Generating superposed terahertz perfect vortices via a spin-multiplexed all-dielectric metasurface

Fan Huang; Quan Xu; Wanying Liu; Tong Wu; Jianqiang Gu; Jiaguang Han; Weili Zhang

doi:10.1364/PRJ.476120

1. INTRODUCTION

Phase and polarization are two nontrivial characteristics of light [1]. Circularly polarized light owns spin angular momentum (SAM) of $σ ℏ$ per photon attributed to the orthogonal handedness [2], where $σ = \pm 1$ represents the spin of left- and right-handed circular polarization (LCP and RCP). On the other hand, orbital angular momentum (OAM) is expressed as $l ℏ$ , where l is the topological charge (TC)—an integer of single quantum number [3]. The beam that carries OAM, named the optical vortex beam (OVB), has an undefined phase in the center of its helical phase structure, which is described by an azimuthal phase dependence $\exp (i l θ)$ ) and enables the intensity distribution to be like a doughnut [4]. The whole angular momentum of a circularly polarized OVB can be expressed by $J = (σ + l) ℏ$ per photon, including both SAM and OAM [5,6]. Over the past 30 years, OVB has been explored in depth and has led to a number of applications in fields such as particle manipulation, high-resolution microscopy, optical communication, and quantum information [7 –11], while traditional OVB generation methods including diffractive optical elements [12], spiral phase plates [13], and spatial light modulators are either bulky or inflexible, hindering the further development of this field.

Recently, the metasurface has emerged as a compact approach for OVB generation through the design of suitable subwavelength meta-atoms and the arrangement of their spatial distributions [14 –21]. In particular, the implantation of the geometric phase endows arbitrary and dispersionless phase engineering for circular polarization by simply tuning the orientation angle, providing great flexibilities for the SAM-to-OAM conversion of light [22 –25]. Further in conjunction with spin-independent design methods, such as the propagation phase, arbitrary optical SAM-to-OAM conversion can be achieved, that is, independent OVB generation under LCP and RCP incidences [26]. Notably, the beam radii of these conventional OVBs always depend on their TC number, limiting their application in some scenarios, for example, coupling multiple OVBs into an optical component with a fixed annular profile for mode division multiplexing. In order to overcome this limitation, the concept of perfect optical vortices (POVs) has been proposed, whose beam radii are independent of the TCs [27 –32]. So far, in the reports of metasurface-based POV generation, both spin-dependent and spin-independent [33 –37], only a single POV can be generated under LCP or RCP incidence, corresponding to a one-to-one SAM-to-OAM conversion.

In this paper, a metasurface-based method for spin-distinguished generation of superposed POVs is proposed. This is achieved by adopting the multiple meta-atoms comprised structure, namely, the meta-molecule as the basic unit to independently manipulate the output phases and amplitudes of orthogonal circular polarizations [21]. In this manner, the designed metasurface can totally tailor and decouple the complex amplitude of two superposed POVs, subsequently imposing the corresponding information to the orthogonal circular polarization. Based on this meta-molecule, we have designed and testified two metasurface samples (Metasurface-I and Metasurface-II) in the terahertz domain and found the measured intensity and phase distribution are well matched with the calculation of the Rayleigh–Sommerfeld diffraction integral. The proposed design strategy opens an avenue toward the ultracompact terahertz functional devices and sophisticated SAM–OAM interaction, which may find applications in terahertz communication, producing complex structured light and exploring photonic spin-Hall effect.

2. RESULTS AND DISCUSSION

A. Design Principle

The envisioned function of the metasurface is to independently convert orthogonal circular polarization to superposed POVs, and these unique responses are illustrated in Fig. 1. The POV can be generated by Fourier transform of the Bessel–Gaussian beam, in which the Bessel–Gaussian beam can be generated by an axicon and the Fourier transform operation can be achieved by a spherical lens. The overall phase profile of a POV, provided by the designed metasurface, has a form of $φ (x, y) = - 2 π \frac{\sqrt{x^{2} + y^{2}}}{d} + l \times \arctan (\frac{x}{y}) + \frac{2 π}{λ} \times (f - \sqrt{x^{2} + y^{2} + f^{2}}),$ (1)where the first item represents the axicon’s phase profile $φ_{axicon}$ , in which $d$ is the axicon period determining the ring radius of the POV. The second item corresponds to the spiral phase profile $φ_{spiral}$ carrying the OAM with a TC of $l$ . The last item is the phase profile of a spherical lens, in which $f$ is its focal length and $λ$ is the operating wavelength [38,39]. The phase profile of the spherical lens is very similar to that of a parabolic lens $φ = - π (x^{2} + y^{2}) / λ f$ .

