1. Introduction

Airy wave packet was firstly theoretically predicted in the domain of quantum mechanics [1] and then it was introduced in the context of optics based on the analogy between Schrödinger equation and the paraxial Helmholtz equation [2]. Except for the nondiffracting and self-healing features, Airy beam has the unique property of accelerating freely along curved trajectories without the presence of external forces. Because of its distinctive properties, Airy beam has been extensively studied and is attractive in potential applications including optical micromanipulation [3], plasma channel generating [4], laser micromachining [5] and signal processing [6]. However, the generation of Airy beam usually requires the Fourier transform of a Gaussian beam on which a cubic phase is imprinted [4, 7–10]. Thus, a spatial light modulator and a lens are involved, which is unfavorable for the compact and integrated optical systems and hinders the implementation and application of Airy beam in the mirco/nano photonics.

The electromagnetic properties of subwavelength metallic structure attract considerable research interest as the demand for miniature and integrated optical systems increases. When light impinges on the subwavelength structure, the light can be transmitted and surface plasmon polaritons (SPPs) which propagate along the 2D metal/dielectric interface can also be excited [11]. The amplitude and phase of the transmitted light [12, 13] and the excited SPPs [14, 15] are determined by the geometry parameters of the structure. By designing the parameters and arrangement of subwavelength units, the wavefront of the transmitted light and SPPs can be almost arbitrarily engineered. Functionalities including focusing [16–19], vortex generation [20–23], hologram [24, 25] and anomalous refraction [12, 26–29] have been demonstrated. Such kind of artificially designed material with subwavelength thickness is defined as metasurface. Based on this idea, Airy beam generation in the nanoscale has been realized. SPPs Airy beam was generated using a specially designed diffraction grating [30] or a nonperiodically arranged nanocave array [31] in 2011. Moreover, SPPs Airy beam with dynamical trajectories [32–34] and hot spots generated by interfering SPPs Airy beams [35] were studied. For the transmitted light, Airy beam in the 3D free space has also been demonstrated using metasurface composed of nanorods [36, 37] or C-shaped apertures [38, 39]. The orientation angles of the nanorods or C-shaped apertures are spatial variant, which can provide the geometry Pancharatnam-Berry (PB) phase for the orthogonally-polarized transmitted light. The geometry PB phase can be achieved by manipulating the different polarization states of light [40–42]. However, the Airy beam generations for SPPs and transmitted light have been considered separately. The metasurface was not fully exploited because either SPPs or the transmitted light was neglected, which affects the efficiency and multifunctionality of the metasurface.

Here, we propose a metasurface consisted of subwavelength slits to launch Airy beam for SPPs and transmitted wave at the same time. The dependent of the generated Airy beams on the handedness of circularly polarized light is analyzed. For left circularly polarized (LCP) and right circularly polarized (RCP) light, the SPPs Airy beams are on the left side and right side of the metasurface respectively, while the Airy beam for transmitted wave is polarization independent. The generated Airy beams are also capable of the nondiffracting, self-bending and self-healing properties. Furthermore, abruptly autofocusing of SPPs and transmitted wave is realized based on the proposed method.

2. Results and discussions

As a proof of concept, the metasurface which can generate Airy beam for SPPs and transmitted wave is designed and studied in the terahertz (THz) frequency range. Subwavelength slits which can modulate the amplitude and phase of both SPPs and transmitted wave are chosen as the unit cell of the metasurface. As shown in Fig. 1(a), the slits are etched on a 100 nm gold film and the substrate is high resistance silicon which is transparency in the THz range. The width of the slit is a fixed value w = 20 μm. The length and the orientation angle of the slit are changed to control the amplitude and phase separately. By using commercially available software FDTD Solutions, 3D finite difference time domain (FDTD) simulations are performed. In the simulation, the frequency of the incident THz wave is set as 0.75 THz which is the central frequency of THz wave experimentally generated by a <110> ZnTe crystal, the gold film is set as perfect electric conductor (PEC) and the refractive index of silicon is 3.42. For the slits with different lengths, the simulated amplitudes of transmitted wave and SPPs are normalized and shown in Fig. 1(b). It can be seen that the amplitudes of transmitted wave and SPPs increase simultaneously as length l changes from 40 μm to 100 μm. For the slits with different orientation angles, the phase modulations of transmitted wave and SPPs are shown in Figs. 1(c) and 1(d). For the transmitted wave, the phase modulation changes linearly and follows $\Phi \text{=2}{\sigma }_{\pm }\alpha$, where $\sigma \text{=}\pm \text{1}$ represents LCP light and RCP incident light, respectively. And it should be noted that the phase modulation is only suitable for the transmitted wave with opposite circular polarization of the incident light [13,41]. The phase modulation of SPPs is symmetrical about the origin and approximately follows the line shape of $\Phi \text{=}{\sigma }_{\pm }\left[\alpha \text{-}\mathrm{sgn}(\alpha )\cdot \frac{\pi }{2}\right]$. For slits with same orientation angle, the phase modulation of SPPs is about half of the phase modulation of the transmitted wave.

