# Circularly Polarized Antenna Array with Decoupled Quad Vortex Beams

^{*}

## Abstract

**:**

_{1}= −1, l

_{2}= 1, l

_{3}= −1, and l

_{4}= 1 in Case I and topological charges of l

_{1}= −1, l

_{2}= 1, l

_{3}= −1, and l

_{4}= 1 in Case II. To further verify the above theory, the planar array in Case I is fabricated and analyzed experimentally. Dual-LP beams are realized by using rectangular patch elements with two orthogonally distributed feeding networks on different layers based on two types of feeding: proximity coupling and aperture coupling. Both the numerical simulation and experimental measurement results are in good agreement and showcase the corresponding quad-vortex-beam characteristics within 8~12 GHz. The array achieves a measured S

_{11}< −10 dB and S

_{22}< −10 dB bandwidth of more than 33.4% and 29.2%, respectively. In addition, the isolation between two ports is better than −28 dB. Our strategy provides a promising way to achieve large capacity and high integration, which is of great benefit to wireless and radar communication systems.

## 1. Introduction

_{1}= −1, l

_{2}= 1, l

_{3}= −1, and l

_{4}= 1. For theoretical design, the approach is further developed for generating decoupled multimode vortex beams which are orthogonal and enable to radiate over a large area. Therefore, the approach will find widespread applications for radars and point-to-point wireless communications and has the potential to be used in communication and imaging systems. The rest of this communication is organized as follows: Section 2 details the decoupled principle for quad vortex beams. In Section 3, the design of the element is presented. Section 4 introduces the feeding line and array design, while the simulated and measured results of the array are given in Section 5. Finally, Section 6 concludes the whole communication.

## 2. Theoretical Background

_{x}and d

_{y}along the x and y axis, respectively. The element can be considered to be composed of two sub-elements that are marked as A

_{mn}and B

_{mn}with different ports, contributing to two orthogonal LP radiations, which can be described as ${E}_{0,m,n}={\alpha}_{m,n}{e}^{j{\beta}_{m,n}}\left({e}_{L}+{e}_{R}\right)$, here m = 1, 2, …, M; n = 1, 2, …, N. The rotation angle φ

_{A(m}

_{,n)}, φ

_{B(m}

_{,n)}and excitation phase β

_{A(m}

_{,n)}, β

_{B(m}

_{,n)}of mn

^{th}element are nonuniformly distributed. Here, we take subarray A as an example and assume there is no coupling between A and B. The gradients of rotation angles and excitation phases along the x and y axis are defined as (φ

_{x}, β

_{x}) (φ

_{y}, β

_{y}), and thus the phase distribution can be represented by φ

_{A(m}

_{,n)}= [m − (M + 1)/2]φ

_{x}+ [n − (N + 1)/2]φ

_{y}and β

_{A(m}

_{,n)}= [m − (M + 1)/2]β

_{x}+ [n − (N + 1)/2]β

_{y}, respectively.

_{L}, AF

_{R}) will be generated from the decoupled LP wave, and AF

_{L}and AF

_{R}are formulated respectively as:

_{L},ϕ

_{L}), and (θ

_{R}, ϕ

_{R}) are azimuthal and elevation angles of LHCP and RHCP beams in free space. To form an efficient beam, the intensity of AF

_{L}and AF

_{R}should be a real number which requires that the combined phase term in each array factor is zero. Therein, the gradients of rotation angle and excitation phase between adjacent elements are derived as a function of the azimuthal and elevation angles.

^{jl}

^{ϕ}, where ϕ = ${\mathrm{tan}}^{-1}\frac{y}{x}$ is the phase-shift factor, ϕ is the azimuthal angle around z axis, and (x, y) are the positions of element. When a linear gradient phase pattern is superimposed with a spiral phase profile, a tilted vortex beam will form and the deflection angle will be determined by the linear gradient phase [37]. Similarly, when a vortex phase profile is superimposed to the linear excitation phase, the deflected vortex beam can be formed, i.e., ${\beta}_{A\left(m,n\right)}^{\prime}={l}_{1}{\mathrm{tan}}^{-1}(\frac{y}{x})+{\beta}_{A\left(m,n\right)}$. In that case, the array factors of dual-CP can be expressed as:

## 3. Dual-LP Element Design

_{r}= 2.65 + j0.001). As shown in Figure 2a, the dual-feed element can be decomposed as two sub-elements sharing an identical rectangular patch and ground plane with an H-shaped slot. To increase the isolation between the two sub-elements and avoid the crosstalk of the feeding networks, two ports are orthogonally arranged and printed in different layers to transmit dual LP wave radiations with orthogonal polarization. The Y-polarization operation of sub-element A is realized by proximity feeding using the upper feeding line, while the X-polarized operation of sub-element B is realized by aperture coupling using the bottom feeding line. The layout and parametric illustration are given in Figure 2b.

