## Abstract

Conventional method to generate autofocusing Airy (AFA) beam involves the optical Fourier transform (FT) system, which has a fairly long working distance due to the focal length of FT lens, presence of spatial light modulator (SLM) and auxiliary total reflection mirrors. Here, we propose an extremely compact design to generate high-efficiency AFA beam at visible frequency by using metasurface which is composed of a single layer array of amorphous titanium dioxide (TiO_{2}) elliptical nanofins sitting on the fused-silica substrate. Numerical simulations show that the designed structures are capable of precisely controlling the deflection of Airy beam and tuning the focal length of AFA beam. We further numerically demonstrate that the phase modulation of AFA beam could combine with the concept of vortex light field to produce vortical AFA beam. We anticipate that such device can be useful in the ultra-compact integrated optic system, biomedical nanosurgery and optical trapping applications.

© 2017 Optical Society of America

## 1. Introduction

The dispersion-free Airy wave packet was first predicted by Berry and Balazs [1] via theoretically demonstrating that the Schrödinger equation has an Airy function form solution. Because the ideal Airy wave packet is associated with infinite energy, in practice it does not exist. In 2007, the approximated finite energy Airy beams were observed by Christodoulides *et al*. [2,3] and have attracted a lot of attentions due to their distinct propagation properties [4–6], such as the ability to self-accelerating without any external potential.

The focusing performance of optical beam has always been an issue of great practical significance. With the observation of autofocusing Airy (AFA) beam [7,8], a different way of focusing has been achieved. The AFA beam has potential in many optical applications, such as optical trapping [6,9,10], laser nanosurgery, and the generation of linear and non-linear intense light bullet [11,12]. In previous studies, the method to generate AFA beam always involves encoding the Fourier transform (FT) of the radially symmetric Airy distribution with a computer-controlled liquid crystal spatial light modulator (SLM) and then Fourier transforming the wavefront again by an external lens [8,13,14]. The main problem of this method is that the optical FT system has a fairly long working distance due to the focal length of FT lens, presence of SLM and auxiliary total reflection mirrors. This is not conducive to the compact integration of autofocusing devices.

Metasurfaces, which are planar optical elements that composed of phase shifters made by subwavelength nanostructures, have drawn a lot of attention due to their excellent electromagnetic performances and ultrathin thicknesses compared to conventional bulk optical components [15,16]. In principle, metasurfaces are capable of controlling phase, polarization, and amplitude of incident beam. Currently, various metasurface-based optical elements have been implemented including wave plates [17], beam deflectors [18], flat lens [19–21], holographic imaging [22,23] and vortex field generators [24,25]. The generation of Airy beam has been theoretically proposed by using plasmonic metasurfaces in near-infrared region [26,27]. However, plasmonic metasurfaces using metallic resonant structures always suffer from strong omhic losses at optical frequencies, especially in visible spectral region. Recently, low-loss all-dielectric metasurfaces have been presented to realize subwavelength resolution imaging using titanium dioxide (TiO_{2}) nanofins array in the visible frequency [21]. The concept of metasurface provides an alternative to generate AFA beam using all-dielectric nanostructures array with compact design.

In this letter, we present an efficient approach for the generation of Airy beam, AFA beam and vortical AFA beam based on phase-only modulation in the visible spectral range. The phase profiles of these beams are created by the arrangement of nanostructures which can generate different space-variant geometric phases. In addition, by introducing an additional phase retardation via changing the nanostructures’ design, it is possible to control the deflection angle of Airy beam and tune the focal length of the AFA beam. Furthermore, we also demonstrate that combining AFA beam with vortex light field is capable of producing vortical AFA beam. The numerical simulation results of beam trajectories agree well with the analytical predictions. The designed optical component maintains an ultrathin thickness and a small geometry size, allowing our devices to be used in various compact optical integration systems.