 

Fig. 1 A unit cell of the metasurface (a), amplitude modulation for slits with different lengths (b), phase modulation realized by changing the orientation angle of the slits for LCP (c) and RCP (d) light.

Download Full Size | PDF

The metasurface designed to generate Airy beam is schematically shown in Fig. 2(a). With light normally incident on the metasurface consisted of subwavelength slits, SPPs propagating along the 2D gold/air interface are excited and the transmitted light propagates in the 3D free space. By properly designing the distribution and geometry parameters of the slits, SPPs Airy beam in the xy plane and Airy beam for the transmitted wave in the yz plane can be simultaneously generated. The amplitude and phase distribution required to generate 1D Airy beam can be expressed as $\varphi \left(y,\theta \right)=Ai(by)\exp (ay+ikby\sin \theta )$, where $Ai(y)$ is the Airy function, b denotes the transverse scale, $a\ll 1$ is a positive value defining exponential apodization of the Airy beam profile, k is the wave number and $\theta$ denotes the bending direction [43]. Thus, the desired amplitude distribution follows $\left|\varphi \left(y,\theta \right)\right|=\left|Ai(by)\exp (ay)\right|$ and the normalized amplitude is shown in Fig. 2(b) which exhibits the oscillatory and exponential decaying features. The desired phase distribution is $\arg [\varphi (y,\theta )]=kby\sin \theta$ for $Ai(by)\ge 0$ and is $\arg \left[\varphi (y,\theta )\right]=\pi +kby\sin \theta$ for $Ai(by)<0$. In the case of $\theta =0$, the bending direction of the Airy beam is perpendicular to the metasurfaces and the phase can be simplified to a binary distribution (0 or π) which is shown in Fig. 2(c). The desired amplitude distribution can be realized by changing the length of the slits, while the phase distribution can be achieved by changing the orientation angle of the slits. The $\alpha =-{45}^{\circ }$ and $\alpha ={45}^{\circ }$ slits are chosen as the building block of the metasurface. The phase difference between $\alpha =-{45}^{\circ }$ and $\alpha ={45}^{\circ }$ slits is π for transmitted wave and is π/2 for SPPs, which can be obtained from Figs. 1(c) and 1(d). To realize the π phase modulation for SPPs, the $\alpha =-{45}^{\circ }$ and $\alpha ={45}^{\circ }$ slits are transversely separated by a distance of $s={\lambda }_{sp}/4$ which corresponds to an additional π/2 phase modulation for SPPs, as shown in the inset of Fig. 2(a). And the longitudinal separation between the slits is d = 100 μm. The wavelength of the excited SPPs ${\lambda }_{sp}$ is 400 μm for incident light with a frequency of 0.75 THz.

 

Fig. 2 Schematic diagram of the metasurface designed for Airy beam generation (a), the normalized amplitude distribution (b) and simplified phase distribution (c) of Airy function.

Download Full Size | PDF

Based on the analyses of subwavelength slits and Airy beam, the metasurface for generating Airy beams is designed and numerically studied. In the simulation and theoretical analyses, the decay factor a and transverse scale b for the desired Airy beam are $a=0.01$ and $b=2.17$, respectively. With the illumination of LCP and RCP light, the generated SPPs Airy beams are shown in Figs. 3(a) and 3(b) respectively. The intensity distribution of SPPs clearly shows the profile of Airy beam, but the position of generated SPPs Airy beam depends on the handedness of the incident circularly polarized light. The generated SPPs Airy beams are on the left and right side of the slits for LCP and RCP light respectively. This can be understood by examining the phase distribution of generated SPPs. The phase of SPPs propagating to the left and right can be written as ${\Phi }^L\text{=}{k}_{sp}s+\phi$ and ${\Phi }^R=-k_{sp}s+\phi$ respectively, where $k_{sp}$ is the wave number of SPPs, $s={\lambda }_{sp}/4$ and $\phi ={\sigma }_{\pm }\pi /2$ is the phase difference between $\alpha =-{45}^{\circ }$ and $\alpha ={45}^{\circ }$ slits. For LCP incident light, phase distributions of SPPs propagating to the left and right are ${\Phi }^L=\pi$ and ${\Phi }^R=0$, respectively. Thus, only SPPs propagating to the left can satisfy the phase distribution shown in Fig. 2(c) which is needed to generate the Airy beam. For the RCP incident light, the phase distribution of SPPs is reversed and thus the SPPs Airy beam is generated on the right side. In addition to the SPPs Airy beam manipulation based on spatial light modulator [29], gradient index structure [30] and graphene [31], here we show that polarization can provide another effective and simple way to dynamically manipulate SPPs Airy beam.

 

Fig. 3 SPPs Airy beam generated by LCP (a) and RCP (b) light, (c) is the transversal intensity distribution along x = −0.7 mm, (d) is the intensity distributions of the main lobe along x = −1 mm, x = −5 mm, and x = −9 mm, (e) and (f) show the self-healing property of the SPPs Airy beam.

