Abstract
Airy beams exhibit intriguing properties such as nonspreading, self-bending, and self-healing and have attracted considerable recent interest because of their many potential applications in photonics, such as to beam focusing, light-sheet microscopy, and biomedical imaging. However, previous approaches to generate Airy beams using photonic structures have suffered from severe chromatic problems arising from strong frequency dispersion of the scatterers. Here, we design and fabricate a metasurface composed of silicon posts for the frequency range 0.4–0.8 THz in transmission mode, and we experimentally demonstrate achromatic Airy beams exhibiting autofocusing properties. We further show numerically that a generated achromatic Airy-beam-based metalens exhibits self-healing properties that are immune to scattering by particles and that it also possesses a larger depth of focus than a traditional metalens. Our results pave the way to the realization of flat photonic devices for applications to noninvasive biomedical imaging and light-sheet microscopy, and we provide a numerical demonstration of a device protocol.
1 Introduction
As a nontrivial solution of the paraxial equation of light [1], an Airy beam exhibits many remarkable features, such as self-bending (even in the absence of any external potential field), nonspreading, and self-healing after diffractions by obstacles [2], [3], [4], [5], [6], [7]. Owing to these attractive properties, Airy beams have many potential applications in photonics, such as for particle manipulation [8], as light bullets [9], [, 10], for super-resolution imaging [11], [, 12], as autofocusing Airy (AFA) beams [13], [14], [15], and for light-sheet microscopy [16]. In 2007, Siviloglou and Christodoulides [17] proposed a truncated form of Airy beam and then realized it experimentally. Such a simplified version of an Airy beam, possessing finite energy, preserves all the key features of an ideal Airy beam and has therefore attracted much attention recently.
The conventional generation of Airy beams utilizes spatial light modulators (SLMs), which are bulky and lack fine spatial resolution [18], [19], [20]. The Airy beams generated in this way do not exhibit good qualities, and the bulky generation systems are unsuitable for practical applications. Recently, plasmonic Airy beams [19], [21], [22] have been successfully generated on metallic surfaces on which are placed nanoscatterers that have been carefully designed to convert impinging light to surface plasmon waves with the desired amplitudes and phases. Although these devices are compact and exhibit improved spatial resolution, the generated Airy beams can flow only on certain planes, and the working efficiencies are quite low owing to the intrinsic losses and nonideal performances of the metallic nanoscatterers that are used. For practical applications, it is highly desirable to have ultracompact and broadband devices that can efficiently generate Airy beams in free space.
Metasurfaces, ultrathin metamaterials consisting of subwavelength microstructures (e.g., meta-atoms) with tailored optical properties, offer a fascinating platform on which to realize planar photonic devices with desired functionalities [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. In the past few years, Airy beams have been successfully generated by carefully designed metadevices constructed from meta-atoms of different types (metallic or dielectric resonators) that can scatter electromagnetic waves with desired amplitudes and phases [20], [,38], [39], [40]. These ultrathin devices can generate Airy beams in free space with relatively high efficiencies. However, since the phase responses of the adopted resonating meta-atoms typically exhibit Lorentz-like frequency dispersion, such metadevices can usually work only at a specific single frequency at which the system precisely exhibits the phase/amplitude distributions required by the analytical formula. Despite several attempts (using, for example, geometric-phase metasurfaces [38], [39], [41], [42] or double-stacked metasurfaces [40]), the constructed metadevices still suffer from chromatic issues [3], [,43], [44], [45], [46], [47], [48], which hinders their practical application.
In this paper, we experimentally realize achromatic Airy beams in the terahertz (THz) regime with a carefully designed dielectric metasurface. Our metadevice consists of a set of silicon posts with different sizes and orientation angles determined by the requirement to generate the desired phase profile for achromatic Airy beams within a broad frequency band of 0.4–0.8 THz. We demonstrate that the generated achromatic Airy beams exhibit self-healing properties. In addition, an Airy-beam-based metalens is constructed by combining two such achromatic Airy beams and possesses a large depth of focus and robustness against scatterers. Our results stimulate us to propose a device protocol for Airy-beam-based metalens microscopy based on the proposed metasurface.