## 2. Unit-cell structure of all-dielectric metasurface

Figure 1(a)-(d) present a unit cell of the designed metasurface which is composed of a single layer array of amorphous titanium dioxide (TiO_{2}) elliptical nanofins sitting on a fused-silica substrate. The TiO_{2} nanofin is placed at the center of square unit cell. We choose TiO_{2} on account of its sufficiently high refractive index, minimal surface roughness and low loss at visible frequency [28]. In addition, because TiO_{2} is a semiconductor with bandgap around 3.2 eV, metasurface made from TiO_{2} can be actively controlled by illuminating the device with a shaped pump-beam above the bandgap, and probing it at the designed visible wavelength below bandgap [29]. Each TiO_{2} nanofin can be regarded as a dielectric waveguide, which operates like weakly-coupled low-quality-factor resonators [30], and light is mainly confined inside the nanofin. To obtain high polarization conversion efficiency, the nanostructure should provide birefringence function similar to half-wave plate, which is used to generate a π phase retardation between two cross-polarized components as well as equal transmission coefficients. Here, the unit cell is designed as an elliptical nanofin to fulfill the requirement of C2 symmetry [31]. Similar to the schematic shown in Fig. 1(e), each nanostructure can be regarded as a pixel and imposed a phase shift to modify the polarization and phase of incident light.

Throughout this letter, all the numerical simulations are performed based on finite difference time domain (FDTD) method. Figure 1(f) shows the simulated transmission coefficients and phase difference of *x*- and *y*-polarized light propagating through the unit cell. At the wavelength of 430 nm, the TiO_{2} nanofin can be considered as an ideal half-wave plate with similar and high transmission coefficients and a nearly π phase retardation for *x-* and *y-*polarized components. As a result, the polarization conversion efficiency from left/right circularly polarization (LCP/RCP) to cross-polarization is above 95% at this wavelength, as shown in Fig. 1(g).

## 3. Generation of Airy beam

In principle, for an incident beam with spatially varying electric field ${E}^{in}$, a designed metasurface can be used to tailor the beam’s spatial distribution and achieve an expected electric field${E}^{out}$, as shown in Fig. 2(a). The electric field envelope of a finite energy Airy beam can be described as [2]:

*a*is a positive parameter to obtain truncated Airy beam and $a\ll 1$,

*x*represents the transverse coordinate,

*b*is the transverse scale,

*ξ*is a normalized propagation distance. Clearly, the initial field envelope of Airy beam is given by $\varphi \left(\xi =0,x\right)=\text{Ai}(bx)\text{exp}(ax)$. A typical amplitude function of one-dimensional (1D) Airy beam with

*a*= 0.05 and

*b*= 1.5 is illustrated in Fig. 2(b). In order to generate Airy beam, the phase profile as a function of

*x*-coordinate along the nanofin array must satisfy: ${\phi}_{airy}\left(x\right)=\text{arg}[\varphi (\xi =0,x)]$. For 1D Airy beam, the designed array is only phase-modulated along the

*x*-axis, and there is no modulation along the

*y*-axis.

To control radiation direction of Airy beam with a specific angle *α*, the designed metasurface requires compensating an additional phase profile for oblique emitted light with a different optical length. Therefore, the additional phase can be described as:

*n*is an integer, $\Delta \phi $(0) is the reference phase when

*x*= 0, and ${\lambda}_{0}$ is the free-space wavelength of 430 nm. As a result, the total phase shift

*φ*of the array to be:

According to the geometric Pancharatnam-Berry phase, the total phase shift generated by spatially rotation of each nanofin with an angle *θ* is given as $\phi =2\theta $, accompanied by polarization conversion to opposite chiral circularly polarized light [32]. Figure 2(c) shows the required phase distribution on initial plane when radiation angle of output beam is *α* = 1°. However, it is difficult to achieve this continuous phase distribution by using nanostructure in practice. Therefore, we intend to take advantage of the arrangement of nanofins to obtain discrete phase distribution for approximation of the Airy beam. Here we only focus on the phase modulation rather than amplitude modulation because previous studies show that Airy beam can be approximated quite well by phase-only modulation [33,34]. The RCP light is normally incident on the designed metasurface array, then converted to the LCP light with the envelope of Airy beam. Figure 2(d) illustrates the propagation of the Airy beam generated from the designed device at the wavelength of 430 nm. The Airy beam is truncated by the number of the nanofins in *x*-direction, which is different from the conventional exponential apodization of the Airy function [3].

