# Theoretical Study on Generation of Multidimensional Focused and Vector Vortex Beams via All-Dielectric Spin-Multiplexed Metasurface

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{m}under LCP illumination. That is, the metasurface implements transformation $\left|LCP\right.\u232a\to \left|FO{V}_{R},{l}_{m}\right.\u232a$. Due to the introduction of the PB phase, the transmitted beam FOV has flipped handedness compared with the incident light. Similarly, the metasurface performs conversion $\left|RCP\right.\u232a\to \left|\left.\u2329FO{V}_{L},{l}_{n}\right|\right.\u232a$ when the incident light is switched from LCP to RCP.

_{2}substrate with a lattice constant P. The refractive indexes of the rectangular nanopillar and square substrate are 3.48 and 1.48, respectively. Due to the angular asymmetry of the unit geometry, it can be considered as a rectangular waveguide exhibiting birefringence property. The phase delays ${\varphi}_{x}$ and ${\varphi}_{y}$ of the meta-atom along the x- and y- axes can be achieved by varying L and W. To investigate the transmission properties of rectangular nanopillars, all simulations are performed based on 3D-finite-difference-time-domain (FDTD). The height H of the rectangular a-Si nanopillar is set as 1000 nm, and the lattice constant is P = 650 nm. The length (L) and width (W) are swept from 50 to 650 nm with an interval of 7.5 nm. By scanning the nanopillars with different geometrical parameters, we calculate the phase shifts and transmission coefficients of rectangular a-Si nanopillars for XLP and YLP light at the design wavelength of λ = 1500 nm. The incident light is plane wave and propagates along the +z direction. Periodic boundary conditions are applied in the x and y directions, and perfectly matched layers (PMLs) are implemented in the z direction. The mesh grids are set as 30 nm × 30 nm × 50 nm. Figure 3b,c describe the transmission coefficient ${T}_{x}$ and phase shift ${\varphi}_{x}$ as a function of the nanopillar size (L, W) upon the XLP light. For YLP incident light, the transmission coefficient ${T}_{y}$ and phase shift ${\varphi}_{y}$ can be regarded as the transposition of ${T}_{x}$ and ${\varphi}_{x}$, as shown in Figure 3d,e. It can be seen from Figure 3b–e that the phase delays can almost span over the full 2π range, so any phase combination (${\varphi}_{x}$, ${\varphi}_{y}$) can be gained by choosing the nanopillars size (L, W) reasonably. A set of 11 nanopillars (black pentagrams in Figure 3b) with an interval of 2π/11 are optimized, and the corresponding transmission coefficients, phase shifts, and polarization conversion efficiencies are shown in Figure 3f. The transmission coefficients (${T}_{x}$, ${T}_{y}$) of 11 nanopillars remain above 85%, and the phase difference between ${\varphi}_{x}$ and ${\varphi}_{y}$ approaches π. At the same time, these nanopillars can achieve high-efficiency polarization conversion. The polarization conversion efficiency is defined at the focal plane by $\eta ={\left|{E}_{RCP}\right|}^{2}/({\left|{E}_{RCP}\right|}^{2}+{\left|{E}_{LCP}\right|}^{2})$. So these selected nanopillars can be regarded as quasi-perfect half-wave plates. In conclusion, a metasurface composed of 11 nanopillars can implement independent and arbitrary phase modulation for LCP and RCP light.

## 3. Results and Discussion

#### 3.1. Generation of Multidimensional FOVs

_{L}and f

_{R}are the focal lengths of LCP and RCP incident light along the z direction, and l

_{m}and l

_{n}represent the topological charges of FOVs for LCP and RCP light, respectively. Specifically, λ = 1.55 μm, f