2 Results and discussions
2.1 Design principle for generation of THz achromatic Airy beams
Rather than using a fixed spatial wavefront to generate a chromatic Airy beam, here we exploit a wavelength-dependent spatial phase based on a silicon metasurface to generate an achromatic Airy beam with the same trajectory. A schematic of the achromatic Airy beam generator is shown in Figure 1(a). The incident THz wave impinges vertically on the metasurface device and the transmitted THz wave follows the trajectory along the white dash line f(z). To realize the achromatic Airy beam, here we study the truncated Airy beam to acquire the wavelength-dependent spatial phase of the achromatic Airy beam. The Airy beam is generated by an initial field distribution in the metasurface plane at z = 0, given by
![Figure 1: Schematic and phase profile of an achromatic Airy beam.(a) Schematic of the experimental generation of terahertz (THz) achromatic Airy beams by a silicon metasurface. The transmitted left circular polarization (LCP) THz intensities in the x–z plane fit the parabolic trajectory f(z) for the right circular polarization (RCP) incident light. (b) To determine the phase distribution φ(x,f)$\varphi \left(x,f\right)$ of the metasurface under design, we draw an auxiliary line (black solid line) tangent to the parabolic asymptotical trajectory f(z) of the desired Airy beam (black dash line) at an arbitrary point, which intersects with x axis exhibiting an angle θ. With θ(x) function obtained by repeating such a process, we then obtain φ(x,f)$\varphi \left(x,f\right)$ based on Eq. (1). (c) Phase profile of an achromatic Airy beam within the frequency range f∈[fmin,fmax]$f\in \left[{f}_{\text{min}},{f}_{\text{max}}\right]$, where fmin=0.4 THz${f}_{\text{min}}=0.4\text{\,THz}$, fmax=0.8 THz${f}_{\text{max}}=0.8\text{\,THz}$, φshift=820∘${\varphi }_{\text{shift}}={820}^{\circ }$ at 0.8 THz, and the arbitrary transverse scale x0 = 650 µm in our design.](/document/doi/10.1515/nanoph-2020-0536/asset/graphic/j_nanoph-2020-0536_fig_001.jpg)
Figure 1:
Schematic and phase profile of an achromatic Airy beam.
(a) Schematic of the experimental generation of terahertz (THz) achromatic Airy beams by a silicon metasurface. The transmitted left circular polarization (LCP) THz intensities in the x–z plane fit the parabolic trajectory f(z) for the right circular polarization (RCP) incident light. (b) To determine the phase distribution
When the analysis in the preceding discussion is simplified in the paraxial approximation, we obtain the following wavelength-dependent phase profile:
where the value of
We now illustrate how to design such a metasurface exhibiting the desired phase profile. With
and
Now our task is to design a series of meta-atoms, which not only yield the required phases required by Eq. (3) at the frequency fmin, but also exhibit the different frequency dispersions as dictated by Eq. (4). However,
Choosing an appropriate α value, we can make the term
We proceed to search a series of meta-atoms exhibiting the desired phases (Eq. (3)) at frequency fmin and frequency-variation slopes as required by Eq. (5). The building structures of the silicon metasurface are presented in Figure 2(a) and (b), consisting of solid and inverse rectangular structures on a silicon substrate. To fulfill the complex requirements on the phase responses, we design the metastructures based on a combination of two mechanisms, namely, the resonance mechanism yielding a frequency-dependent transmission phase and the Pancharatnam-Berry (PB) one yielding a frequency-independent phase. Since the PB mechanism is adopted, we assume the incident THz wave to exhibit RCP, and employ finite-difference time-domain (FDTD) simulations to determine the width w and length l of our metastructures (with fixed lattice period p and etching depth t) based on two criterions: 1) transmission phases exhibit linear dependences on frequency with desired slopes; and 2) they possess relative high conversion efficiencies to LCP within the frequency band of interest. The section of method provides more simulation details. Finally, we carefully select 32 different structural parameters containing solid and inverse rectangular structures in Table S1 and present the individual phase responses and conversion efficiencies in Figure S1.To directly present the coversion efficiencies, we provide an efficiency map (as shown in Figure S2) of the selected meta-atoms with respect to different frequencies. Here as an example, Figure 2(c) presents the simulated phase differences of the solid structure (sequence number 15, 580°) and the inverse structure (sequence number 27, 725°) with a same etching depth t = 350 μm, showing the linear phase response as a function of f (or 1/λ). Note that the inverse structures in Table S1 have a wider phase response than the solid structures in Table S1 and can compensate for a larger phase difference, so that the positions of the inverse structures are close to x = 0 and those of the solid structures are close to the left edge of the sample, as shown in Figure 1(a).