Based on this generation method, we further demonstrate three particular attributes of Airy beam, namely self-healing after passing through obstacles, non-diffracting and self-accelerating trajectory. Figure 2(e) illustrates that the Airy beam is scattered by a *λ*_{0} × *λ*_{0} square barrier at the path of the main Airy beam lobe and recovers with the evolution of the electric field, arising from the intrinsic attributes to healing itself even in a complex environment. Therefore, it is a promising feature that can be used in atmospheric propagation or microscopy biological tissues.

Figure 2(f) shows the intensity full-width at half-maximum (FWHM) of the first beam lobe as a function of propagation distance. Although the FWHM exhibits an oscillatory pattern at the initial stage of propagation due to interference, it is obvious that the excellent non-diffraction nature of the Airy beam can be observed by polynomial fitting curve, especially at a stable propagation distance from 16 um to 30 um. The non-diffraction propagation zone is determined by the truncation of Airy wave packet due to finite transverse number of the nanofins.

The main Airy beam lobe deflection ${x}_{d}$ versus propagation distance *z* is commonly described by the theoretical relation ${x}_{d}\cong {\lambda}_{0}^{2}{z}^{2}/(16{\pi}^{2}{x}_{0}^{3})$ [2], where ${x}_{0}={b}^{-1}\approx 0.67$μm. Since the additional phase profile used to control the radiation direction is introduced in our design, their relationship should be described as:

*m*is parabolic trajectory coefficient and should be adjusted according to the deflection angle of the beam. Equation (4) is still valid until the diffraction effect plays a dominant role. The black solid line in Fig. 2(g) corresponds to the analytical calculation and shows a smooth parabolic trajectory. The deflection of the Airy main lobe extracted from numerical data shown in red hollow circle has a good agreement with the theoretical prediction. Since diffraction effect takes over after a propagation distance about 30 μm, small discrepancy between two curves is observed.

Next, we investigate the control of radiation direction under the paraxial approximation, by choosing appropriate emergence angle *α* in Eq. (2) to tailor the phase profile of metasurface array. Figure 3(a) and (b) show the intensity distributions for two special cases where *α* = 10° and *α* = −5°, respectively. We also show the white dashed line as reference parabolic trajectory without deflection (*α* = 0). It can be clearly seen that the Airy beams, including the main lobe and all the sidelobes, are deflected. Figure 3(c) and (d) show that main Airy beam lobe deflection as a function of the propagation distance and indicate a good accuracy of our theoretical prediction calculated from Eq. (4) with different deflection angles.

## 4. Generation of mirror-symmetric AFA beam

To realize mirror-symmetric AFA beam by utilizing the proposed design scheme, a metasurface array composed of two symmetrical structures is designed to generate two counter-propagating Airy beams. Two Airy beams will get convergence when they accelerate to the symmetry axis. The phase profile is implemented using $\varphi \left(x\right)={\phi}_{total}\left(x\right)+{\phi}_{total}\left(-x\right)$ where ${\phi}_{total}\left(x\right)$ is the phase profile given in Eq. (3). One of the key issues in a self-focusing device is the adjustability of the focal length, and the controllability of the radiation direction of Airy beam just provides an effective solution.

