Generation of Polarization Independent Ring-Airy Beam Based on Metasurface

Abstract

In this paper, we generated polarization-independent ring-Airy beams by designing metasurfaces that can realize modulations of both phase and amplitude. In numerical simulation, such metasurfaces are designed by placing subwavelength rectangular slits in Au film uniformly. Two orthogonal types of slits, with orientation angles of 45 and −45 degrees, are used to obtain the binary phase profile in the light transmitted from the metasurface under illumination with either right circular polarization (RCP) or left circular polarization (LCP). This satisfies the phase required for Airy beam generation. Meanwhile, the difference between the phase profile under RCP illumination and that under LCP illumination is right 2π, which can be regarded as the same. This makes the metasurface available to generate Airy beams regardless of incident polarization. We also analyzed the auto-focusing, self-healing, and frequency-response properties of the generated Airy beams with different parameters. This work opens up more opportunities for applications of Airy beams.

1. Introduction

Airy wave packets were first predicted by Berry and Balazs in the context of quantum mechanics [1] and then demonstrated in the domain of optics by Siviloglou et al. based on the similarity between the Schrödinger equation and the paraxial Helmholtz equation [2,3]. Due to the non-diffracting, self-accelerating, and self-healing properties, Airy beams have been of great interest in optics. Tremendous applications have been proposed and experimentally demonstrated in the past few years, such as optical manipulation [4,5,6], laser micro-machining [7], plasma channel generation [8], and high-resolution optical microscopy [9,10], etc. However, traditional methods for generating Airy beams, as well as other interesting beams, usually involve spatial light modulators and lenses [11,12], which is unfavorable to applications in small-scaled or integrated optical devices.

Metasurfaces, composed of an artificially engineered array of metallic or dielectric subwavelength antennas, have been proposed to reduce the size and complexity of conventional optical systems [13,14,15]. By designing the geometric parameters (shapes, lengths, or orientations) of the antennas, one can control the amplitude, phase, and polarization of the wave transmitted or reflected from the metasurfaces in a wide spectrum band from microwave to light. This provides an alternative approach for producing nano-devices with various functionalities such as hologram imaging [14,16,17,18,19], anomalous refraction or reflection [13,20,21,22], light focusing and directing [23,24,25,26,27,28,29], leaky-wave antenna [30], and special beam generation [31,32,33,34], including Airy beams [35,36,37]. However, most proposed metasurfaces are either polarization-sensitive due to the non-rotational symmetry of unit antennas or with fractional frequency-response bandwidth, and this confines the metasuface functionality to particularly polarized light illumination [38]. Moreover, the polarization-independent properties of metasurface-based Airy beams have been seldom discussed in the transmitted waves, although polarization-independent Airy beam has been obtained in the near-field of the metasurface [39,40]. More comprehensive studies in this aspect are needed.

In this paper, polarization-independent metasurfaces are designed to generate and study Airy beams in numerical simulation. The basic unit of the metasurface is a metallic rectangular slit antenna of subwavelength, and the phase and the amplitude of the transmitted light wave can be modulated by adjusting the orientation angle and the length of each antenna, respectively. To keep the metasurface independent from the incident light polarization, two types of slit antenna (θ = −45°and θ = 45°) are chosen to obtain the binary phase modulations (0 and π). The abrupt autofocusing, focus length, frequency -response, and self-healing features of the generated Airy beams are all studied in this paper. Such a metasurface-based Airy beam may be of interest in polarization-independent integrated photonic systems.

2. Results and Discussions

Figure 1a schematically shows the metasurface designed for generating polarization independent ring-Airy beams and its basic subwavelength slit with length l, width w = 50 nm and orientation angles θ are shown in the inset. Here, in our work, the light distributions behind the metasurface are numerically simulated using commercially available software FDTD Solutions (Lumerical Fdtd Solutions 2016a), while the consistency between the simulation results and experiments has been validated by previous works. Figure 1d shows the normalized amplitude profiles as a function of length l for both the LCP and RCP incident lights. It can be seen that the transmitted light amplitude increases as l changes from 60 nm to 180 nm, which can be fitted by a Gaussian curve $T=A\exp (-{((x-B)/C)}^2)$, where the coefficients are A = 0.9, B = 180.3, and C = 66.8, respectively. Figure 1e shows the phase modulation in the transmitted wave as a function of the orientation angle θ. We see that the phase increase $\phi =2\theta$ but decrease $\phi =-2\theta$ linearly with increasing θ under LCP and RCP incident light, respectively.