Figure 1.Functional schematic of designed metasurface. (a) Phase and amplitude manipulation for LCP incidence. The generated multiplex amplitude $E_{1} (x, y) \exp [i φ_{1} (x, y)]$ is the superposition of POVs $| {POV}_{L}, l_{a} = 1 ⟩$ and $| {POV}_{L}, l_{b} = - 1 ⟩$ . (b) Phase and amplitude manipulation for RCP incidence. The generated multiplex amplitude $E_{2} (x, y) \exp [i φ_{2} (x, y)]$ is the superposition of POVs $| {POV}_{R}, l_{p} = 2 ⟩$ and $| {POV}_{R}, l_{q} = - 2 ⟩$ . For all the cases, only the flipped handedness to the incidence is considered in the output, and $E_{1} (x, y) \exp [i φ_{1} (x, y)]$ is independent of $E_{2} (x, y) \exp [i φ_{2} (x, y)]$ .

Download full size

View all figures

In conjunction with the above phase distribution of the POV, the ultimate complex amplitude formed by the superposition of two POVs can be calculated. When LCP is incident on the metasurface [Fig. 1(a)], the RCP output is expressed as a complex amplitude distribution $E_{1} (x, y) \exp [i φ_{1} (x, y)]$ , which can be decomposed into two POVs $\exp [i φ_{a} (x, y)]$ with TC of $l_{a}$ and $\exp [i φ_{b} (x, y)]$ with TC of $l_{b}$ . Similarly, as the input is altered to RCP [Fig. 1(b)], the LCP output from the metasurface is $E_{2} (x, y) \exp [i φ_{2} (x, y)]$ comprising two POVs $\exp [i φ_{p} (x, y)]$ with a TC of $l_{p}$ and $\exp [i φ_{q} (x, y)]$ with a TC of $l_{q}$ . By adopting Jones vectors $| LCP ⟩ = {[1, i]}^{T}$ and $| RCP ⟩ = {[1, - i]}^{T}$ to represent the normal LCP and RCP incidences, respectively, the metasurface can be described by a Jones matrix $J (x, y)$ calculated as $\frac{1}{2} (\begin{matrix} E_{1} (x, y) e^{i φ_{1} (x, y)} + E_{2} (x, y) e^{i φ_{2} (x, y)} & - i [E_{1} (x, y) e^{i φ_{1} (x, y)} - E_{2} (x, y) e^{i φ_{2} (x, y)}] \\ - i [E_{1} (x, y) e^{i φ_{1} (x, y)} - E_{2} (x, y) e^{i φ_{2} (x, y)}] & - [E_{1} (x, y) e^{i φ_{1} (x, y)} + E_{2} (x, y) e^{i φ_{2} (x, y)}] \end{matrix}) .$ (2)