Figure 2:
Phase distributions and conversion efficiencies of the solid and inverse structure.
Side views of (a) a solid and (b) an inverse silicon structures with length l, width w, etching depth t, and lattice constant p on a silicon substrate. (c) Simulated phase responses (blue curves) and RCP-to-LCP polarization conversion efficiencies (red curves).
Before closing this section, we emphasize that only the LCP component of the transmitted wave can acquire the PB phases under RCP incidence [49], [, 50], so that only this wave component can generate the desired achromatic Airy beam. Moreover, the metadevice thus constructed does not work for linear-polarization incidence under which the meta-atoms do not generate any PB phase.
2.2 Numerical calculation of THz achromatic Airy beams
To confirm the desired phase profile φ (x,f) in Eq. (5) derived by the geometrical construction method and reveal the underlying mechanism of the achromatic Airy beam, we performed numerical calculations based on Fraunhofer diffraction integration. In our numerical results, the structural region ranged along x axis from x = −7 mm to x = +4 mm. Figure 3(a) shows the phase profiles at frequencies fmin = 0.4 THz, f = 0.6 THz, and fmax = 0.8 THz. Note that the phase profile at 0.8 THz is smaller than that at 0.4 THz, which is in conflict with the relation between transmission phase response and frequency. This impasse is broken by introducing an additional phase shift

Figure 3:
Fraunhofer diffraction integration of the phase profiles in Eq.(2) realized by PB phase.
(a) Phase profile of an achromatic Airy beam with additional phase shift
2.3 Experimental setup and fabricated samples
To validate the above theoretical analysis, we designed a silicon metasurface device for generating a broadband THz achromatic Airy beam. However, the theoretical design of achromatic Airy beams requires a phase compensation of 1350°, which cannot be compensated owing to the large aspect ratio in experimental fabrication. Therefore, we reduced the sample size, which ranges along the x-axis from −3.2 to 4 mm. In our design, we chose the additional phase shift
To verify the reliability of our designed metasurface devices, we fabricated samples to generate achromatic Airy beams and explore the corresponding characteristics using THz near-field scanning microscopy (NFSM), as shown in Figure 4(a). The polarizer controls the polarization of the light output by the fiber laser, which is focused by the lens and reflected by the mirror, and finally illuminates the light on the probe. In the real system shown in Figure S5 in the Supplementary material, the collimated THz waves radiating from a 100 fs (λ=780 nm) laser-pulse-pumped photoconductive antenna emitter are modulated with an appropriately polarized state. A commercial THz near-field probe is positioned 2 mm away from the sample to detect the Ex of the transmitted LCP light, with the RCP light illuminating the metasurface device. The sample is fabricated on a silicon wafer by conventional lithography together with deep reactive ion etching. Further details of sample fabrication are given in methods. Figure 4(b) shows an optical microscope image of the whole of the fabricated sample. Figure 4(c) and (d) shows partially enlarged scanning electron microscope (SEM) images of the inverse and solid structures, respectively. Note that the inverse structures with a larger phase compensation are located toward the right of the metasurface device, while the solid structures with a smaller phase compensation are located toward the left side.

Figure 4:
Experimental setup of the terahertz (THz) near-field scanning microscopy (NFSM) and sample images.
(a) Schematic of THz NFSM. (b) Optical image of the whole sample. (c) Partial enlargement on the right side (red desh box) of the inverse structures. (d) Partial enlargement on the left side (red box) of the solid structures.