Download Full Size | PDF

Transversal intensity distribution along x = −0.7 mm for SPPs Airy beam generated by LCP light is extracted and shown in Fig. 3(c). The transversal profile of the SPPs Airy beam agrees well with the theoretical results. As shown in Fig. 3(d), the intensity distributions of main lobe along x = −1 mm, x = −5 mm and x = −9 mm are also extracted to analyze the nondiffracting feature of SPPs Airy beam. The full widths at half maximum (FWHM) of the main lobe are 711.2 μm, 717.6 μm and 788 μm, respectively. The FWHMs only increase slightly (77 μm) when SPPs Airy beam propagates from x = −1 mm to x = −9 mm. A hole with a diameter of 250 μm is artificially created in front of the main lobe to examine the self-healing property of SPPs Airy beam. The hole which is represented by a green circle in Figs. 3(e) and 3(f) only affects the field distribution locally and the SPPs Airy beam recovers to its shape after the holes.

For the transmitted wave, the generated Airy beams are shown in Figs. 4(a) and 4(b), respectively. The nondiffracting and self-bending features of the generated Airy beam can be clearly seen. And the generation of the Airy beam is not influenced by the handedness of the incident circularly polarized light. That’s because the π phase difference between $\alpha =-{45}^{\circ }$ and $\alpha ={45}^{\circ }$ slits for LCP light is equivalent to the –π phase difference for RCP light. Transversal intensity distribution along z = 0.7 mm is plotted in Fig. 4(c), which shows the oscillatory and exponential decaying features and agrees well with the theoretical results. The intensity distributions of the main lobe along z = 1 mm, z = 5 mm, and z = 9 mm are extracted and shown in Fig. 4(d). The FWHMs of the main lobe are 659.7 μm, 727.2 μm and 832.2 μm, respectively. The increase of the FWHM is less than ${\lambda }_{sp}/2$ after propagating a distance of $20{\lambda }_{sp}$, which quantitatively demonstrates the nondiffracting feature of the Airy beam. The self-healing property of the Airy beam is studied by placing an obstacle 1.2 mm away from the metasurface in front of the main lobe. The simulated Airy beam intensity distributions with an obstacle are shown in Figs. 4(e) and 4(f). The green circle represents the obstacle with a diameter of 250 μm. It can be seen that the obstacle only disturbs the local field distribution and the Airy beams recover after passing the obstacle.

 

Fig. 4 Airy beam generation for transmitted wave with the illumination of LCP (a) and RCP (b) light, transverse intensity distribution along z = 0.7 mm (c), the intensity distributions of the main lobe along z = 1 mm, z = 5 mm, and z = 9 mm (d), the self-healing property of the Airy beam (c) and (d).

Download Full Size | PDF

Furthermore, abruptly autofocusing of both SPPs and transmitted wave can be realized by interfering two Airy beams. This feature is analogous to the autofocusing effect of ring Airy beam in the free space. The ring Airy beam can remain relatively low intensity profiles and suddenly autofocus following a parabolic trajectory [44]. The required electric filed to realize autofocusing can be expressed as $\varphi \left(y,0\right)=Ai\left[b\left(y_0-\left|y\right|\right)\right]\exp \left[a\left(y_0-\left|y\right|\right)\right]$, where $y_0$ is the distance between the main lobe and the origin [39]. As shown in Figs. 5(a) and 5(b), for LCP and RCP light, the autofocusing of the transmitted waves are indistinguishable and the focal lengths are about 5.1 mm. The autofocusing of SPPs are shown in Figs. 5(c) and 5(d). The SPPs are focused on the left side for LCP light and are focused on the right side for RCP light, which is caused by the polarization-controlled generation of SPPs Airy beam discussed above. The focal length can be modulated by changing the distance between the two SPPs Airy beams [35]. Compared with the focusing of light with lenses or zone plates, an advantage of the designed metasurface is that the focusing is less disturbed by obstacles [43], which inherits from the self-healing property of Airy beam.

 

Fig. 5 Abruptly autofocusing of transmitted wave for LCP (a) and RCP (b) incident light, (c) and (d) are the corresponding results for SPPs.

Download Full Size | PDF

3. Conclusions

In conclusion, a metasurface which can realize Airy beam generation for both SPPs and transmitted wave is demonstrated. The metasurface is consisted of spatial-variant subwavelength slits which can achieve amplitude and phase modulation. The excited SPPs and transmitted wave share the common properties of Airy beams including nondiffracting, self-bending and self-healing, but their responses to the polarization differ a lot. The handedness of circularly polarized light can control the position of SPPs Airy beam, but it has no effect on the Airy beam generated for the transmitted wave. Thus, besides the traditional applications of Airy beams, the chiral property of SPPs Airy beam can enable the metasurface to be applied to polarization-controlled nano-objects manipulation.