_{L}= 12 μm, f

_{R}= 20 μm, l

_{m}= −2, and l

_{n}= 1. The focused vortex metalens composed of 61 × 61 micropillars is employed to generate spin-dependent FOVs at different focal planes under the orthogonally circularly polarized light. The diameter of vortex metalens is about 39.5 μm (D = 39 μm). To reduce the simulation time and ensure accuracy, we establish a 3D FDTD simulation model with a simulation region of 20 µm × 20 µm × 2 µm, and a mesh size of 30 nm × 30 nm × 50 nm is employed in the multifunctional metadevices. The PML boundary conditions are applied along all the three axes for the simulations. Here, we can get the required far-field information by projecting the near-field data to the far-field. The numerical simulation results are shown in Figure 4. Figure 4a

_{1}displays the donut-shaped intensity distribution of electric field at the focal plane for LCP illumination. The electric field intensity profile of the x-z plane is given in Figure 4a

_{2}. It can be seen that the FOV is located at z = 12.4 μm, which is close to the preset value (f

_{L}= 12 μm). Figure 4a

_{3}demonstrates the phase distribution of the FOV along the x-y plane at z = 12.4 μm, where the number and direction of the spiral represent the number and sign of topological charge, implying that a FOV with topological charge l = −2 is generated under LCP incident light. The focusing efficiency of the transmitted beam FOV is 62%. (The focusing efficiency is given by the ratio of the focusing power to the total incident power). Correspondingly, Figure 4b

_{1}–b

_{3}show the electric field intensity distributions and phase profile of the output beam FOV under RCP incidence. That is, a FOV with topological charge of l = 1 is generated at z = 20.2 μm with a focusing efficiency of 65% for RCP light. According to the calculation of the numerical aperture (NA) = sin[tan

^{−1}(D/2Fi)], the NA of the two FOVs are 0.844 and 0.695 for LCP and RCP incident light. It can be found from Figure 4a

_{1}–b

_{3}that the vortex metalens can independently modulate the phases of LCP and RCP. When the incident light is converted into XLP, two FOVs can be observed at z = 12.4 μm and z = 20.2 μm, respectively, as shown in Figure 4c

_{1}–c

_{5}, which successfully realizes the multiplexed spin-dependent FOVs.

_{1}–a

_{3}. The corresponding focusing efficiencies are 27.4% and 16.2%, respectively. Figure 5a

_{4},a

_{5}show the phase profiles of the produced FOVs with topological charges l = −2 and l = −1 in the x-y plane. Similarly, two FOVs with focusing efficiencies of 35.3% and 20.1% are observed in the left half space, and the focusing positions are (−7.8, 0, 12.5) and (−14.7, 0, 20.3), respectively (see Figure 5b

_{1}–b

_{3}). The phase distributions of FOVs with the topological charges l = 2 and l = 1 along the x-y plane for RCP are presented in Figure 5b

_{4},b

_{5}. Therefore, the metasurface can implement spin-dependent and multidimensional FOVs under LCP and RCP incident light. From the Figure 5c

_{1}–c

_{3}, we can observe four FOVs mentioned above under XLP incident light. In order to analyze the phase of output beams intuitively, only the phase profiles of FOVs at (7.8, 0, 12.5) and (−14.7, 0, 20.3) are given here (see Figure 5c

_{4},c

_{5}). Additionally, some deviations (the inconsistency of the focus position and the imperfect phase distribution) between the simulation results and preset values can be attributed to the discrete phase profile of the metasurface for approximation of the continuous phase distribution, which leads to the optical response of selected nanopillars not exactly matching the required phases. However, these small deviations will not fundamentally affect the metadevice to generate multidimensional spin-multiplexed FOVs.

