Introduction

The vortex beam carrying orbital angular momentum (OAM) has received much attention recently for its potential in the field of optical communications1, high-capacity communications2, the radar field of detection and imaging3, etc. The OAM vortex waves feature a helical phase front of \({e}^{-jl\varphi }\), where l is the OAM mode order and \(\varphi \) is the azimuthal angle. The OAM modes are orthogonal to each other, which enables multiple signals to be transmitted simultaneously at the same frequency. Therefore, OAM can be a useful degree of freedom for increasing the channel capacity and the spectral efficiency4. Although the OAM-based radio systems are considered equivalent to conventional multiple-input-multiple-output (MIMO) systems5, an experiment demonstrates that OAM-mode division multiplexing systems have low receiver complexity and high capacity compared with conventional line-of-sight MIMO systems6. Furthermore, an OAM-based synthetic aperture radar imaging method is experimentally verified, whose azimuth resolution is higher than that of a conventional one7. Thus, the potential of OAM in various fields remains to be discussed.

Different methods for generating OAM beams have been reported untill now, yet the generation of broadband and high-purity OAM beams is still considered as a big challenge. Spiral phase plates (SPPs)8,9 are widely used in the field of optics due to their simple structures. The SPPs can only generate a single mode of OAM at a time. Also, in the low frequency radio domain, SPPs are of bulky structure which increases the cost. The metallic traveling-wave ring resonant cavity antennas10,11 are proposed to generate linearly polarized OAM waves carrying ±3 modes with the purity of 87% and the attenuation coefficient of 0.03612. The sizes of these models are large and the bandwidth is quite narrow. Several small-size microstrip antennas have been reported as well, such as the circular patch antenna13, the concentric patches antenna14, and the half-mode substrate integrated waveguide (HMSIW) antenna15. These microstrip antennas are limited to a narrow impedance bandwidth of about 2% and a low OAM purity of about 72%16. In addition, the array antennas17,18,19,20,21 are proposed to generate OAM beams by controlling a phase shift of the successive element. But the number of the array elements limits the order of the OAM mode and more elements are needed to ensure that the main lobe is in the pure mode region17. The OAM purity of 80% with the mode +1 can be achieved at the maximum gain angle by the uniform circular array with 16 elements. Consequently, formerly reported 2 × 2 array antennas18,19,20,21 can only generate ±1 OAM modes with the main lobe not being in the pure mode region. To increase the channel capacity, a dual linearly polarized (LP) microstrip antenna array is proposed with an impedance bandwidth of 3.6%18. A high-profile dual LP bow-tie dipole array antenna achieves a wide bandwidth of 25%19. Some dual circularly polarized (CP) patch array antennas20,21 are proposed to mitigate the multipath interference, but their feeding networks are quite complex. Several metasurfaces without complex feeding networks are proposed to generate dual-mode OAM beams22,23. However, most of these metasurfaces are excited by additional feed horn which makes them difficult to integrate and increase the profile of the antenna. Recently, an innovative OAM generation method based on transformation optic using all-dielectric lens is presented24. The lens can generate an OAM mode of +1 from 8 GHz to 16 GHz. In this method, the physical realization of the device is a challenge and radiation patterns of the proposed lens have a slight divergence effect. A phase-modulation converging lens25 based on the concept of the optical converging axicon excited by a patch array source can achieve the OAM beam converging from 9.3 GHz to 10 GHz, which is advantageous for improving the propagation performance of the OAM beam.

In this paper, we propose a new strategy to generate broadband high-purity dual-mode OAM beams using the four-feed circular patch antenna. The theoretical formula of the radiation field shows that a four-feed patch antenna at TMn1 mode can produce ±n OAM modes. A prototype of the four-feed circular TM21-mode patch antenna is fabricated and measured to verify the effectiveness of the theoretical analysis. The simulated and measured results are well matched with the theoretical ones. The measured 3-dB axial ratio (AR) bandwidth is about 28.33% from 5.00 to 6.70 GHz for both ports. The OAM beams with ±2 modes are generated in a wide band from 5.50 to 6.50 GHz with high mode purity.

