformed beam can be approximated as a Fourier transform of a shifted Gaussian function with a cubic phase, ultimately forming an Airy function in the radial direction.

On the other optical path, the pulse is dechirped using a grating pair to produce the transform-limited (reference) pulse. The 3D intensity measurement is performed by overlapping the object and reference wave packets utilizing the diagnostic technique described in Ref. [35]. As the object and reference wave packets are combined at a slight tilt angle, an interference pattern is generated. Since the reference pulse is substantially shorter, the interference pattern reveals the intensity distribution of the object wave packet at the reference pulse temporal position. This process is repeated to capture numerous interference patterns and therefore the intensity distributions at various reference pulse delays. By combining these intensity distributions, the 3D intensity profile of the wave packet can be reconstructed.

The experimentally generated Airy ring beam-dual Airy pulse wave packet is shown in Fig. 5. Figure 5(a) denotes the cross-section of the wave packets, while the 3D iso-intensity profile of the wave packet is shown in Fig. 5(b). The spatial profile reveals the ring beam pattern with tails extending in radial direction, as predicted by the Airy ring beam structure. In the temporal domain, the cross-sections of the wave packets clearly show two pulse structures with tails extending in opposite directions, as predicted by the dual Airy pulse configuration. The Airy ring beam-dual Airy pulse wave packet propagates through a 4-inch-long SF11 glass. This material induces enough diffraction and dispersion effect, causing the wave packet to autofocus in both space and time. By controlling the phase in SLM 1 for the pulse, and SLM 2 for the beam, the propagation of both the beam and the pulse can be adjusted to ensure that spatial and temporal autofocusing occur precisely at the same location, thereby achieving the 3D autofocusing. The measured autofocused wave packet is shown in Figs. 6(a) and 6(b). In the experiment, an initial Airy ring beam with a diameter of ∼1.8mm is autofocused to a beam size of ∼130μm. Meanwhile, Airy pulses separated by ∼500fs are temporally autofocused to a pulse with a duration of ∼110fs, due to propagation through the SF11 glass. The measurement clearly shows a high-intensity spot appearing at the center of the wave packet, providing evidence of the abrupt autofocusing phenomenon.

In the experiment, the abruptly autofocused focal spot size and the minimum pulse duration are limited by the angular frequency bandwidth and the spectral bandwidth of the wave packet. The former one is proportional to the beam spot size before SLM2 (3 mm) and is inversely proportional to the focal length of the Fourier lens (400 mm). The spectral bandwidth, on the other hand, is limited by the finite spectral bandwidth of the pulses from the laser. The final pulse duration of the autofocused wave packet cannot be shorter than the original laser pulse duration.

This 3D abruptly autofocusing effect occurs in the spatial and temporal domains. As the pulse feature does not modify the beam profile, this approach maintains the same spatial resolution, versatility, and focal spot size as in the autofocusing beam case. However, by adding the pulse features, it is possible to achieve high-peak-power autofocusing. The wave packet can be precisely controlled for the spatial and temporal autofocusing occurring at a desired location simultaneously to achieve a high peak power with a more enhanced autofocusing effect than that of the beam-only case. Meanwhile, since we employ beam shaping and pulse shaping techniques using SLMs with finite pixel sizes, there are fundamental limitations. Spatial/temporal window sizes will be constrained, and there is the potential appearance of a remote beam/pulse. This can be a limitation compared to other methods, such as a phase mask.

We have demonstrated the 3D abruptly autofocusing phenomenon by the Airy ring beam-dual Airy pulse wave packet. As this wave packet propagates through the dispersive material, a high-intensity spot forms at the center, providing evidence of the abruptly autofocusing phenomenon. This wave packet holds potential for a variety of applications, such as micromachining and other precision material processing^[36,37].