Therefore, the transformations $J (x, y) | LCP ⟩ = E_{1} (x, y) \exp [i φ_{1} (x, y)] | RCP ⟩$ and $J (x, y) | RCP ⟩ = E_{2} (x, y) \cdot \exp [i φ_{2} (x, y)] | LCP ⟩$ are simultaneously satisfied, meaning the spin-distinguished superposed POV generation can be achieved. Despite $J (x, y)$ being a non-unitary matrix that cannot be directly diagonalized, we can transform $J (x, y)$ to $[J_{M} (x, y) + J_{S} (x, y)] / 2$ , where $J_{M} (x, y)$ and $J_{S} (x, y)$ are unitary and can be rewritten as $J_{M} (x, y) = R_{M} Λ_{M} R_{M}^{- 1}$ and $J_{S} (x, y) = R_{S} Λ_{S} R_{S}^{- 1}$ , where $Λ_{M}$ and $Λ_{S}$ are the diagonal matrices, and the matrices $R_{M}$ and $R_{S}$ correspond to the rotation matrix for the $Λ_{M}$ and $Λ_{S}$ [Appendix D, Eqs. (D1) and (D2)]. This indicates that the metasurface includes two linear birefringent elements, in which the fast-axis orientation angle ( $θ$ ) and the phase shift ( $δ_{x}$ and $δ_{y}$ ) of each element are ascertained by the eigenvector and eigenvalue of $J_{M} (x, y)$ and $J_{S} (x, y)$ , respectively. For the given LCP and RCP inputs, $δ_{M, x}$ , $δ_{M, y}$ , and $θ_{M}$ corresponding to $J_{M} (x, y)$ and $δ_{S, x}$ , $δ_{S, y}$ , and $θ_{S}$ corresponding to $J_{S} (x, y)$ can be calculated as [Appendix D, Eqs. (D4) and (D5)] [21] ${\begin{cases} δ_{M, x} = [φ_{1} (x, y) + φ_{2} (x, y) + \arccos E_{1} (x, y) + \arccos E_{2} (x, y)] / 2 \\ δ_{M, y} = [φ_{1} (x, y) + φ_{2} (x, y) + \arccos E_{1} (x, y) + \arccos E_{2} (x, y)] / 2 - π \\ θ_{M} = [φ_{1} (x, y) - φ_{2} (x, y) + \arccos E_{1} (x, y) - \arccos E_{2} (x, y)] / 4 \end{cases},$ (3) ${\begin{cases} δ_{S, x} = [φ_{1} (x, y) + φ_{2} (x, y) - \arccos E_{1} (x, y) - \arccos E_{2} (x, y)] / 2 \\ δ_{S, y} = [φ_{1} (x, y) + φ_{2} (x, y) - \arccos E_{1} (x, y) - \arccos E_{2} (x, y)] / 2 - π \\ θ_{S} = [φ_{1} (x, y) - φ_{2} (x, y) + \arccos E_{2} (x, y) - \arccos E_{1} (x, y)] / 4 \end{cases} .$ (4)

Referring to the meta-atoms for implementing $J_{M} (x, y)$ and $J_{S} (x, y)$ , a series of subwavelength silicon pillars is designed to provide $δ_{M, x}$ , $δ_{M, y}$ , and $θ_{M}$ , and another series of silicon pillars is selected to satisfy $δ_{S, x}$ , $δ_{S, y}$ , and $θ_{S}$ . Figure 2(a) exhibits the schematic diagram and optical microscope image of the first demo sample—Metasurface I. All the silicon pillars are constructed on silicon substrate with square lattice constant $P = 230 μm$ and possess a uniform height $(H$ ) of 300 μm. The building block, named meta-molecules, contains two silicon pillars arranged at the main diagonal (silicon pillars M) and two silicon pillars arranged at the secondary diagonal (silicon pillars S) on a $2 \times 2$ square grid. The dimensions $D_{M, x}$ , $D_{M, y}$ of silicon pillar M and $D_{S, x}$ , $D_{S, y}$ of silicon pillar S specify the propagation phase shift, and their orientation angles $θ_{M}$ and $θ_{S}$ ascertain the geometry phase. In order to find appropriate silicon pillars that not only satisfy Eqs. (3) and (4) but also cover the phase range from 0 to $2 π$ , we perform simulation using commercially available software computer simulation technology (CST) Microwave Studio. By sweeping the transmitted phase and amplitude with varied $D_{x}$ and $D_{y}$ , 24 silicon pillars with phase gradient $π / 12$ are obtained at 0.6 THz. As one can observe in Fig. 2(b), the phase difference between $δ x$ and $δ y$ in the selected unit structures is $π$ , and the phase shift ( $δ x$ and $δ y$ ) increases linearly by $π / 12$ from 0 to $2 π$ , which matches well with Eqs. (3) and (4). Additionally, the polarization conversion efficiency of selected silicon pillars is also validated to be higher than 80% as illustrated in Fig. 2(c).