#### 3.2. Generation of Multidimensional Vector Vortex Beams

_{m}and l

_{n}, respectively. cos(α/2) and sin(α/2) denote the amplitude of RCP and LCP FOV, and β is the relative phase difference between them. To more clearly explain the implementation of the vector vortex beam, we introduce a hybrid-order Poincaré Sphere (HyOPS) in Figure 6. $\left|E\right.\u232a$ in Equation (12) describes any point with spherical coordinates (α, β) on the HyOPS, and each point represents a vector vortex beam with space-dependent polarization distribution and phase profile, where the polarization distribution can be determined by the polarization order p = (l

_{m}− l

_{n})/2, and the phase profile can be characterized by the topological Pancharatnam charge l

_{p}= (l

_{m}+ l

_{n})/2. Specifically, five points are selected on the HyOPS surface, which indicates five polarization states of the incident light, including LCP, left-handed elliptically polarized (LEP), LP, right-handed elliptically polarized (REP), and RCP, as shown in Figure 6. Among them, only a single FOV is generated at the specific focal plane for LCP and RCP. In particular, the vector vortex beam will be produced by the superposition of FOVs with equal power when the LP impinges on the metasurface.

_{1}–a

_{3}). The corresponding phase distributions along the x-y plane are shown in Figure 7a

_{4},a

_{5}. When the incident light is converted into RCP, two FOVs with focusing efficiencies of 41.8% and 35.4% can be seen at the same positions from Figure 7b

_{1}–b

_{3}. The NA of the two beams are 0.845 (z = 12.5 μm) and 0.611 (z = 25.3 μm), respectively. It is noteworthy that the sign of topological charge of the output beam FOV generated by RCP is opposite to that produced by LCP light in Figure 7b

_{4},b

_{5}. We can conclude that FOVs with a topological charge of l = $\mp $2 and l = $\mp $1 are gained at z = 12.5 μm and z = 25.3 μm under LCP and RCP incident light, respectively. When a LP light is incident onto the metasurface, the equal components of $\left|FO{V}_{R},{l}_{-2}\right.\u232a$ and $\left|FO{V}_{L},{l}_{2}\right.\u232a$ will further superimpose at the focal plane z = 12.5 μm to produce a vector vortex beam with the polarization order p = (−2–2)/2 = −2. Similarly, the linear combination of equal-weighted $\left|FO{V}_{R},{l}_{-1}\right.\u232a$ and $\left|FO{V}_{L},{l}_{1}\right.\u232a$ can generate a radial vector vortex beam with the polarization order p = (−1–1)/2 = −1 at z = 25.3 μm for XLP light.

_{1},a

_{5},b

_{1},b

_{5}. Figure 8a

_{3}shows that a vector vortex beam can be realized due to the superposition of FOVs at z = 12.5 μm for XLP light. The transmitted beam presents a pretty petal-like pattern through the horizontal linear polarizer, and the total number of lobes of the pattern is equal to $2\left|p\right|=4$. In the case of LEP and REP light, the gap between the intensity decreases due to the non-equal superposition of FOVs (see Figure 8a

_{2},a

_{3}). Similarly, it can be seen from Figure 8b

_{1},b

_{5}that both LCP and RCP incident light can generate annular light intensity distributions at z = 25.3 μm. The intensity profiles exhibit two petals due to the polarization order $2\left|p\right|=2$ for LEP and REP light from Figure 8b

_{2},b

_{4}, and a gap between the two petals becomes more clear under XLP illumination (see Figure 8b

_{3}). Thus, the multidimensional vector vortex beams with different modes can be implemented by a metasurface with phase-only modulation, which provides more opportunities and possibilities for high-capacity data transmission and communication. In addition, different polarized light is normally incident onto the metasurface. The introduction of angular-multiplexed is considered in further studies to realize multifunctional integrated and tunable metadevices.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic diagram of multidimensional focused vortex beams generated based on the dielectric spin-multiplexed metasurface.

**Figure 2.**The relationship between propagation, PB, and combined phases. The combined phase is obtained by simultaneously adjusting the propagation phase (only changing the size of meta-atoms) and the PB phase (only changing the rotation angle of meta-atoms).

**Figure 3.**(

**a**) Perspective and top views of the meta-atom which consists of a rectangular a-Si nanopillar patterned on the square SiO

_{2}substrate. (

**b**,

**c**) The transmission coefficients and phase shifts for XLP illumination. (

**d**,

**e**) The transmission coefficients and phase shifts under YLP incident light. (

**f**) The corresponding transmission coefficients, phase shifts, and polarization conversion efficiencies of all selected 11 nanopillars.