To generate ring-Airy beam, the required amplitude and phase distributions of the transmitted waves on the initial plane right behind the metasurface should follow the equation in [15,36]

$$E(r)=Ai\left({\displaystyle \frac{r_0-r}{b}}\right)\exp \left[a\left({\displaystyle \frac{r_0-r}{b}}\right)\right]$$

(1)

where $Ai(r)$ is the Airy function, $r_0$ is the distance between the main lobe and the origin, $r$ is the radius, b is the scaling length, and $a<<1$ is the truncation factor. In FDTD simulation, the parameters in Equation (1) are set to be a = 0.01, $b=0.6 \mathsf{\mu }\mathrm{m}$, $r_0=1.55 \mathsf{\mu }\mathrm{m}$ and incident wavelength $\lambda =0.633 \mathsf{\mu }\mathrm{m}$, with the designed focal length $f\approx 17.9(b^2/\lambda )\sqrt{r_0/b-1}$ correspondingly to be about 11 μm. The calculated amplitude and phase distributions for such ring-Airy beam are given in Figure 1b and Figure 1c, respectively. We can see from Figure 1c that the phases contains values of 0 and π. Such a binary phase distribution is also the obvious requirement of Equation (1), that is, the phase distribution $\phi =\arg (E(r))$ could be 0 for $Ai\left({\displaystyle \frac{r_0-r}{b}}\right)>0$ and π for $Ai\left({\displaystyle \frac{r_0-r}{b}}\right)<0$, respectively.

Herein, we place two orthogonal sets (θ = −45°and θ = 45°, indicated by the dotted circles in Figure 1e of the slit antenna on the metasurface to obtain the required binary phase profile. It should also be pointed out that the phase difference π (between θ= −45°and θ = 45°) for LCP light plays the same role with the phase difference −π for RCP incident light. Thus, when the metasurface is illuminated by either LCP or RCP light, the binary phase distribution required for generating ring-Airy beams can be simultaneously satisfied. Moreover, the proposed metasurface can generate Airy beams under illumination of arbitrary polarization because light of any arbitrary polarization can be seen as linear composing of orthogonal LCP and RCP. Thus, we can build up polarization-independent metasurfaces by arranging the well-designed slit antennas to obtain ring-Airy beams.

We now verify the ring-Airy beams generation with our proposed metasurface. In the FDTD simulation, the parameters of the required Airy beam is set as that above, and nanoslits are placed uniformly in a 6 μm × 6 μm area with each unit area 200 nm × 200 nm. The simulation results are shown in Figure 2. Figure 2a–f shows the longitudinal field distributions of the generated ring-Airy beam, corresponding to the incident beams with LCP, RCP, x-linearly polarization (XLP), y-linearly polarization (YLP), left-hand and right-hand elliptically polarization (LEP and REP), respectively. We can see that the main lobe of the generated Airy beam follows a parabolic trajectory, and the beam intensity is concentrated into a point as the propagation distance increases. The intensity near the maximum value at the center increases rapidly. This abruptly auto-focusing effect is due to the lateral acceleration of the ring-Airy beam itself. Furthermore, we see that the intensity distributions are almost the same, no matter whether the polarization of the incident light is CP, LP, or EP. This phenomenon indicates that the proposed metasurface is polarization-independent.

Figure 2(a1–f1) show the transverse field distributions on a x-y plane 1 μm behind the metasurface, respectively, corresponding to Figure 2a–f. It can be seen that some concentric rings are formed, with the brightest ring located in the inner side, which is in consistent with intensity pattern of ring Airy beam. The measured radius of the peak intensity ring is about 2.07 μm, which agrees well with the theoretical value 2.10 μm. To study the auto-focusing properties of the ring-Airy beam, the corresponding transverse field distributions at the focal plane (z = 11 μm) are also given in Figure 2(a2–f2), respectively. We can all observe a clear focal spot. The intensity distributions along y = 0 and x = 0 through the focus point are shown and compared in Figure 2g and Figure 2h, respectively. As can be seen by comparison, all the curves are nearly the same, matching each other exactly and further proving the polarization-independent property of the metasurface. The depth of the acoustic focusing (DOF) is 2480 nm ($\approx 4\lambda$) and the full width half maximum (FWHM) of the focusing spot is roughly 614 nm.

