## Abstract

Airy beams are widely used in various optical devices and optical experiments owing to their unique characteristics such as self-acceleration, self-recovery, and non-diffraction. Here we designed and demonstrated a metasurface capable of encoding two phase distributions independently in dual circular polarization channels. We experimentally observed the generated Airy beam arrays loaded on the metasurface in the real and K spaces. Compared with the traditional method, such method provides a more efficient solution to generate large capacity Airy beam arrays with switchable working modes in the broadband spectrum. The results may pave the way for the integration and miniaturization of micro-nano devices and provide a platform for information processing, particle manipulation, space–time optical wave packets, and Airy lasers.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Airy beams have useful characteristics such as self-acceleration, self-recovery, and non-diffraction in free-space transmission. Unlike the Bessel beam, the Airy beam does not rely on simple conic superposition of plane waves, although both are defined as undiffracted light. Since Berry and Balazs predicted that the Airy functions can be solutions to the Schrödinger equation [1], and Siviloglu et al. first experimentally demonstrated the Airy beams with finite energy [2], many breakthroughs have been achieved using the Airy beams, such as particle manipulation [3,4], space–time wave packets [5,6], and the Airy lasers [7,8]. To expand the potential application of the Airy beams, several methods have been proposed to generate and control the Airy beams [9–15]. However, some of these methods, such as mechanical modulation, usually complicate the generation system significantly. Other methods are not suitable for optical micromanipulation because of the bulky optical modulator. Therefore, the design of a simple and effective method for generating and controlling the Airy beam remains an ongoing area of research.

In recent years, metasurfaces consisting of subwavelength structures and capable of flexibly manipulating the amplitude, phase, and polarization of incident light waves have been developed rapidly. Metasurfaces have been widely studied, and planar antenna arrays have been proven to effectively reduce the loss of light while passing through artificially designed materials [16–20]. When compared with traditional bulk optical elements, the metasurface has the advantages of flexible design, relatively low processing difficulty, and low manufacturing cost. These characteristics make it useful for further minimized, integrated metadevice designs. Because of their unique physical properties, such as the generation of abrupt phase discontinuity within ultrashort distances, metasurfaces have been widely used in various applications, such as optical holography [21–23], structure color [24–26], and ultrafast optical pulse shaping [27,28]. Note that several important works have reported the use of metasurfaces for generating Airy beams based on synthetic-phase metasurface design [29], Huygens’ metasurface design [30,31], complex amplitude metasurface design [32] and achromatic metasurface design [33]. However, many of these designs can only achieve the generation of a single Airy beam, rather than efficient array design [30–33]. Meanwhile, these designs can only operate in a single working mode and cannot go to the flexibility to change according to requirements [29–33].

Here, we propose and demonstrate a metasurface with dual circular polarization channels, through which one can encode various phase distributions. The design of the metasurface is based on the principle of the summarization of both geometric (Pancharatnam–Berry (PB) phase) and transmission phases. The geometric phase principle is attributed to the polarization conversion evolution history and can be explained by the path envelope on the Poincare sphere. In a special case, it corresponds to twice the azimuthal angle rotation of nanoantennas when the light with a circular polarization state changes its chirality. Each nanoantenna on the metasurface acts as a half-wave plate; therefore, it can completely change the polarization state while simultaneously realizing the required wavefront modulation. Owing to our precise design, the Airy metasurfaces can generate the Airy beams in the real and K spaces under left circular polarization(LCP) and right circular polarization (RCP) light incidences, respectively. Moreover, we also use the phase optimization method of the Dammann grating to generate an array of the Airy beams for improved efficiency so that the Airy beams can be used in parallel to meet various demands. In general, our study has a significant impact on the development of micro-nano devices, in terms of their integration, flexibility, and potential applications in optical communications, manipulations, and so on.

## 2. Methods and results

In the fields of micro-nano processing and optical detection, optical focus arrays can significantly reduce the scanning time of optical devices to improve their efficiency [34,35]. Due to the ability to produce uniform light field array, Dammann gratings have a wide range of applications in the field of optical fiber communication, three-dimensional measurement, laser direct writing and so on [12,36–38]. In this study, we adopt the Dammann grating to generate an Airy beam array. As depicted in Fig. 1, when an LCP beam is illuminated on the designed metasurface, a 3 × 3 Airy beam array located 1 mm away from the metasurface can be detected in the real space under the orthogonal chirality of the incident light, namely, the RCP light. Because of the distinct self-accelerating property of the Airy beam, the transmitted light travels along a curved trajectory, resulting in the detected optical field of the Airy beam being off-axis and staggered with the location of the metasurface in the horizontal direction (the location of the metasurface is marked using a white frame). When the RCP light is incident, another 3 × 3 Airy beam array with an LCP state in the K-space (corresponds to infinity or to the focal plane behind the lens) can be captured.