Figure 4 (a)-(c) show the normalized intensity distributions for different focal length where *α* = −3°, *α* = −5° and *α* = −7°, respectively. The interference of the two Airy main lobes leads to a localized energy enhancement at different intersection point, and the corresponding focus positions are *f*_{1} = 20 μm, *f*_{2} = 25 μm, and *f*_{3} = 31 μm, respectively. Although TiO_{2} has a relatively high refractive index, the nanofin structures with subwavelength unit-cells reduce the impendence mismatches at SiO_{2}-TiO_{2} and TiO_{2}-air boundaries [21]. As a result, for the designed devices, the calculated focusing efficiencies ranging from 65%-75% are achieved. The intensity profiles also reveal an interesting depth of focus (DOF) feature. The DOF is proportional to the focal length, which originates from the longer interference region for the two Airy beams with larger deflection angle. Figure 4 (e)-(g) show line profiles at focal point along the *x*-axis. The retrieved FWHMs of the normalized intensity on the focal planes are approximately 423 nm, 498 nm, 525 nm for cases *α* = −3°, *α* = −5° and *α* = −7°, respectively. Figure 4(d) shows that the two main lobes of AFA beams are blocked by a 3μm × 1μm rectangular barrier at the path of their propagation. Due to self-acceleration properties, two beams still converge almost at the same position with a similar FWHM (Fig. 4h).

## 5. Generation of rotation-symmetric AFA beam and vortical AFA beam

The phase profile computed for airy beam can be expediently combined with additional rotational phase terms so as to generate rotation-symmetric and vortical AFA beams [35,36]. We numerically demonstrate these results in Fig. 5, which shows that two different styles of optical beams can be readily achieved using metasurface array. Our nanofins are arranged on a fused-silica disk, and the required phase is created using the same phase discretization method as mentioned above. The diameter of disk is about 9 μm and contains 41 nanofins. To obtain the required phase of the rotation-symmetric AFA beam, we apply rotational symmetry operation to the computed phase profile ${\phi}_{total}(r)$ which versus radius *r* of the disk, as shown in Fig. 5(a). In the simulation, by employing perfectly matched layers (PML) conditions, the designed device with a total area of 63.6 μm^{2} is simulated in a space of 9.3 × 9.3 × 25 μm^{3} simulation region. Electric field intensity distributions are respectively monitored in *x-y* plane at the designed focal position and in *x-z* plane along the path of propagation. Same as the mirror-symmetric AFA beams simulation, the normal incident light is RCP. As shown in Fig. 5(b) and (c), the autofocusing phenomenon is observed at the focal plane. By comparing with the former mirror-symmetric AFA beams, Fig. 5(c) shows a longer DOF and different convergence point due to optical beam is truncated by less transverse number of nanofins.

To obtain vortical AFA beam, we imprint the phase profile of AFA beams within a vortex phase mask similar as previously reported technology in Refs [13,37]. This phase profile is implemented in Fig. 5(d) by using $\varphi \left(r,{\theta}_{r}\right)={\phi}_{total}(r)+{\phi}_{vortex}({\theta}_{r})$, where ${\phi}_{vortex}\left({\theta}_{r}\right)=l\Delta {\theta}_{r}$ describes the vortex phase. Here, ${\theta}_{r}$ can be given as ${\theta}_{r}=\text{arc}\mathrm{tan}(y/x)$, and *l* represents the charge of the vortex with respect to the orbital-helicity of the beam [38] and it can be positive, negative or fractional values. Figure 5(e) and (f) demonstrate that the AFA vortex beam can be realized as expected. It is obvious that additional vortex phase assists to achieve a longer transmission distance and it will be useful for optical trapping. Significantly, compared with the conventional FT systems, the designed metasurface devices are capable of generating rotation-symmetric and vortical AFA beams in a straightforward manner and have an ultrathin thickness (~1.4*λ*_{0}) and a small geometry size (*r* = 4.5 μm).

## 6. Summary

In conclusion, we propose and design an extremely compact component to generate Airy beam, AFA beam and vortical AFA beam at the visible frequency. FDTD simulation results show that the designed components have the flexibility of controlling the deflection of Airy beam and tuning the focal length of AFA beam. Furthermore, we also demonstrate that combining AFA beam with vortex light field is capable of producing vortical AFA beam. Our design provides a simple way to generate and control various non-diffraction beams. This device may be used in the applications such as integrated optics system, biomedical nanosurgery and optical trapping techniques.

## Funding

Key Research and Development Program from Ministry of Science and Technology of China under Grant No. 2016YFA0202100; National Natural Science Foundation of China under Grant No. 61575092.

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