Self-healing is an intrinsic feature of Airy beam; that is, the beam would reconstruct itself without influence in the propagation after it overcomes the obstacles. In order to verify the self-healing feature of the Airy beams, we place sphere obstacles with a diameter of 800 nm along the y axis, as shown in Figure 3, where the obstacles are represented by green circles. Figure 3a,b show the intensity distributions with one obstacle placed on the main lobe, under incidence of LCP and RCP light, respectively. We can see that the beams are scattered by the obstacle but then reconstructed after passing it. In Figure 4a,b, four obstacles are placed with a spacing distance of 3 μm. Due to the self-healing ability, both scattering and reconstructing of the beams are still being observed. Figure 3e,f show the intensity profiles along the z-axis and y-axis in the corresponding focal planes. The black, blue, and red lines represent the intensity profiles with no obstacle, one obstacle, and four obstacles, respectively. Both the focal length of the metasurface and the FWHM of the beams show no visible difference under the three cases. These phenomena indicate the robust self-healing feature of the generated beam and the proposed metasurface.

Then, we investigate the frequency -response properties of the Airy beam, that is, to see whether the metasurface work under illumination with a broad wavelength range. The intensity distributions behind the metasurface with incident of other four arbitrary wavelengths, e.g., 450 nm, 531 nm, 730 nm, and 800 nm, are shown in Figure 4a–h. Comparing the LCP cases in the left column with the corresponding RCP case in the right column, we see that good focusing is realized and that the intensity patterns are nearly the same, although focal length changes slightly under a certain wavelength. Their intensity profiles along the x-axis at the focal planes and that along the z-axis are presented in Figure 4i and Figure 4f, respectively. The case for 633 nm is also presented for comparison. The FWHMs of the focus spot at the five wavelengths remain almost unchanged along the x-axis, and the intensity profiles show good Gaussian-type, although the focal length increases along with the wavelength. That is, the proposed metasurface possesses the polarization-independent properties during a wide bandwidth, at least from 450 nm to 800 nm.

In addition, we realize the flexible control of the focal length of the ring-Airy beam by adjusting the parameters (r₀, b, a = 0.01). Two metasurfaces with focal lengths f₁ = 9 µm and f₂ = 13 µm are designed with parameters $r_0=1.15 \mathsf{\mu }\mathrm{m}$, $b=0.56 \mathsf{\mu }\mathrm{m}$ and $r_0=1.6 \mathsf{\mu }\mathrm{m}$, $b=0.66 \mathsf{\mu }\mathrm{m}$, respectively. Figure 5a–d shows the intensity distributions generated by the two types of metasurface, under LCP and RCP incidence, respectively. The insets show the corresponding intensity distributions at the focal planes. Focal spots at z = 9 μm and 13 µm are obtained for the two cases, which is as expected in the design. For LCP and RCP light, the autofocusing of the transmitted waves shows little dependence on the incident polarizations. It is obvious that polarization-independent ring-Airy beams with different focal lengths can be generated by changing parameters (r₀, b, a) of the metasurface.

3. Conclusions

In conclusion, we demonstrated the design of a polarization-independent metasurface for generating ring-Airy beam in numerical simulation. The amplitude profile needed for Airy beam generation is obtained by adjusting the lengths of the rectangular nanoslits. The required binary phase profiles were obtained by using two orthogonal types of rectangular nanoslits with orientation angles of 45 and −45 degrees, respectively. Because the difference between the phase profile under RCP illumination and that under LCP illumination is right the ignoble 2π and that any polarization can be regarded as linear composing of orthogonal RCP and LCP, the metasurface can generate ring-Airy beams under arbitrary light illumination regardless of polarization. The autofocusing, self-healing, and frequency -response properties of the generated ring-Airy beam have been investigated in detail, and the full control of the focal length of the ring-Airy beam have been realized by changing the parameters of the metasurface. Furthermore, such metasurfaces can be fabricated by etching subwavelength-scaled slits in Au-SiO₂ film using Focused Ion Beam Etching Technology, and many previous works have validated the consistency of simulated and experimental results. The results herein may be used in applications such as integrated optics systems and optical trapping techniques.