_{11}| and |S

_{22}| below −10 dB are observed over the bandwidths of 9.81–10.35 GHz and 9.56–10.67 GHz, and the isolation between the two ports is better than −38 dB.

## 4. Design of Quad-Vortex-Beam Planar Array

_{(A,}

_{L}

_{)}= 30°, ϕ

_{(A,}

_{L}

_{)}= 72°; θ

_{(B,}

_{L}

_{)}= 30°, ϕ

_{(B,}

_{L}

_{)}= 150°; θ

_{(B,}

_{R}

_{)}= 30°, ϕ

_{(B,}

_{R}

_{)}= 252°; θ

_{(A,}

_{R}

_{)}= 30°, and ϕ

_{(A,}

_{R}

_{)}= 330°. In this case, the distributions of the rotation angle and excitation phase are shown in Figure 5a.

_{A}, β

_{B}) that are introduced by feeding lines of two different sub-elements. According to Equations (4) and (5), the final quad-vortex-beam will obtain different topological charges (l

_{(A}

_{,}

_{L)}, l

_{(A}

_{,}

_{R)}, l

_{(B}

_{,}

_{L)}, l

_{(B}

_{,}

_{R)}) when the vortex phase profile with topological charges of l

_{1}, l

_{2}, and l

_{3}is superimposed to the above three phase patterns, respectively. Table 1 summarizes all of the scenarios of topological charges of the final vortex beams relative to three DoFs.

_{x}= d

_{y}= 18 mm. The distributions of the rotation angles and excitation phases are illustrated in Figure 5b,c. For Case I, l

_{1}= 1, l

_{2}= l

_{3}= 0 and then l

_{(A}

_{,}

_{L)}= l

_{(B}

_{,}

_{L)}= −1, l

_{(A}

_{,}

_{R)}= l

_{(B}

_{,}

_{R)}= 1 is obtained according to Table 1; for Case II, l

_{1}= 1, l

_{2}= 0, l

_{3}= 1 is chosen, and thus l

_{(A}

_{,}

_{L)}= −1, l

_{(B}

_{,}

_{L)}= 0, l

_{(A}

_{,}

_{R)}= 2, l

_{(B}

_{,}

_{R)}= 1. Figure 5d,e plot the corresponding far-field patterns that were simulated in HFSS, where the patterns are obtained by an API process which can automatically construct all the metallic patterns through program codes introducing 128 excitation phases to 128 ports with specific rotation angles, rather than using real feeding lines.

_{(A}

_{,}

_{L)}= 30°, ϕ

_{(A}

_{,}

_{L)}= 72°, and θ

_{(B}

_{,}

_{L)}= 30°, ϕ

_{(B}

_{,}

_{L)}= 150° with topological charges of l

_{(A}

_{,}

_{L)}= l

_{(B}

_{,}

_{L)}= −1, while two RCP vortex beams are located at θ

_{(B}

_{,}

_{R)}= 30°, ϕ

_{(B}

_{,}

_{R)}= 252°; θ

_{(A}

_{,}

_{R)}= 30°, and ϕ

_{(A}

_{,}

_{R)}= 330° with topological charges of l

_{(A}

_{,}

_{R)}= l

_{(B}

_{,}

_{R)}= 1.

## 5. Results and Discussion

_{11}≤ −10 dB is 3.34 GHz (8.12–11.46 GHz), with a relative bandwidth of 33.4%; the impedance bandwidth of S

_{22}≤ −10 dB is 2.92 GHz (8.66–11.58 GHz), corresponding to a relative bandwidth of 29.2%. The transmission coefficient of S

_{21}is lower than −28 dB over 8–12 GHz, indicating an elegant isolation. Slight differences of results between simulations and measurements are due to small gaps between dielectric substrates and some errors of relative positions of the dielectric substrates when they are assembled. Figure 10b shows the simulated axial ratios of l

_{1}= −1 (Theta = 21°, Phi = 72°), l

_{2}= 1 (Theta = 20°, Phi = 150°), l

_{3}= −1 (Theta = 20°, Phi = 252°), and l

_{4}= 1 (Theta = 22°, Phi = 330°). The simulated axial ratio (AR) bandwidth of quad modes is about 17%.