2.4 Experimental detection of THz achromatic Airy beams
Figure 5(a)–(c) shows the normalized abs(Ex) distributions of the transmitted LCP light obtained from the THz NFSM at the three frequencies. Note that there are no data for the first 2 mm in the experimental results because of a safety distance imposed during the experiment to prevent collisions. In these figures, the normalized abs(Ex) distributions show a good match with the parabolic trajectory (the white dash line) at 0.4 THz. To verify the experimental results, Figure 5(d)–(f) presents the normalized intensity distributions Ex simulated by FDTD at frequencies 0.4, 0.6, and 0.8 THz, respectively. Furthermore, in Figure S6 in the Supplementary material, we present the results of Fraunhofer diffraction integration at more frequencies, the results of simulation, and experimental data, all of which verify the achromatism of the Airy beams. For a more intuitive comparison between the simulation results and experimental data, we obtain the lateral offsets x of the maximum normalized intensity (with error bars corresponding to 95% of the maximum normalized intensity) at different propagation distances z and compare them with the numerical deflection. Figure 5(g)–(i) shows the results of this comparison at the three frequencies, and it can be seen that there is good agreement. In the simulation, we calculated the working efficiency of the device, as shown in Figure S7 in the Supplementary material. The working efficiency ranges from 20 to 60% and is higher at frequencies near 0.6 THz. Furthermore, we present the simulation results for the self-healing properties of the achromatic Airy beam in the case of scattering by silicon particles in Figure S8 in the Supplementary material.

Figure 5:
Achromatic Airy beam experiments.
(a)–(c) Experimental results on near-field detections at the three frequencies, showing that the intensity distributions of the left circular polarization (LCP) field transmitted through the metasurface devices match well with the white dash line indicating the parabolic trajectory. (d)–(f) Simulation results at the three frequencies. (g)–(i) Comparison of experiment, simulation, and theoretical deflections, where the error bars correspond to 95% of the maximum normalized intensity.
2.5 THz achromatic Airy-beam-based metalens
Airy-beam-based metalenses have many advantages for detection, such as large depth of focus (DOF) and self-healing properties, and so here we design a metalens based on autofocusing achromatic Airy beams. To realize this, the achromatic Airy-beam-based metalens, a metasurface device composed of two symmetrical structures is designed to generate two counter-propagating achromatic Airy beams. Figure S9 in the Supplementary material shows optical and SEM images of the metasurface device. To highlight the robustness of the Airy-beam-based metalens, an aluminum foil cover is placed over a 1 mm region (ranging along the x-axis from x = −0.5 mm to x = +0.5 mm) around the center of the experimental sample. Figure 6(d)–(f) shows the normalized abs (Ex) distributions of the metalens at frequencies 0.4, 0.6 and 0.8 THz, respectively. Experimental results are presented on the left and FDTD results on the right. Experimental results at other frequencies are shown in Figure S10 in the Supplementary material. In particular, it should be noted that there is less energy in the region covered by aluminum foil, which is quite different from what occurs with a traditional metalens. The intensity distributions in the vertical direction at the center of the two cases can be found in Figure S11 in the Supplementary material. A comparison shows that the Airy-beam-based metalens has the advantage of a larger DOF and detection with a high signal-to-noise ratio. Interestingly, the starting points of the autofocusing effect fit well with the intersection of the parabolic trajectory. When two counter-propagating achromatic Airy beams begin to intersect, constructive interference occurs, and the trajectory curves coincide with the lower edge of the normalized intensity distributions of the focal spots. Furthermore, because the full width at half maximum (FWHM) of the focal spots is the main indicator to evaluate the performance of Airy-beam-based metalenses, we compared the experimental and theoretical distributions at a focal length f = 10 mm and measured the FWHM at the three frequencies, as shown in Figure 6(a)–(c). The FWHM is 0.84 mm at 0.4 THz (λ = 0.75 mm), 0.52 mm at 0.6 THz (λ = 0.5 mm), and 0.38 mm at 0.8 THz (λ = 0.375 mm). The experimental results agree well with those of the simulations, and the FWHMs are approximately equal to the wavelength of incidence of the THz wave. The self-healing property is typically verified by placing an obstacle on the travel path of the parabolic trajectory. The trajectory recovers quickly if two silicon obstacles of diameter 400 μm are located at (x, z) = (−0.8 mm, 6 mm) and (0.8 mm, 6 mm), indicating a rather robust self-healing property of the generated Airy-beam-based metalens. The self-bending, diffraction-free, and self-healing achromatic properties are illustrated in Figure S12 in the Supplementary material.