The Dammann grating can modulate the amplitude or phase distribution of the pixel unit within the superlattice period, resulting in equal or ratio diffraction intensity distributions of various diffraction orders. The Dammann grating depicted in Fig. 2(a) is a beam array with a target diffraction order of 3 × 3, and each period is composed of 12 × 12 pixels. The area of the metasurface is 400 × 400 µm^{2}, and the lattice constant of each structural unit is 500 nm × 500 nm, as depicted in Fig. 2(a). To obtain a uniform light intensity distribution of the target diffraction order, a genetic method is used to optimize the phase distribution within each pixel to match the diffraction intensity distribution of the target. By dividing the continuous phase from 0 to 2π into 16 orders with equal intervals, the initial value of the random phase distribution was put into the optimization process of the genetic algorithm for optimized calculation. Using an appropriate number of iterations, an ideal phase distribution result can be obtained.

To generate the Airy beams in the K-space, we can obtain the spatial frequency spectrum of the Airy beam, by limiting the infinite energy Airy beam using an exponential decay function, as follows:

*a*represents the damping coefficient and

*k*represents the spatial frequency coordinate. The spectrum distribution of the Airy beam with finite energy contains a cubic phase, which is formed by the Fourier transform of the Airy function itself. This type of spectrum distribution has guiding significance for the generation of the Airy beams in experiments. Based on the cubic phase distribution in formula (1), one can generate an Airy beam with finite energy. Subsequently, to directly generate an Airy beam through phase modulation in real space, we apply the approximate Airy formula as follows: $\textrm{Ai}(x )\cong {x^{ - 1/4}}\textrm{exp}({iC{x^{3/2}}} )$, where

*C*is a constant. Through observation, one can find that when we ignore the variation in the amplitude, the intensity of the Airy beam changes as the phase changes, which directly provides a method to generate the Airy beam in real space. We can obtain the formula for the phase distribution of the Airy beam transmitted along a parabola through a simple mathematical equation: where (

*ɛ,ζ*) represents the coordinates of one specific phase plane,

*λ*represents the wavelength, and

*b*is a constant related to the trajectory of propagation. By selecting the appropriate phase modulation depth, a similar optical field distribution of the Airy beam can be obtained. Note that using this design idea, one can generate self-accelerating beams traveling along various trajectories in free space, such as the logarithmic and higher-order exponential trajectories [15]. Figure 2(b) and 2(c) depict the phase distributions of the Airy beam array calculated using Eqs. (1) and (2), respectively. Next, we normalize the phase of the Dammann grating, the cubic phase and the 3/2-order phase into 0-2π range, and then we apply dot multiplication of Dammann grating phase profile with the other two phases, respectively. Then we can obtain the phase distributions of the Airy beam array in which one is in the real space and the other is in the K-space. The phase distributions of the resulting Airy beam array are depicted in Fig. 2(d) and 2(e). Note that we wrapped all the phase distributions in the 0–2π range.

We adopted the design method of combining the PB and propagation phases to realize the two independent phase profiles $({\Phi_1\; \textrm{and}\;{\Phi_2}} )\; $of the dielectric metasurface in two different circular polarization channels. The Jones matrix of the metasurface to illustrate its multiplexing principle is as follows [9,39,40]:

*π*. We select the dielectric nanofin as the building block. A single pixel of the metasurface is depicted in Fig. 3(a). The materials of the nanofin and substrate are amorphous Si(a-Si) and SiO

_{2}, respectively. The working wavelength is 800 nm; therefore, the loss of a-Si is very low. The period is set to 500 nm to prevent higher-order diffraction and the height of the nanofin is 600nm. To optimize the geometric parameters of nanofins (

*Dx*and

*Dy*), we used the finite-difference time-domain method to sweep the geometric parameters of nanofins to calculate the transmission coefficient and the phase under both x-polarized and y-polarized illumination (Figs. 3(b)–3(e)). Using the minimum-value matching algorithm, 16 nanofins with high cross-polarization conversion efficiencies were selected, as depicted in Fig. 3(f). Evidently, their conversion efficiencies are more than 80%, and the phase for either

*x*or

*y*polarization components (

*t*,

_{xx}*t*) can cover the range of 0–2

_{yy}*π*, which meets our design requirements. Finally, the phase modulation of two opposite polarization channels can be realized after mapping the nanofins of the 16 different sizes to specific spatial coordinates and rotating the specific angles based on the phase distributions of both the Airy beams.

To further verify the proposed concept, we fabricated a metasurface on a glass substrate following the processes of deposition, patterning, lift-off, and etching. Figure 4(a) and 4(b) depict the scanning electron microscopy (SEM) image of the sample. It is evident that all the nanofins on the sample have a fine appearance and ideal angular rotation. The dimensions of the entire sample were 400 µm × 400 µm (800 × 800 cells). The experimental optical path is depicted in Fig. 4(c). The laser is collimated and switched to circularly polarized light after passing through a linear polarizer and a quarter-wave plate, finally illuminating the metasurface. Next, the generated Airy beam passes through the objective lens (Nikon, NA = 0.6, 40 ×) and a tube lens (Thorlabs, *f* = 200 mm) to be collected in a charge-coupled device (CCD) for the working mode of the real space. By changing the tube lens to a Fourier lens, we can observe the working mode of the K-space; that is, the two different working modes are switched by adjusting the distance between the sample and microscope objective. The details of the experiment under both modes are described below.

