Airy beams have inspired tremendous research for their exotic properties, such as diffraction-free, self-accelerating, and self-healing¹. The demand for generating tunable Airy beams has steadily increased since their discovery, driven by the controlled trajectories^2-4. It sparked growing research interest across diverse applications⁵, including optical manipulation^{6, 7} and laser fabrication⁸. In the traditional configuration, an optical tweezer is propelled by a tightly focused Gaussian beam to produce an optical gradient force, which serves to confine particles within a limited range of just a few micrometers, primarily because of the relatively short Rayleigh lengths⁷. In contrast, the precisely controlled propagation trajectory of the Airy beam enables the optical manipulation of particles in air or liquid along designated paths and provides the capability to navigate obstacles. These beams can also facilitate the creation of surfaces with desired curvatures and enhance the adaptability of laser fabrication. While the feasibility of demonstrating these applications has been established, the Airy beam still requires certain tunable characteristics to enhance its practical utility.

Airy beams were traditionally generated by bulky and costly optical devices, such as complex optical lens system⁹, spatial light modulators (SLM)¹⁰, etc. These methods offer the capability to achieve tunable Airy beams. Utilizing a complex optical lens system generates tunable Airy beams by manipulating the tilt angle of cylindrical telescopic systems. SLMs are diffractive devices engineered to create tunable Airy beams by performing pixel-level phase changes. While these methods offer a degree of Airy beam control, each method comes with its challenges. The intricate optical setup inevitably encounters difficulties in achieving precise alignment. SLMs face challenges such as low conversion efficiency, restricted resolution, and limitations related to incident polarization and power. These techniques also struggle to enable the realization of compact and integrated optical systems due to the utilization of bulky components.

Metasurface is an advanced flat optical device composed of artificial nano-antennas^11-16. It offers an unusual way to engineer the light’s amplitude^{17, 18}, phase^{19, 20}, wavelength^21-23, polarization^24-26, orbital angular momentum²⁷, etc. Numerous functions and applications have been developed based on ultrathin optical devices^28-35, such as meta-lenses^36-39, nonlinear meta-lens⁴⁰, orbital angular momentum manipulation^41-43, logic gate⁴⁴, polarization control and measurement²⁵, structural color⁴⁵, single-shot phase-imaging devices^46-48 and other novel meta-devices⁴⁹. It is thus an ultra-compact and flat technique for Airy beam generation^{4, 50-55}. For example, Lei et al. demonstrated a series of Airy beams with different trajectories by combining the cubic phase with different lens profiles⁴; Lu et al. showed a way to control the trajectories of the Airy beams by integrating the cubic phase and a gradient phase into a single metasurface⁵⁰. While the currently demonstrated works offer a variety of methodologies for manipulating the propagation trajectory or the focal spot position of the Airy beam in a compact way, real-time tuning of the generated Airy beam remains challenging.

In this work, we demonstrate a novel method for generating tunable Airy beams using a bilayer all-dielectric meta-device. It is meticulously crafted by integrating and rotating various phase profiles, including the cubic phase and two off-axis Fresnel lens phase profiles. By leveraging the transmission phase properties of the nanoantennas at specific positions, the meta-device effectively functions as a tunable optical wavefront modulator, showing the capacity of tunable trajectory manipulation. This inventive approach allows us to dynamically manipulate the trajectories of Airy beams by rotating these two metasurfaces, as shown in Fig. 1. We conduct experimental investigations on a proof-of-concept meta-device to validate its viability and flexibility. Our experimental findings align well with the theoretically predicted intensity profiles and propagation dynamics of the Airy beam.

We implement the all-dielectric meta-device as follows: a combination of a cubic phase and one of the Fresnel holographic lens phases is applied to one metasurface, while the other metasurface is equipped with another Fresnel holographic lens phase. Note that the two Fresnel holographic lens phases are both off-axis and have opposite focal lengths. The generation of Airy beams involves performing a Fourier transformation on Gaussian beams that propagate through the metasurface exhibiting cubic phase modulation. A tunable gradient phase can be achieved by superimposing these two off-axis Fresnel holographic lenses. The tunable gradient phase enables the flexible modulation of the Airy beam in response to varying propagation trajectories in the longitudinal direction and focal spot locations in the transverse plane. We opted to utilize Fresnel holographic lenses instead of directly employing two gradient phases¹⁹ due to the benefits of focusing phases. Focusing phases can mitigate losses caused by the divergence of the light beam during propagation, ensuring optimal transmission through the first metasurface and facilitating its subsequent passage to the second metasurface.