**Figure 4.**The simulated electric field intensity distributions of the FOV at the focal plane for (

**a**) LCP, (

_{1}**b**) RCP, and (

_{1}**c**,

_{1}**c**) XLP illumination. The electric field intensity profiles of the FOV in the x-z plane upon the illumination with (

_{2}**a**) LCP, (

_{2}**b**) RCP, and (

_{2}**c**) XLP input light. The corresponding phase distributions in the x-y plane at the focal plane under (

_{3}**a**) LCP, (

_{3}**b**) RCP, and (

_{3}**c**,

_{4}**c**) XLP incident light.

_{5}**Figure 5.**The simulated electric field intensity distributions of FOVs at the focal plane for (

**a**,

_{1}**a**) LCP, (

_{2}**b**,

_{1}**b**) RCP, and (

_{2}**c**,

_{1}**c**) XLP illumination. The electric field intensity profiles of FOVs in the x-z plane upon the illumination with (

_{2}**a**) LCP, (

_{3}**b**) RCP, and (

_{3}**c**) XLP input light. The corresponding phase distributions at the focal plane in the x-y plane under (

_{3}**a**,

_{4}**a**) LCP, (

_{5}**b**,

_{4}**b**) RCP, and (

_{5}**c**,

_{4}**c**) XLP incident light.

_{5}**Figure 6.**Schematic view of a vector vortex beam with space-dependent polarization distribution and phase profile on the HyOPS.

**Figure 7.**The simulated electric field intensity distributions of FOVs at the focal plane for (

**a**,

_{1}**a**) LCP and (

_{2}**b**,

_{1}**b**) RCP. The electric field intensity profiles of FOVs in the x-z plane upon the illumination with (

_{2}**a**) LCP and (

_{3}**b**) RCP. The corresponding phase distributions in the x-z plane at the focal plane under (

_{3}**a**,

_{4}**a**) LCP and (

_{5}**b**,

_{4}**b**) RCP.

_{5}**Figure 8.**The electric field intensity profiles of generated beams corresponding to five po-larized states of the HyOPS after transmission through a horizontal linear polarizer at different focal planes for (

**a**,

_{1}**b**) LCP, (

_{1}**a**,

_{2}**b**) LEP, (

_{2}**a**,

_{3}**b**) XLP, (

_{3}**a**,

_{4}**b**) REP, and (

_{4}**a**,

_{5}**b**) RCP illumination. The analyzing polarizer is depicted by the white double arrow.

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## Share and Cite

**MDPI and ACS Style**

Liu, Y.; Chen, L.; Zhou, C.; Guo, K.; Wang, X.; Hong, Y.; Yang, X.; Wei, Z.; Liu, H. Theoretical Study on Generation of Multidimensional Focused and Vector Vortex Beams via All-Dielectric Spin-Multiplexed Metasurface. *Nanomaterials* **2022**, *12*, 580.
https://doi.org/10.3390/nano12040580

**AMA Style**

Liu Y, Chen L, Zhou C, Guo K, Wang X, Hong Y, Yang X, Wei Z, Liu H. Theoretical Study on Generation of Multidimensional Focused and Vector Vortex Beams via All-Dielectric Spin-Multiplexed Metasurface. *Nanomaterials*. 2022; 12(4):580.
https://doi.org/10.3390/nano12040580

**Chicago/Turabian Style**

Liu, Yue, Li Chen, Chengxin Zhou, Kuangling Guo, Xiaoyi Wang, Yuhan Hong, Xiangbo Yang, Zhongchao Wei, and Hongzhan Liu. 2022. "Theoretical Study on Generation of Multidimensional Focused and Vector Vortex Beams via All-Dielectric Spin-Multiplexed Metasurface" *Nanomaterials* 12, no. 4: 580.
https://doi.org/10.3390/nano12040580