# Realization of Multimode OAM Beams with Almost the Same Divergence Angles

Authors: Yunfan Zhang; Xianling Liang; Weihao Qi; Hongliang Wu; Qian Chen; Junping Geng; Wei Wang
Journal: International Journal of Antennas and Propagation
 (2021)

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## Abstract

This paper proposes a new method of realizing multimode OAM beams with almost the same divergence angles. The theoretical relationship between the divergence angle of the OAM beam and its radiation source is presented, and the radiation source distributions for various mode OAM beams with the same divergence angle range are discussed. In order to verify this method, an eight-mode OAM antenna constructed by a bifocal parabolic reflector and dual OAM feeds is designed and simulated. The simulation results show that the divergence angle ranges of 3 dB beamwidth for OAM modesl=±1,±2,±3,±4 are 2.2°,4.6°,2.2°,5.0°,2.4°,4.4°,and2.6°,4.8°, the divergence angles corresponding to the maximum beam directions are 3.5°,3.6°,3.5°,and3.8°, respectively, and the maximum difference is within 0.3°.


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## Body

## 1. Introduction

Since Allen et al. first discovered OAM (orbital angular momentum) in 1992 [1] and Thide et al. verified the multiplexing transmission capability of OAM electromagnetic waves in the radio frequency band in 2007 [2], the research on OAM antennas has attracted widespread attention [3–5]. From the published literature, many methods were proposed to generate OAM waves at radio frequency, such as crafted antennas [6], loop antennas [7], OAM phase plates [8], and circular antenna arrays [9, 10]. It is worth noting that the essential feature of OAM antennas is the beam divergence, which results in the phase space distribution expansion with the increase of distance and makes it difficult to receive and demodulation. Literature studies [11, 12] introduced lens and reflectors to decrease the beam divergence angles of OAM antennas, but the beam divergence angles of various OAM modes are different. To achieve the same divergence angle for different OAM modes, Zhang et al. proposed an OAM antenna based on a ring slit resonator, which can generate omnidirectional OAM beams [13], this method effectively realizes the consistency of beam divergence angles for various OAM modes, but the beam divergence angle is 90° due to its omnidirectional radiation. Qin et al. proposed a four-mode UCA (uniform circular array) antenna design [14], through optimization of the element positions, and the beam with OAM modes of l = −1, −2, and −3 and the divergence angle of 18° was generated simultaneously.In short, there is little literature discussing in principle how to control the divergence angles of multimode OAM beams. Different from the work mentioned [13, 14], in this paper, the theoretical relationship between the divergence angle of the OAM beam and its radiation source is presented, and the radiation source distributions for various mode OAM beams with the same divergence angle range are discussed. Based on this, a bifocal parabolic reflector antenna with dual OAM feeds is proposed as an example to achieve the corresponding equivalent source distributions and verify the design of eight-mode OAM beams with almost the same divergence angles.