Figure 2.Design of the meta-molecules of the metasurface. (a) Left, schematic diagram and optical microscope image of Metasurface-I made up of silicon pillars. Scale bar, 300 μm. Right, a typical meta-molecule including two diagonal silicon pillars M and two anti-diagonal silicon pillars S. (b) Twenty-four levels of discrete phase in the full range [0, $2 π$ ] for exact phase level $δ_{x}$ and $δ_{y}$ (described by blue and yellow bars, respectively) and simulated phase level $P_{x}$ and $P_{y}$ (described by triangles and pentagons symbols, respectively). Each selected silicon pillar is an effective half-wave plate. (c) Calculated polarization conversion efficiency at 0.6 THz for varied dimensions ( $D_{x}$ and $D_{y}$ ) ranging from 40 μm to 200 μm. The selected silicon pillars are indicated as black dots.

Download full size

View all figures

With the 24 selected pillars, the design flow of the metasurface is depicted in Figs. 3(a)–3(d). For the first demo metasurface, the phase profiles of four POVs with TCs of 1, $- 1$ , 2, and $- 2$ are calculated, respectively, as shown in Fig. 3(a). Subsequently, the POVs with TCs of 1 and $- 1$ are superposed as the encoded information for RCP output while the superposition of POVs with TCs of 2 and $- 2$ is given to the LCP output. The summarized amplitude and phase profiles consist of the target complex-amplitude distributions $E_{1} (x, y) \exp [i φ_{1} (x, y)]$ and $E_{2} (x, y) \exp [i φ_{2} (x, y)]$ as depicted in Fig. 3(b). In order to accomplish this manipulation through the metasurface, appropriate silicon pillars in every local pixel are selected and properly rotated to best meet the requirements given by Eqs. (3) and (4) [Fig. 3(c)]. After the combination between silicon pillar M and silicon pillar S, the final structure as shown in the middle of Fig. 3(d) is simulated in CST and the vortex purities under different incidences are calculated based on the simulated results. As displayed in Fig. 3(d), the designed metasurface shows perfect vortex purity that matches well with the setting TCs.

Figure 3.Design flow of the metasurfaces. (a) Phase profiles of four POVs with TCs of 1, $- 1$ , 2, and $- 2$ from top to bottom. (b) Target amplitude and phase profiles of superposed POVs, which are encoded on the metasurface. (c) Silicon pillars M and S, which depend on the setting parameters ( $δ_{M, x}$ , $δ_{M, y}$ , $δ_{S, x}$ , $δ_{S, y}$ , $θ_{M}$ , and $θ_{S}$ ). (d) Top, vortex mode purity of the superposition of $| {POV}_{R}, l_{a} = 1 ⟩$ and $| {POV}_{R}, l_{b} = - 1 ⟩$ . Middle, arrangement of the silicon pillars on the designed metasurface. Bottom, vortex mode purity of the superposition of $| {POV}_{L}, l_{p} = 2 ⟩$ and $| {POV}_{L}, l_{q} = - 2 ⟩$ .

Download full size

View all figures

B. Independent Generation of Superposed POVs for LCP and RCP Incidences

Figure 4.Theoretical and experimental demonstration of Metasurface-I generating superposed POVs with different TCs and having an identical ring radius for LCP and RCP incidences. (a) Schematic diagram under LCP incidence. (b) Calculated and measured intensity and phase distributions under LCP incidence at 0.6 THz. The outer and inner radii of the ring pattern are $r$ and $r_{0}$ , respectively. (c) Calculated and measured purity of OAM modes for LCP incidence. The sampling radius in corresponding intensity profiles is 4.0 mm. (d) Schematic diagram under RCP input. (e) Calculated and measured intensity and phase distribution under RCP input states at 0.6 THz. (f) Calculated and measured purity of the OAM mode for RCP incidence. The sampling radius in the corresponding intensity profile is 4.6 mm.