The Airy beams are generated by the Fourier transformation of Gaussian beams combined with cubic phase, and the transverse field distribution, h(u,v), can be written as⁴

$h\left(u\mathrm{,}v\mathrm{,}z=0\right)=Ai\left(\frac{u}{u_0}\right)\text{exp}\left(a\frac{u}{u_0}\right)\cdot Ai\left(\frac{v}{v_0}\right)\text{exp}\left(a\frac{v}{v_0}\right)\mathrm{,}$(1)

where Ai(u, v) is the Airy function, a is a constant. (u, v) are dimensionless transverse coordinates of the Airy beam. (u₀, v₀) is an arbitrary transverse scale. By careful calculation, the Fourier transform of the above Airy beam at the origin, H(x, y), could be written as

$H\left(x\mathrm{,}y\right)=b^2\text{exp}\left[\frac{\text{i}{\left(2\pi b\right)}^3\left(x^3+y^3\right)}{3}\right]\mathrm{,}$(2)

where b is a constant with units of m^–1; i is the imaginary unit; (x,y) is spatial frequencies in the Fourier frequency domain. Now we get the cubic phase ϕ(x_meta, y_meta) for Airy beam generation with metasurface, which is

$\varphi =\frac{{\left(2\pi b\right)}^3\left(x_{\text{meta}}{}^3+y_{\text{meta}}{}^3\right)}{3}\mathrm{,}$(3)

where, (x_meta, y_meta) is the coordinate of the metasurface phase plane.

The tunable gradient phase could be obtained using two Fresnel holographic lens phases, which are both off-axis and have opposite focal lengths. One phase profile has a positive focal length of f_d and a displacement of d to +x direction. The other has a corresponding negative focal length and a displacement to the left. We also assume that the first Fresnel holographic lens phase is rotated counter-clockwise by an angle of θ/2 around the central axis, the other symmetrically clockwise by an angle –θ/2. Then, the corresponding phase profiles P₁ and P₂ can be written as⁵⁶

$P_1=-\frac{\pi }{{\lambda }_d{f}_d}\left({\left[x_{\text{meta}}+d\cos \frac{\theta }{2}\right]}^2+{\left[y_{\text{meta}}-d\sin \frac{\theta }{2}\right]}^2\right)\mathrm{,}$(4)

$P_2=\frac{\pi }{{\lambda }_d{f}_d}\left({\left[x_{\text{meta}}-d\cos \frac{\theta }{2}\right]}^2+{\left[y_{\text{meta}}+d\sin \frac{\theta }{2}\right]}^2\right)\mathrm{,}$(5)

where λ_d is the design wavelength of the two Fresnel holographic lenses. Adding the two phases, we could get the total phase profile P_tot, which is

$P_{\text{tot}}=-4d\frac{\pi }{{\lambda }_d{f}_d}\cos \frac{\theta }{2}{x}_{\text{meta}}\mathrm{,}$(6)

We get a phase gradient with a grating vector G,

$G=4d\frac{\pi }{{\lambda }_d{f}_d}\cos \frac{\theta }{2}{e}_x\mathrm{,}$(7)

where e_x is a unit vector in the x-direction. Then, the beam deflection angle β in x-direction of a normal incident light will be,

$\beta =\arcsin \left[\frac{\left|G\right|\lambda }{2\pi }\right]=\arcsin \left[\frac{2d\lambda }{{\lambda }_d{f}_d}\cos \frac{\theta }{2}\right]\mathrm{.}$(8)

Here, λ is the working wavelength. We can find that the magnitude of the phase gradient depends on the metasurface's rotation angles, and parameters of the off-axis Fresnel holographic lens phases determine the maximum magnitude. The theoretically achievable zone is constrained once the meta-device has been manufactured. These achievable zones can be tailored to suit specific applications by tuning the parameters of the phase profiles before fabrication. We account for more realistic conditions in the simulation and experiment section. Initially, we compute the complex amplitude (E₁) when light is about to impinge on the second metasurface (1 mm away from the first one), and subsequently determine the complex amplitude (E₂) after passing through the second metasurface. We then employ E₂ as the new source for calculating the Airy beam distribution. Without any mathematical approximation, the experiment setup is consistent with the simulation to obtain excellent agreement.

For a compact meta-device design, we set the phase profile P₁ for the first metasurface (near the incident light) and combine the cubic phase profile ϕ and the off-axis Fresnel holographic lens profile P₂ to form the phase of the second metasurface. When the phases of two metasurfaces superimpose, the squared terms of the respective focused phases nullify each other, leading to a gradient phase related to the rotational angle. As a result, the outcome comprises a cubic phase profile and a tunable gradient phase, as shown in Fig. S1. We set b to 2.5 and π/(f_dλ_d) to 220 and subsequently fabricated the sample as a proof-of-concept meta-device. With the tunable grating, the Airy beam can be tuned dynamically as designed, i.e., the propagation trajectories and focal spots of the Airy beams can be controlled flexibly.