Figure 6:
Achromatic Airy beam-based metalens.
(a)–(c) Comparison of simulated and experimental results for the intensity distributions at the focal length. The full width at half maximum (FWHMs) of the intensity distribution are 0.84 mm at 0.4 THz, 0.52 mm at 0.6 THz, and 0.38 mm at 0.8 THz. (d)–(f) Demonstration of the generation of two counter-propagating achromatic Airy beams in experiments and simulations. The focal length f = 10 mm.
3 Conclusion
We have experimentally demonstrated achromatic Airy beams in the THz regime using carefully designed silicon metasurface devices. Furthermore, we have shown numerically that a metalens based on an achromatic Airy beam has the advantages of a larger DOF and a self-healing property compared with a traditional achromatic metalens. The achromatic Airy beam developed in this paper has important potential applications to light-sheet microscopy, self-healing metalens bio-imaging, and high signal-to-noise ratio detection.
4 Methods
4.1 Simulation details
FDTD was used to find and simulate the structural units and the transmission Ex field of the metasurface devices. In the simulation of the unit, the mesh grid size was set to 10 nm. In the simulation of the whole metasurface device, the grid size was set to 1 μm
4.2 Sample fabrication
All the silicon metasurface devices were fabricated using conventional lithography together with deep reactive ion etching. The refractive index n of silicon is 3.45 and its resistance R is higher than 104 Ω cm. First, a 2-μm-thick silica layer was grown on a 500-μm-thick double-side-polished high-resistivity silicon wafer. Next, by conventional photolithography, the left photoresist and the silica became a double layer of protection above the silicon posts. Conventional deep reactive ion etching was then employed to make silicon posts of thickness t = 350 μm. Finally, the remaining photoresist and the remaining silica were cleaned off separately.
4.3 Experimental characterization
THz near-field scanning microscopy was employed in the experimental tests owing to its high scanning speed and high resolution. Here, only the x-polarized electric field Ex component was measured with RCP illumination. The electric field was detected at 0.2 mm intervals from −3.2 to 4 mm in both the x and z directions. Note that we reserved 2 mm to avoid collisions between the probe and the metasurface devices.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11874266
Award Identifier / Grant number: 11604208
Award Identifier / Grant number: 11734007
Award Identifier / Grant number: 91850101
Award Identifier / Grant number: 11674068
Award Identifier / Grant number: 11874118
Funding source: National Key Research and Development Program of China
Award Identifier / Grant number: 2017YFA0303504
Award Identifier / Grant number: 2017YFA0700201
Funding source: Chenguang Program
Award Identifier / Grant number: 17CG49
Funding source: Natural Science Foundation of Shanghai
Award Identifier / Grant number: 20JC1414601
Award Identifier / Grant number: 18ZR1403400
Author contributions: Q.C., J.W. and L.M. contributed equally to this work. J.C., L.M., Z.S., J.Z., and X.Z. carried out simulations, fabricated the samples and conducted part of the measurements; J.W., T.C., Y.Y., and D.Y. did the theoretical calculations and designed the samples; J.W. and Z.S. built the experimental setup and conducted part of measurements; Q.H. and W.H. provided technical supports for simulations and data analyses. Q.C., S.Z. and L.Z. organized the project, designed experiments and analyzed the results. All the authors contributed to the preparation of the manuscript, and have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This work was funded by National Natural Science Foundation of China (No. 11874266, No. 11604208, No. 11734007, No. 91850101, No. 11674068, No. 11874118), National Key Research and Development Program of China (No. 2017YFA0303504 and No. 2017YFA0700201), Chenguang Program (17CG49), Natural Science Foundation of Shanghai (No.20JC1414601 and No.18ZR1403400).
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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Supplementary Material
The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2020-0536).
© 2020 Qingqing Cheng et al., published by De Gruyter, Berlin/Boston
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