Download full size

View all figures

In addition to the spin-multiplexed and equal-radius POV generation and superposition demonstrated by Metasurface-I, our design scheme of Metasurface-II with the same size as Metasurface-I can generate superposed POVs with different radii for LCP and RCP incidence [Figs. 5(a) and 5(b)], enabling novel vortex beam-based complex structured light, such as the radial twisted beam. Different from the conventional TC-dependent radial twist, the radial twist generated by Metasurface-II is TC-independent and spin-multiplexed, which has potential application in high-resolution refractive index sensing and optical tweezers [43,44]. From the design parameters of POVs in Table 1, the TCs and radii of two superposed POVs for LCP or RCP incidence are different while the radii of POVs in the two sets of superpositions for opposite circular polarization are correspondingly equal. The intensity and phase profiles of Metasurface-II at 0.6 THz are subsequently characterized computationally and experimentally, as displayed in Figs. 5(b) and 5(e). The lobe numbers of the intensity distributions, under LCP and RCP incidences, are in accord with the calculation (L, $l_{p 1} - l_{q 1}$ ; R, $l_{p 2} - l_{q 2})$ , and the numbers of discontinuities in the phase profiles are also identical to the lobe numbers. Besides, the vortex purity retrieved from the calculation and measurement is illustrated in Figs. 5(c) and 5(f). It is demonstrated that the mode weight of the scheduled TCs of $- 1$ and 2 for LCP incidence remarkably exceed others. As for RCP incidence, the OAM modes of $- 3$ and 1 are much higher than others as designed. We further compared the radii of the surrounding lobes under LCP and RCP incidences. The intensity profiles substantiate that the resulting lobes for both incidences will have nearly equal radius as long as the scheduled $d$ for different incident polarizations are correspondingly equal. Here, the measured radii are 5.55 mm ( $r$ ) and 2.50 mm ( $r_{0}$ ), which only slightly deviate from the radii of calculated intensity ( $r = 5.95 mm$ , $r_{0} = 2.32 mm$ ). Meanwhile, the inner circle rings ( $r_{0}$ ) of the superposed POVs are approximately between the theoretical radii of two POVs ( $r_{0} = 2.3 mm$ , $r_{0} = 2.8 mm$ ) with the different radii. Moreover, the response independence of Metasurface-II from 0.5 to 0.7 THz is also evaluated (Appendix F, Fig. 10), and the results of Metasurface-II are similar to those of Metasurface-I.

Figure 5.Theoretical and experimental demonstration of Metasurface-II transforming LCP and RCP incidence into the superposition of two POVs with completely different TCs and radii. (a) Schematic diagram under LCP incidence. (b) Calculated and measured intensity and phase distributions under LCP incidence at 0.6 THz. The outer and inner radii of the ring pattern are $r$ and $r_{0}$ , respectively. (c) Calculated and measured purity of OAM mode for LCP incidence. The sampling radius in corresponding intensity profile is 3.7 mm. (d) Schematic diagram under RCP incidence. (e) Calculated and measured intensity and phase distributions under RCP incidence at 0.6 THz. (f) Calculated and measured purity of OAM mode for RCP incidence. The sampling radius in corresponding intensity profile is 3.7 mm.

Download full size

View all figures

3. CONCLUSION

In summary, we have proposed and demonstrated a spin-multiplexed all-dielectric metasurface to generate superposed terahertz POVs independently and simultaneously for LCP and RCP incidences. This adjustment endows input LCP and RCP states with tailored complex amplitudes of superposed POVs through changing the dimension and rotation angle of silicon pillars of every meta-molecule in the metasurface platform. As a proof-of-concept demonstration, Metasurface-I and Metasurface-II were designed, fabricated, and characterized to verify this sophisticated transformation. Not only the measured intensity and phase distributions but also the experimental purity of the vortex modes are consistent with the theoretical results, which confirm the function of the metasurfaces. This work paves a novel path toward investigating the photonic spin-Hall effect and developing an ultracompact and terahertz functional device for terahertz communication and complex structured generation.

Category: Optical and Photonic Materials

Received: Sep. 20, 2022

Accepted: Jan. 11, 2023

Published Online: Feb. 24, 2023

The Author Email: Quan Xu (quanxu@tju.edu.cn), Jianqiang Gu (gjq@tju.edu.cn)

DOI:10.1364/PRJ.476120