The meta-device comprises arrays of amorphous titanium dioxide (TiO₂) cylindrical nanorods of different diameters and fixed height (800 nm) on a glass substrate. The period (p) is 300 nm and the diameters of the metasurfaces are both 1 mm. We choose TiO₂ due to its high refractive index, minimal surface roughness, and high transmission at visible frequency^57-59. Figure 2(a) shows the optical characteristics of the designed nanoantenna of the meta-device obtained through COMSOL Multiphysics^®. The working wavelength is 532 nm, the phase can cover 2π when the diameter varies gradually from 50 nm to 113 nm, and the efficiency of the selected nanoantennas is higher than 90%. Insert shows the schematic of the nanoantenna. Figure 2(b−d) shows the designed phase profile, optical image, and the measured phase profile of the first metasurface, respectively. Figure 2(e−g) shows the designed phase profile, optical image, and the measured phase profile of the second metasurface, respectively. The phase profile is measured using MetronLens from Ideaoptics Co., Ltd, Shanghai. Figure 2(h−i) shows scanning electron microscope (SEM) images of the meta-device. The fabrication process is shown in Fig. S2. Our results substantiate a high degree of congruence between design and fabrication.

We first provide two sets of detailed comparisons between simulation and experimental results to validate our design and fabrication, as shown in Fig. 2(a−c) and Fig. 2(d−f), respectively. The simulation method is based on the Fresnel diffraction algorithm¹⁹. The experimental setup can be found in Fig. S3. We will methodically present the comparative analysis of simulation and experimental results from three perspectives: the intensity distribution along the propagation of the Airy beam, the intensity distribution in the focal plane, and the intensity distribution along a single line passing through the focal spot. The rotation angles of the two metasurfaces are both zero in the case shown in Fig. 2(a). Figure 2(b) and 2(c) show the simulated and experimental intensity of the longitudinal distribution (left column), focal plane (upper right), and a line in the focal plane (lower right). The simulation and measured results match well with each other in all these aspects. Figure 2(d−f) illustrates the cases where the rotation angles for the two metasurfaces are -π/2 and π/2, respectively. The presented results further validate the accuracy of our design. The measured efficiency in the generation of Airy beams is approximately 15%. Upon practical implementation of the proposed meta-device, the separation between these two metasurfaces can be further minimized to mitigate diffraction losses during transmission. In both cases, the Airy beams were measured and exhibited a strong and clear agreement with the predicted results, showcasing distinct focal spots and controlled trajectories.

Based on the simulation and experimental results depicted in Fig. 3, it is evident that the Airy beam exhibits the remarkable ability to sustain its focused state over a propagation distance of up to 0.5 mm, which is nearly 1000 times the working wavelength. This underscores the non-diffraction attribute of the Airy beam. In contrast to the conventional Gaussian beam, which, focusing, yields only a single focused spot that rapidly diverges, the non-diffraction property of the Airy beam guarantees the maintenance of focus and preservation of its initial shape and intensity distribution during propagation. This characteristic holds promise for applications in optical manipulation, such as nanoparticle transportation along its transmission path. The self-accelerating feature of Airy beams denotes the phenomenon wherein an Airy beam inherently bends and accelerates while traversing a curved trajectory, setting it apart from traditional Gaussian beams. This self-accelerating attribute is of great practical value in specific scenarios, especially in situations requiring obstacle avoidance or the tracing of complex light pathways. We posit that it can be applied effectively in laser manufacturing processes for nanostructures with curved surface profiles.

We present comprehensive experimental results in Fig. 4. The focal spots of the Airy beams achieved by rotating the metasurfaces are illustrated in Fig. 4(a). The relationship between the rotation angles and focal spot positions is listed in Table S1. From Fig. 4(a), we observe the flexibility of steering focal spots within the focal plane, implying diverse propagation trajectories. We have chosen several cases labeled with colored squares to demonstrate the intensity distribution across the focal planes, as depicted in Fig. 4(b), and the scalar bar is 20 µm. Airy beams with side lobes of different orientations are observed, increasing the manipulation flexibility of the propagation trajectories. This is reasonable because the rotation of the second metasurface determines the orientations of the side lobes of the Airy beams.

In summary, we have demonstrated a new method for tuning the Airy beam’s focal spots and propagation trajectories for optical manipulation and laser manufacturing. The Airy beams can be tuned dynamically by rotating the meta-device once encoding a cubic phase profile and two off-axis Fresnel lens phase profiles into two cascading metasurfaces. The manipulation capacity of an Airy beam's propagation trajectory and coverage range can be easily achieved in demand by tuning the parameters of these phase profiles. Without enlarging the device footprint, our approach effectively enhances the modulation flexibility of Airy beams. Real-time rotation of metasurfaces can be performed by piezoelectricity, further strengthening the tunability and flexibility of our meta-devices. Compared with the traditional tilted cylindrical telescopic system or the SLM, the demonstrated meta-device significantly reduces the volumetric thickness and operational complexity and can be easily transferred to other working bands without polarization or other limitations. The proposed meta-device, taking advantage of miniaturization and easy control, can be compatible with other optical devices and holds promise for various applications.