_{(A,}

_{L}

_{)}= 72°, θ

_{(A,}

_{L}

_{)}= 30°, and ϕ

_{(B,}

_{L}

_{)}= 150°, θ

_{(B,}

_{L}

_{)}= 30°. In contrast, the RCP wave is formed in directions of ϕ

_{(B,}

_{R}

_{)}= 252°, θ

_{(B,}

_{R}

_{)}= 30°, and ϕ

_{(A,}

_{R}

_{)}= 330°, θ

_{(A,}

_{R}

_{)}= 30°. In each beam, there is an intensity null at the corresponding main lobe. The sidelobe levels are observed as −12 dB, −14 dB, −15 dB, and −18 dB for the co-polarization component in an azimuthal angle of ϕ

_{(A,}

_{L}

_{)}= 72°, ϕ

_{(B,}

_{L}

_{)}= 150°, ϕ

_{(B,}

_{R}

_{)}= 252°, and ϕ

_{(A,}

_{R}

_{)}= 330°, respectively.

_{(A,}

_{L}

_{)}= −1, l

_{(B,}

_{L}

_{)}= −1, while it varies clockwise from 180° to −180° in Figure 13c,d corresponding to l

_{(A,}

_{R}

_{)}= l

_{(B,}

_{R}

_{)}= 1. Moreover, the spatial phase fronts are slightly distorted with an irregular doughnut shape, which is caused by the deflection and the side-lobe interference of other beams. Nevertheless, all the results have verified our proposed method in generating decoupled quad vortex beams with topological charges of l

_{(A,}

_{L}

_{)}= l

_{(B,}

_{L}

_{)}= −1, l

_{(A,}

_{R}

_{)}= l

_{(B,}

_{R}

_{)}= 1.

## 6. Conclusions

_{(A,}

_{L}

_{)}= l

_{(B,}

_{L}

_{)}= −1, l

_{(A,}

_{R}

_{)}= l

_{(B,}

_{R}

_{)}= 1 at 10 GHz is fabricated and measured. All the results agree well and show that the proposed strategy was capable of realizing multi-mode decoupled vortex beams in broadband. The proposed device features compactness, wide bandwidth, and large channel capacity, promising potential engineering applications in radars and point-to-point wireless communications.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Geometry of the proposed dual-LP element. (

**a**) Schematic diagram. (

**b**) Layout and parametric illustration. The optimized dimensions are: a = 8.15 mm, b = 6 mm, w

_{1}= 0.56 mm, w

_{2}= 0.67 mm, l

_{1}= 5.5 mm, l

_{2}= 3.9 mm, l

_{3}= 1.8 mm, l

_{4}= 5.7 mm, h

_{1}= 0.8 mm, h

_{2}= 0.8 mm, and h

_{3}= 0.5 mm.

**Figure 4.**Normalized co-polarization and the cross-polarization patterns for LP element when the (

**a**) sub-element A and (

**b**) sub-element B working in ϕ = 72°, 150°, 252°, 330°.

**Figure 5.**Rotation angles and excitation phases of sub-elements A and B for (

**a**) quad CP beams without OAM and (

**b**,

**c**) quad-vortex-beam with (

**b**) l

_{(A}

_{,}

_{L)}= l

_{(B}

_{,}

_{L)}= −1, l

_{(A}

_{,}

_{R)}= l

_{(B}

_{,}

_{R)}= 1 in Case I and (

**c**) l

_{(A}

_{,}

_{L)}= −1, l

_{(B}

_{,}

_{L)}= 0, l

_{(A}

_{,}

_{R)}= 2, l

_{(B}

_{,}

_{R)}= 1. in case II (

**d**,

**e**) Simulated far-field patterns (

**d**) in Case I and (

**e**) in case II.

**Figure 6.**Layouts of the (

**a**) upper and (

**b**) bottom feeding network for the quad-vortex-beam array in Case I.

**Figure 8.**Simulated 3-D radiation pattern at 10 GHz for the quad-vortex-beam in Case I with different OAM modes (l

_{(A}

_{,}

_{L)}= l

_{(B}

_{,}

_{L)}= −1, l

_{(A}

_{,}

_{R)}= l

_{(B}

_{,}

_{R)}= 1) at predicted deflection angles of (θ, ϕ) = (30°, 72°), (30°, 150°), (30°, 252°), and (30°, 330°).