## 2. Theory and Analysis

Figure1 shows an OAM radiation source distribution, where the current loop is with radius R and current density JR. It is assumed that the amplitude of the current density JR in the loop is uniform, and the phase is periodically distributed with ejlφ, where l is the OAM mode. The vector magnetic potential of its radiation is expressed as follows:(1)A⇀r≈a⇀φ−jl+1μ0RJRejkrejlφ4πr⋅J|l|+1kRsinθ−J|l|−1kRsinθ,l=±1,±2….Figure 1
OAM radiation source.According to the far-field approximation conditions [15], the corresponding electric field can be approximately expressed as(2)E⇀≈a⇀φ−jlωμ0RJRejkrejlφ4πr⋅J|l|+1kRsinθ−J|l|−1kRsinθ,l=±1,±2….To obtain the maximumE⇀, we can get(3)J|l|+1kRsinθ−J|l|−1kRsinθ′=0,l=±1,±2….According to the property of Bessel function, the above formula can be simplified to(4)Jl″kRsinθ=0,l=±1,±2….The value ofkRsinθ is the eigenvalue of the second-order Bessel function, k is the wave number, and θ is the beam divergence angle. Table 1 shows eigenvalues of the second-order Bessel function for l = 1, 2, 3, and 4. Firstly, from the expression of kRsinθ, it is noted that the larger R, the smaller θ. Secondly, the values of kRsinθ are different for various l; the higher order l, the larger θ, as shown in Figure 2.Table 1
Eigenvalues of second-order Bessel’s equation.
l1234kRsinθ1.552.633.684.70Figure 2
Radius of source vs. divergence angle.Assuming a target OAM beam, its angle range of 3 dB beamwidth isθ∈3°,4° and the corresponding divergence angle range is also θ∈3°,4°. Then, the radiation source distributions for various OAM modes can be obtained, R1∈(3.5,4.7) λ0, R2∈(6.0,8.0) λ0, R3∈(8.4,11.2) λ0, and R4∈(10.7,14.3) λ0, as shown in Figure 3, where λ0 is the operation wavelength in free space. This means that the beams of these modes will have the same divergence angle range as long as each mode is distributed according to the above radiation source.Figure 3
Source distributions of various OAM modes.In order to achieve the abovementioned OAM source distributions of modesl=1,2,3,4, we propose a bifocal parabolic reflector with dual OAM feeds, as shown in Figure 4. The reflector is composed of two paraboloids with different focal lengths (f1,f2 ) and aperture sizes (D1,D2 ), which are denoted by part A and part B. The geometric expressions of the two paraboloids are as follows:(5)x2+y2=4f1z+Δz,x2+y2∈R1min,R2max,x2+y2=4f2z,x2+y2∈R3min,R4max,where f1 is the focal length of part A and OAM feed 1 is located at this focal position. f2 is the focal length of part B and OAM feed 2 is located at that focal position.Figure 4
Schematic diagram of bifocal parabolic reflector antenna.The equivalent radiation source distributions of the proposed bifocal parabolic reflector antenna can be explained as follows. OAM feed 1 is used to generate OAM beams ofl=1 and 2, and OAM feed 2 is used to generate OAM beams of l=3 and 4. According to the nature of the parabolic reflector antenna, the radiation formed by feed 1 beams incident on the reflecting surface part A can be regarded as the equivalent source radiation at the position of directrix 1. The radiation formed by feed 2 beams incident on the reflecting surface part B can be regarded as the equivalent source radiation at the position of directrix 2. Therefore, the source distribution required by the above modes l=1, 2, 3, 4 can be formed by optimizing sizes of the two paraboloids and beamwidth of dual OAM feeds. According to Figure 3, the aperture sizes of two paraboloids need to meet the conditions: R3min≥D1/2≥R2max and D2/2≥R4max, where D1 is the diameter of part A and D2 is the diameter of part B.Then, the OAM feed beams can be generated from the geometric relationship between the parabolic reflector and the dual OAM feeds, as follows:(6)θl′=2arccot4f1Rl,l=1,2,θl′=2arccot4f2Rl,l=3,4.We chooseD1 = 16.8 λ0, D2 = 28.6 λ0, f1 = 12.6 λ0, and f2 = 10.3 λ0 based on the above analysis. The feed beam range corresponding to source distribution of each mode can be calculated according to formula (6): θ1′∈ (16.0°, 21.2°), θ2′∈ (26.8°, 35.2°), θ3′∈ (44.4°, 57.1°), and θ4′∈ (55.0°, 69.6°), as shown in Table 2.Table 2
Feed beams for various OAM modes.
lTarget beam rangeθSource distributionRlFeed beam rangeθl′1(3°,4°)(3.5, 4.7)λ0(16.0°,21.2°)2(6.0, 8.0)λ0(26.8°,35.2°)3(8.4, 11.2)λ0(44.4°,57.1°)4(10.7, 14.3)λ0(55.0°,69.6°)


