1. Introduction

The non-spreading Airy wave packet with a curved trajectory was first predicted by Berry and Balazs in 1979. Within the context of quantum mechanics, the Schrödinger equation was demonstrated to have an Airy function form solution [1]. However, in the last few decades, the ideal Airy wave packet associated with infinite energy had not been realized in practice. In 2007, the Airy function solution in the domain of optics was introduced by Siviloglou et al. combining with the Schrödinger equation and the paraxial Helmholtz equation due to both similarities [2,3]. Account for its peculiar properties and applications, Airy beam has attracted considerable research interest in forming optical bullet [4,5]. optical micro-manipulation [6], microchip lasers emitting [7], and so on.

Initially, A specially designed optical system generated the Airy beam with two-dimensional cubic phase mask displayed on a spatial light modulator [8]. Although several different methods of generating Airy beam have followed [9–11]. they all require bulky optical system, no matter how flexible they are to manipulate the beams. This is unfavorable to its application for small-scale or integrated optical devices.

Recently, metasurface which has attracted extensive attention is a fascinating sub-wavelength structure that can control the amplitude and phase of the wave to achieve purposes such as focusing [12–16], beam deflection [17,18], and special beam generation [19–23]. Instead of the existing complex and large optical system, metasurface is a simple planar construction overcoming the block of limited functionality and difficulty of integration. Credibly, it will play an important role in the development of the research of future miniature and high-precision equipment. From the first metasurface was used to generate the Airy beam in 2011 [24], many plasmon-based planar structures were well-designed, but polarization-insensitive Airy beam generated by these metasurface has great optical losses in the visible light [25]. With the introduction of the all-dielectric metasurface [26], this problem has been solved effectively. Here, we designed an all-dielectric metasurface array that can generate Airy beam with self-bending and self-healing properties. For the purpose of generating one-dimensional(1D) Airy beam with polarization-insensitive, we used square nanopillar to form the metasurface array. They can be thought of as truncated short-wave guides, where plane waves travel at a rate that changes as they pass through nanopillars of different widths. In addition, the insensitive characteristics of polarization make the metasurface more widely used [27], compared to the structure based on geometric phase which may be limited with its polarization conversion efficiency [28].

2. Structure design

Designing the phase and amplitude of a unit-atom at the same time is troublesome and sometimes does not meet the precise design requirements. Fortunately, previous research has demonstrated that the phase-only (binary phase) modulation Airy beam can be approximated quite well [29]. In order to pursue high enough energy transmission, the titanium dioxide (TiO₂), having good characteristics in the visible band, was selected to build the unit cell of the all-dielectric metasurface in our design. Under this premise, for the reason of obtaining the required phase, it is necessary to optimize the nanopillars’ parameters: width, period, and height. Also, the square cross-section with polarization-insensitive operation possesses higher filling factor than circle in our design [30]. Meanwhile, the square structure is easier to be processed and manufactured. For example, in some micron-scale lithography machines, the etched pattern consists of hundreds of nanometers of pixels. The square structure can have a smooth side, while the circle requires smaller pixels. In other words, a smooth round side requires a more accurate lithography machine, usually at the nanometer level. As illustrated in Figs. 1(a), 1(b), the period of the square nanopillar based on SiO₂ substrate is $P\; = \; 400\; nm$ with both heights are $H\; = \; 550\; nm$. Two different types of width are needed in this design, respectively are $206{\; \textrm{nm}}$ and $150{\; \textrm{nm}}$. Figures 1(c), 1(d) show that two types of all-dielectric nanopillar with substrate present high transmission efficiency, approximately $93\%$ at the wavelength of $635\; nm$, and at this wavelength their phase difference is $\pi $. At $620\; nm$, the light resonates with the structure, causing a reduction in transmittance and a rapid change in phase.

 

Fig. 1. Schematic of the unit-atom structure. (a) The perspective of the square nanopillar. (b) Top view of the building block. (c) Simulated transmission for incident light and (d) phase.

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At the incidence of both linear and circularly polarized light, the results show that the transmission intensity and phase have no obvious change. This will result in the polarization insensitivity of the designed Airy beam. Before designing the array structure to generate Airy beam, two key points need to be identified. One is that the ideal Airy beam would have to be generated with a suitable number of nanopillar in one direction. Also, the number of nanostructures would affect the bending of the beam [31].

3. Simulation and discussion

In principle [2], the Airy beam wave packet of finite energy can be described as:

$Ai$ is Airy function, $0 < a < 1$ is a parameter that truncates the Airy beam, x is the transverse coordinate, b is the transverse scale, $\mathrm{\xi } = \textrm{z}{b^2}/\textrm{k}$ is a normalized propagation distance, where k is the wave number. The initial field envelope of 1D Airy beam can be described as $\varPhi \textrm{(}\xi = 0\textrm{, }x\textrm{)} = Ai({bx} )\exp (ax)$, and the phase profile must satisfy $\varPhi (x )= phase[{\varPhi ({\xi = 0,x} )} ]$. In the design, each lobe that produces an Airy beam requires at least two unit-cells controls to meet sampling criterion. As a result, the half-width of the main lobe ${b^{ - 1}}$ is set as $1.2{\; }\mathrm{\mu}\textrm{m}$. According to Fig. 2(a), it is possible to clearly recognize the process of generating Airy beam. When plane wave passes over the specially designed metasurface, it evolves into the Airy beam that decay from the main lobe and bend gradually as it propagates. Based on the two key points mentioned above, we successfully generated the ideal Airy beam with the structure of 48 nanopillars in the x-axis. As clearly illustrated in Fig. 2(b), the absolute value of Φ is used to represent the amplitude distribution of the Airy beam. The Red fork marks the position of the unit-atoms (in x direction). In addition, the orange structure represents the 150 nm unit-atoms and the yellow represents the 206 nm unit-atoms. The multiple structures on each lobe are identical. At the same time, such a structure distribution conforms to the phase distribution shown in Fig. 2(c). As a result, the position of the two different unit structures is placed according to the phase requirement that produce Airy beam. Since we only regulate the phase, so we do not have to care about the state ($|\mathrm{\varPhi } |$) of the red fork. It is intuitive to see that the half-width of the main lobe from the position 0 is equal to the size of the three unit-atoms. When illuminated by plane wave, the structure produces Airy beam that is almost identical to the curved path of the theoretical beam. The simulated result is clearly shown in Fig. 2(d). The efficiency of generating Airy beam is approximately $58\; \%$, comparing 56% in Ref. [27] and 65%-75% in Ref. [28]. Figure 2(e) shows the full-width at half-maximum (FWHM) of the main-lobe beam as a function of propagation distance. The non-diffraction nature of the Airy beam can be observed near the solid green line. We define that the zone below 1.5 times the constant $FWHM = 3{\; }\mathrm{\mu}\textrm{m}$ is valid by referring to the theory that a Gaussian beam diffracted after traveling a distance of $\sqrt 2 $ times of FWHM. It can be clearly observed that the FWHM remains relatively stable near the solid green line. From the propagation distance $\; 90\; \mu m$, the beam begins to diffraction. In fact, the cut-off Airy beam with limited energy will inevitably occur diffracting when it continues propagating. At the initial stage of propagation, due to there is no adjustment of the transmission amplitude, the FWHM can fluctuate greatly. However, from propagation distance almost $15\; \mu m$ to $90\; \mu m$, the beam becomes ideal. Besides, Airy beam has a special self-bending characteristic. The main Airy beam lobe deflection ${x_d}$ and propagation distance z are commonly described by the theoretical relation, ${x_d} \cong {\lambda _0}^2{z^2}/16{\pi ^2}{x_0}^3$, where ${x_0} = {b^{ - 1}}.$ Fig. 2(f) shows the deflection of the main-lobe beam as a function of propagation distance, and simulated values are marked by orange dots. In the picture, simulated values that could be further optimized are almost consistent with the theoretical curve. This property can be used to manipulate particles to bypass obstacles to reach the target position. It has important application prospects and significance in biological science, medical field.

 

Fig. 2. Generation of Airy beam. (a) Design schematic. (b) Value of the amplitude function of the Airy beam. The red cross marks show the distribution of the structures. (c) The phase distribution that produces Airy beam. (d) The simulated electric intensity distribution of the Airy beam. The excitation wavelength is ${\lambda _0} = 635{\; }nm$. (e) FWHM of the main-lobe beam as a function of propagation distance. (f) Theoretical deflection of the main-lobe beam as a function of propagation distance (blue curve) and simulated values (red circles).

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In addition, in order to verify the self-healing properties of the Airy beam, we set an obstacle located in the main lobe. As clearly shown in Fig. 3(a), the main lobe of the Airy beam was reconstructed after it was blocked by an opaque ball with a radius $r\; = \; 0.635\; \mathrm{\mu}m$. Similarly, if other side lobes are blocked, they will revert to the original propagation path over time. This self-healing ability is resistant to the destruction of the Airy beam by the outside environment. Over the years, the auto-focusing Airy beam have also attracted wide attention. According to its propagation characteristics, we use two planar structures arranged in opposite ways to form a planar structure. After the plane light passes through the structure, the focal point is successfully formed as expected which is shown in Fig. 3(b). The energy on the side lobes flows to the center and eventually converges a little. It is superior to conventional optical focus because the converged Airy beam does not diverge quickly, but rather degenerates into a Bessel beam. We can also see that it has a large focus depth ($Depth\; of\; focus\; \cong \; 15.6\; \mathrm{\mu}\textrm{m}$) and a high energy density. This advantage has a good application prospect in laser cutting.

 

Fig. 3. Schematic of the self-healing property and auto-focusing Airy beams. (a) a perfect electric-conductor ball whose radius is equal to incident wavelength was set in the distance of $20\; \mu m$ at the path of the main Airy beam lobe. (b) Focus on a distance of $60{\; }\mu m$ from the structure with the distance of two main lobes $d = 10.65{\; }\mu m$.

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4. Conclusion

We propose and design an extremely compact component to generate Airy beam at the wavelength of $635\; nm$. Simulation results show that the designed components can generate the Airy beam with self-bending behavior. In addition, the efficiency of generating Airy beam is approximately 58%. Furthermore, we also demonstrate that the Airy beam possesses the self-healing property and the structure can be used to generate auto-focusing Airy beams. Our design provides a simple way to generate non-diffraction beams. This device may be used in the applications such as integrated optics system, biomedical nano-surgery, and optical trapping techniques.