**Figure 9.**Photographs of the final fabricated quad-vortex-beam planar microstrip array. (

**a**) Assembled model, (

**b**) radiating elements, (

**c**) upper feeding network, (

**d**) ground plane with H slot, and (

**e**) bottom feeding network.

**Figure 10.**(

**a**) Simulated and measured S-parameters and (

**b**) simulated AR of the microstrip quad-vortex-beam planar array.

**Figure 12.**Numerical far-field patterns of the LHCP and RHCP beams in the ϕ

_{(A,}

_{L}

_{)}= 72°; ϕ

_{(B,}

_{L}

_{)}= 150°; ϕ

_{(B,}

_{R}

_{)}= 252°; and ϕ

_{(A,}

_{R}

_{)}= 330° planes at 10 GHz.

**Figure 13.**The measured magnitude and phase distributions at 10 GHz for four beams of different OAM modes at different directions. (

**a**) l

_{(A,}

_{L}

_{)}= 1 at θ

_{(A,}

_{L}

_{)}= 30°, ϕ

_{(A,}

_{L}

_{)}= 72°. (

**b**) l

_{(B,}

_{L}

_{)}= 1 at θ

_{(B,}

_{L}

_{)}= 30°, ϕ

_{(B,}

_{L}

_{)}= 150°. (

**c**) l

_{(B,}

_{R}

_{)}= −1 at θ

_{(B,}

_{R}

_{)}= 30°, ϕ

_{(B,}

_{R}

_{)}= 252°. (

**d**) l

_{(A,}

_{R}

_{)}= −1 at θ

_{(A,}

_{R}

_{)}= 30°, ϕ

_{(A,}

_{R}

_{)}= 330°.

l_{(A,}_{L}_{)} | l_{(A,}_{R}_{)} | l_{(B,}_{L}_{)} | l_{(B,}_{R}_{)} | |
---|---|---|---|---|

l_{1}ϕ in φ | −l_{1} | l_{1} | −l_{1} | l_{1} |

l_{2}ϕ in β_{A} | l_{2} | l_{2} | / | / |

l_{3}ϕ in β_{B} | / | / | l_{3} | l_{3} |

Final topological charge | −l_{1} + l_{2} | l_{1} + l_{2} | l_{1} + l_{3} | l_{1} + l_{3} |

Z_{1} | Z_{2} | Z_{3} | Z_{4} | Z_{5} | |
---|---|---|---|---|---|

Z (Ω) | 50 | 67 | 92 | 65 | 92 |

Width (mm) | 2.25 | 1.25 | 0.56 | 1.3 | 0.56 |

Length (mm) | 5 | 4.7 | 3.97~80.27 | 4.75 | / |

Z_{6} | Z_{7} | Z_{8} | Z_{9} | Z_{10} | |
---|---|---|---|---|---|

Z (Ω) | 50 | 61 | 75 | 53 | 75 |

Width (mm) | 1.43 | 0.86 | 0.67 | 1.23 | 0.67 |

Length (mm) | 10 | 5.08 | 3.6~74.5 | 5.06 | / |

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## Share and Cite

**MDPI and ACS Style**

Xu, S.; Xu, H.-X.; Wang, Y.; Xu, J.; Wang, C.; Pang, Z.; Luo, H.
Circularly Polarized Antenna Array with Decoupled Quad Vortex Beams. *Nanomaterials* **2022**, *12*, 3083.
https://doi.org/10.3390/nano12173083

**AMA Style**

Xu S, Xu H-X, Wang Y, Xu J, Wang C, Pang Z, Luo H.
Circularly Polarized Antenna Array with Decoupled Quad Vortex Beams. *Nanomaterials*. 2022; 12(17):3083.
https://doi.org/10.3390/nano12173083

**Chicago/Turabian Style**

Xu, Shuo, He-Xiu Xu, Yanzhao Wang, Jian Xu, Chaohui Wang, Zhichao Pang, and Huiling Luo.
2022. "Circularly Polarized Antenna Array with Decoupled Quad Vortex Beams" *Nanomaterials* 12, no. 17: 3083.
https://doi.org/10.3390/nano12173083