## 3. Simulation and Discussion

If a circular microstrip antenna works inTMl+11 mode and is excited by two feeding points with a certain phase, it can produce OAM beam with the mode of l [16, 17]. In order to achieve the mentioned feed beams, feed 2 adopts a dual-ring microstrip antenna [18, 19], as shown in Figure 5.Figure 5
Structure of feed 2.The inner side of each ring is grounded with a set of metallized via holes, which is equivalent to a shorting wall, and the outer side is equivalent to a magnetic current loop as the radiation source [20]. Each ring is simultaneously excited by four feeding ports separated by a certain angle, where p5 to p8 are used for exciting the inner ring patch and p9 to p12 are used for exciting the outer ring patch. The excitation amplitude and phase are shown in Table 3, which can be achieved by two feeding networks with four Wilkinson dividers and two 90° hybrids. The inner ring patch works in the TM41 mode to generate OAM beams of l=±3, and the outer ring patch works in the TM51 mode to generate OAM beams of l=±4. Particularly, an additional grounded ring is used to reduce the back radiation. Based on the full-wave HFSS simulation, the optimized parameters are as follows: rb3 = 15.3 mm, ra3 = 19.5 mm, rb4 = 21.1 mm, ra4 = 24.9 mm, rg3 = 26.5 mm, R34 = 30 mm, ϵr = 3.66, h1 = 0.762 mm, and the diameter of the via hole is 0.2 mm and the spacing is 1.1 mm. The simulated reflection coefficients of all ports are shown in Figure 6.Table 3
Excitation amplitude and phase of various modes.
lFeeding portsAmplitudePhase (°)+1p1,p21,190 0−1p1,p21,10 90+2p3,p41,190 0−2p3,p41,10 90+3p5,p6,p7,p81,1,1,190 90 0 0−3p5,p6,p7,p81,1,1,10 0 90 90+4p9,p10,p11,p121,1,1,190 0 0 90−4p9,p10,p11,p121,1,1,10 90 90 0Figure 6
Simulated reflection coefficients of feed 2. (a)l=±3. (b) l=±4.
(a)(b)The 10 dB impedance bandwidth of model=±3 is from 9.91 GHz to 10.14 GHz, and that of mode l=±4 is from 9.67 GHz to 10.19 GHz. The corresponding radiation patterns of feed 2 are shown in Figure 7, which can cover the feed 1 beam requirements of θ±3′ and θ±4′, but wider than required. The main reason is that the OAM beam divergence is related to its beamwidth, and it is not easy to change the beamwidth under the fixed beam direction.Figure 7
Radiation patterns of feed 2.Considering that the beam divergence angle range of feed 1 is much smaller than that of feed 2, it is hard to satisfy the feed beam requirements by using the microstrip antenna alone. Therefore, the combination of the microstrip antenna and the Fabry–Perot cavity is used [21], as shown in Figure 8.Figure 8
Structure of feed 1.Each patch is simultaneously excited by two feeding ports separated by a certain angle, wherep1 and p2 are used for exciting the inner circular patch and p3 and p4 are used for exciting the outer ring patch. Similar to feed 2, the inner circular patch works in the TM21 mode to generate OAM beams of l=±1 and the outer ring patch works in the TM31 mode to generate OAM beams of l=±2. The excitation amplitude and phase are also shown in Table 3, which can be achieved by two 90° hybrids. The principle is that the microstrip antenna radiates OAM electromagnetic waves, some of them are directly transmitted from the partially reflective surface (PRS), while others are reflected. When the reflected OAM waves reach the ground and the metal wall, they will be reflected to the PRS with the same phase. After multiple reflections and transmissions, the radius of the radiation source increases to a certain extent, thereby narrowing the beam divergence angle range. The optimized parameters are as follows: rb1 = 3.8 mm, ra1 = 8.3 mm, rb2 = 15.2 mm, ra2 = 18.9 mm, rg1 = 9 mm, rg2 = 19.4 mm, ϵr = 3.66, h2 = 0.762 mm, h0 = 14.7 mm, the diameter of the via hole is 0.2 mm, and the spacing is 0.95 mm. The simulated reflection coefficients of all ports are shown in Figure 9.Figure 9
Simulated reflection coefficients of feed 1.The 10 dB impedance bandwidth of model=±1 is from 9.96 GHz to 10.02 GHz, and that of mode l=±2 is from 9.97 GHz to 10.02 GHz. It is noted that the impedance bandwidth of feed 1 is much smaller than that of feed 2. The main reason is the Fabry–Perot cavity limits the impedance bandwidth due to the phase synchronization of the OAM wave multiple reflections. The radiation patterns of feed 1 are shown in Figure 10, which can cover feed 1 beam requirements of θ±1′ and θ±2′.Figure 10
Radiation patterns of feed 1.Based on the above dual OAM feeds, the bifocal parabolic reflector antenna is designed. Its optimized parameters areD1 = 480 mm, D2 = 816 mm, f1 = 360 mm, and f2 = 294 mm. The configuration of the bifocal parabolic reflector antenna is shown in Figure 11. The dual OAM feeds and the bifocal reflector are fixed by a nylon support frame. Figure 12 shows the simulated radiation patterns of various OAM modes. The divergence angle ranges of 3 dB beamwidth for OAM modes l=±1,±2,±3,±4 are 2.2°,4.6°,2.2°,5.0°,2.4°,4.4°,and2.6°,4.8°, respectively, which are slightly wider than the target’s 3 dB OAM beamwidth. The main reason is that the beamwidth of feed 1 and feed 2 is wider than required. The divergence angles corresponding to the maximum beam directions for OAM modes l=±1,±2,±3,±4 are 3.5°,3.6°,3.5°,3.8°, respectively, and the maximum difference is within 0.3°, which means that eight-mode OAM beams almost have the same divergence angles. Moreover, the gains and side lobe levels for OAM modes l=±1,±2,±3,±4 are 28.2 dBi, 27.6 dBi, 28.0 dBi, and 26.0 dBi, and −14.1 dB, −11.1 dB, −12.0 dB, and −7.6 dB, respectively. Compared with literature [13, 14], the proposed antenna has more OAM modes, smaller divergence angles, and lower side lobe levels.Figure 11
Configuration of bifocal reflector antenna.Figure 12
Radiation patterns of bifocal parabolic reflector antenna.


## 4. Conclusion

A new method of realizing multimode OAM beams with almost the same divergence angles has been presented in this paper. By establishing the relationship between the divergence angle of the OAM beam and its radiation source, the radiation source distributions of various mode OAM beams for the same divergence angle range are discussed. To verify this method, an eight-mode OAM antenna constructed by a bifocal parabolic reflector and dual OAM feeds has also been proposed, designed, and simulated. The simulation results show that the divergence angle ranges of 3 dB beamwidth for OAM modesl=±1,±2,±3,±4 are 2.2°,4.6°,2.2°,5.0°,2.4°,4.4°,and2.6°,4.8°, the divergence angles corresponding to the maximum beam directions are 3.5°,3.6°,3.5°,and3.8°, respectively, and the maximum difference is within 0.3°. This method is also suitable for other forms of multimode OAM antennas to achieve the same divergence angles.



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*Source: 6683622-2021-02-18.xml*