First, we used the microscope objective and tube lens to image the magnified Airy beam array in real space and recorded the result using a CCD. The tube lenses used in this study were for aberration correction of infinite distance imaging. Figure 5(a) and 5(b) depict the 3 × 3 Airy beam arrays (imaged 1 mm away from the metasurface) obtained in the simulation and in the experiment, respectively. Both are in good agreement with each other, and the characteristics of the main and side lobes of the Airy beam are well preserved. The phase change of the Airy beam is relatively slow because of self-acceleration in the real space (Fresnel diffraction zone). In the experiment, we realized detection by adjusting the distance between the microscope objective and sample. By changing the axial position (*z*) of the sample, we obtained a clear reconstructed image in the CCD (we defined the location as *z* = 0 mm). We then continued to adjust the precision 3D stage at an interval of 0.05 mm, finally capturing 31 light field images from *z* = 0 to *z* = 1.5 mm. The maximum *z* value is limited by the CCD field of view. Therefore, we obtained a clear observation of the evolution of the Airy beam in real space (See Visualization 1). Because of its excellent non-diffraction and self-accelerating characteristics, the Airy beam can bend within a distance when propagating in the real space while maintaining its unique field distribution. In other words, we can obtain the transmission trajectory of the Airy beam by detecting the distribution of the light intensity at various locations. By extracting the position of the main lobe of the central order of the Airy beam in each frame, we successfully plotted the 2D propagation trajectory of the Airy beam in the *XZ* plane (blue and red triangle curves). This 2D propagation trajectory is in good agreement with the parabolic trajectory predicted theoretically (blue and red curves), as depicted in Fig. 5(e). By switching the chirality of the incident circularly polarized light, we observed the disappearance of the Airy beam array
(See Visualization 2) in real space, which proved our design theory of polarization multiplexing.

Next, we replaced the tube lens with a normal lens to detect the Fourier plane. We captured the Airy beam array in the K-space using the CCD. Similar to the above demonstration, we also present the simulation and experimental results of the Airy beam array in this working mode. The main and side lobe appearances can also be clearly seen in Fig. 5(c) and 5(d). By changing the chirality of the incident light, the light field distribution of the Airy beam in the K-space also changed (See Visualization 3).

Note that although the Airy beam propagating behind the Fourier plane has little deformation, it still shows good non-diffraction and self-acceleration characteristics, such as in real space, over a long distance compared with the Gaussian beam, as depicted in Fig. 6(a). Therefore, the Full-Width Half-Maximum (FWHM) of the Airy beam can be obtained, which in K space is 84.3 µm and 59.4 µm in real space. We also detected our designed metasurface by changing the incident wavelength for the two working modes, as depicted in Fig. 6(b). The upper and lower rows depict the results captured in the real and K spaces, respectively. In the columns, the results are detected under various incident wavelengths from 680 nm to 800 nm at intervals of 20 nm. It is evident that the Airy beam size becomes larger with an increase in the incident wavelength, because the phase profile of the Airy beam is almost retained despite the variation of different wavelengths. It is evident that the Airy beam array retains its unique intensity distribution characteristics in both the real and K spaces. This result verifies the broadband operating characteristics of our metasurface. Finally, to evaluate the modulation efficiency of the designed metasurface, we obtained the efficiency spectrum using a Fourier transform infrared spectrometer, in which the maximum transmissivity could reach 80% at a wavelength of 800 nm.

## 3. Conclusion

In summary, we designed and demonstrated a high-efficiency dielectric metasurface with dual circular polarization channels, which can encode two independent phase distributions of the Airy beam array in the real and K spaces. Furthermore, we present a very simple design theory for the generation and manipulation of the 3 × 3 Airy beam arrays. For the more, we can use optimization method based on Dammann grating principle to increase the array number. We believe that the results of this study pave the way for the integration and miniaturization of micro-nano devices. Such metasurfaces provide a convenient platform for processing, particle manipulation, spatiotemporal optical wave packets, and Airy lasers. Furthermore, the proposed metasurface can operate in complex and challenging environments owing to its broadband property and its switchable working mode.

## Funding

National Key Research and Development Program of China (2017YFB1002900); Fok Ying-Tong Education Foundation of China (161009); Beijing Municipal Natural Science Foundation (JQ20015); National Natural Science Foundation of China (52075041, 61775019, 61861136010, 92050117); Beijing Outstanding Young Scientist Program (BJJWZYJH01201910007022).

## Disclosures

The authors declare no conflict of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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