Results

Theoretical analysis of a four-feed circular patch antenna with CP OAM modes

The geometries of the circular patch antennas excited by two feeds and four feeds are shown in Fig. 1(a,b). Circular antennas are fed with equal amplitude, a relative 90° phase shift, and the proper angular spacing to obtain CP radiation. In the case of TM21-mode patch antenna, the feed angular spacing α is set to 135°. We employ two additional feeds 3, 4 located diametrically across from the feeds 1, 2. The four-feed antenna is fed with equal amplitude and a phase shift of \([{0}^{\circ },{90}^{\circ },{0}^{\circ },{90}^{\circ }]\) for the TM21 mode. As shown in Fig. 1(d), a more symmetric current distribution is obtained by using four-feed structure. This is because most of the cluttered current due to the mutual coupling between the feeds is suppressed by the two opposing feeds. Moreover, the effect of the remaining cluttered current is significantly eliminated by the opposite current. Thus, the four-feed structure achieves a lower cross-polarization level compared to the two-feed structure, as shown in Fig. 1(e). Additionally, Fig. 1(f,g) compares the electric field distributions for the two-feed antenna and the four-feed antenna. The four-feed structure has a more uniform and symmetrical distribution of the TM21 mode because of the low cross polarization. This illustrates that the radiation of the working mode (TM21 mode) is enhanced and the fields of the unwanted adjacent modes (TM11 and TM31 modes) are suppressed. As a result, the CP bandwidth is widened and the mode purity is improved.

Figure 1
figure 1

Geometries, current and electric field distributions of the circular patch antennas excited by different feeds. (a,c,f) Two feeds. (b,d,g) Four feeds. (e) Simulated normalized radiation patterns. The black arrow represents the direction of the current of the TM21 mode. The pink arrow signifies the direction of the clutter current.

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Next, a theoretical analysis of a four-feed circular microstrip patch antenna generating CP OAM beams is presented as follows. Based on the cavity-model approximation, different TMnm modes can be excited by different circular patch sizes. The symbols n and m are the number of the angle mode and the radial mode, respectively. The electric field components of the circular patch excited by a single feed in the spherical coordinate are given as follows26

$$\begin{array}{rcl}{E}_{\theta n}(\varphi ,\theta ) & = & {j}^{n}\frac{{e}^{-j{k}_{0}r}}{2r}{k}_{0}{a}_{e}h{E}_{0}{J}_{n}({a}_{e}{k}_{0}\sqrt{{\varepsilon }_{r}})\,\cos \,n\varphi \,[{J}_{n+1}(\gamma )-{J}_{n-1}(\gamma )]\\ {E}_{\varphi n}(\varphi ,\theta ) & = & {j}^{n}\frac{{e}^{-j{k}_{0}r}}{2r}{k}_{0}{a}_{e}h{E}_{0}{J}_{n}({a}_{e}{k}_{0}\sqrt{{\varepsilon }_{r}})\,\sin \,n\varphi \,[{J}_{n+1}(\gamma )+{J}_{n-1}(\gamma )]\,\cos \,\theta \end{array}$$
(1)

where \(\gamma ={k}_{0}{a}_{e}\,\sin \,\theta \), k0 is the phase constant in a vacuum, ae is the equivalent circular patch radius, h is the substrate thickness, E0 is the electric field amplitude at the edge of the patch, Jn is the Bessel function of the first kind for order integer n, and \({\varepsilon }_{r}\) is the relative permittivity. Moreover, the equivalent circular patch radius can be derived by \({a}_{e}={\lambda }_{0}{X}_{nm}/(2\pi \sqrt{{\varepsilon }_{r}})\)27, where λ0 denotes the free-space wavelength at the center operating frequency (f0 = 6 GHz), and Xnm is the mth zero of the derivative of Jn, which is a constant and independent of frequency. Some values of Xnm are shown in Table 1. Because of the fringe fields, the physical patch radius ap is slightly smaller than the equivalent one.

Table 1 Some Values of Xn1.
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For the case of four feeds, the excitation amplitudes are same and mutual coupling is neglected. The feed angular spacing is given by \(\alpha =(2k+1)\pi /2n,k=0,1,2,\ldots ,n-1\)15. Additionally, the four-feed antenna has a phase shift of   \([{0}^{\circ },{90}^{\circ },{0}^{\circ },{90}^{\circ }]\) for the even-order modes and  \([{0}^{\circ },{90}^{\circ },{180}^{\circ },{270}^{\circ }]\) for the odd-order modes27. For simplicity, only the radiated electric field of the right-hand circular polarization (RHCP) is given. The radiated electric field of the left-hand circular polarization (LHCP) can also be obtained in a similar way. The total electric field components generated by the four orthogonal modes are as follows:

$$\begin{array}{rcl}{E}_{\theta n}^{t} & = & {E}_{\theta n}^{1}(\varphi ,\theta )+j{E}_{\theta n}^{2}(\varphi +\alpha ,\theta )+{(-1)}^{n}[{E}_{\theta n}^{3}(\varphi +180^\circ ,\theta )\\ & & +\,j{E}_{\theta n}^{4}(\varphi +\alpha +180^\circ ,\theta )]\\ {E}_{\varphi n}^{t} & = & {E}_{\varphi n}^{1}(\varphi ,\theta )+j{E}_{\varphi n}^{2}(\varphi +\alpha ,\theta )+{(-1)}^{n}[{E}_{\varphi n}^{3}(\varphi +180^\circ ,\theta )\\ & & +\,j{E}_{\varphi n}^{4}(\varphi +\alpha +180^\circ ,\theta )]\end{array}$$
(2)

where superscripts 1, 2, 3, and 4 correspond to individual electric fields excited by the four feeds. By combining Eqs (1) and (2), the total electric field components are written as

$$\begin{array}{rcl}{E}_{\theta n}^{t} & = & {j}^{n}\frac{C}{r}{e}^{-j{k}_{0}r}{e}^{-jn\varphi }[{J}_{n+1}(\gamma )-{J}_{n-1}(\gamma )]\\ {E}_{\varphi n}^{t} & = & {j}^{n+1}\frac{C}{r}{e}^{-j{k}_{0}r}{e}^{-jn\varphi }[{J}_{n+1}(\gamma )+{J}_{n-1}(\gamma )]\,\cos \,\theta \end{array}$$
(3)

where C is the amplitude constant. Since the vortex EM waves dynamically rotate around the z-axis, the electric field components should be obtained in the cylindrical coordinate system.

The \(\rho \)-, \(\varphi \)-, and z-components of the total electric field in the cylindrical coordinate system are

$$\begin{array}{rcl}{E}_{\rho n}^{t} & = & \cos \,\theta \,{E}_{\theta n}^{t}=-\,{j}^{n}\frac{C}{r}{e}^{-j{k}_{0}r}{e}^{-jn\varphi }[{J}_{n-1}(\gamma )-{J}_{n+1}(\gamma )]\,\cos \,\theta \\ {E}_{\varphi n}^{t} & = & {E}_{\varphi n}^{t}={j}^{n+1}\frac{C}{r}{e}^{-j{k}_{0}r}{e}^{-jn\varphi }[{J}_{n-1}(\gamma )+{J}_{n+1}(\gamma )]\,\cos \,\theta \\ {E}_{zn}^{t} & = & -\sin \,\theta \,{E}_{\theta n}^{t}={j}^{n}\frac{C}{r}{e}^{-j{k}_{0}r}{e}^{-jn\varphi }[{J}_{n-1}(\gamma )-{J}_{n+1}(\gamma )]\,\sin \,\theta .\end{array}$$
(4)

The radial electric field can be divided into two orthogonal LP components or CP components, which can be expressed as

$$\overrightarrow{{E}_{ln}^{t}}={\overrightarrow{e}}_{\rho }{E}_{\rho n}^{t}+{\overrightarrow{e}}_{\varphi }{E}_{\varphi n}^{t}={\overrightarrow{e}}_{L}{E}_{Ln}^{t}+{\overrightarrow{e}}_{R}{E}_{Rn}^{t}$$
(5)

where \({E}_{Ln}^{t}=({E}_{\rho n}^{t}-j{E}_{\varphi n}^{t})/\sqrt{2}\), \({E}_{Rn}^{t}=({E}_{\rho n}^{t}+j{E}_{\varphi n}^{t})/\sqrt{2}\).

By combining Eqs (4) and (5), the LHCP and RHCP components of the radial electric field are

$$\begin{array}{rcl}{E}_{Ln}^{t} & = & 2{j}^{n}\frac{C}{r}{e}^{-j{k}_{0}r}{e}^{-jn\varphi }\,\cos \,\theta \,{J}_{n+1}(\gamma )/\sqrt{2}\\ {E}_{Rn}^{t} & = & -\,2{j}^{n}\frac{C}{r}{e}^{-j{k}_{0}r}{e}^{-jn\varphi }\,\cos \,\theta {J}_{n-1}(\gamma )/\sqrt{2}.\end{array}$$
(6)

The calculated AR is defined in decibels as

$$AR=20\,{\rm{lg}}|\frac{|{E}_{Ln}^{t}|+|{E}_{Rn}^{t}|}{|{E}_{Ln}^{t}|-|{E}_{Rn}^{t}|}|=20\,{\rm{lg}}|\frac{|{J}_{n-1}(\gamma )|-|{J}_{n+1}(\gamma )|}{|{J}_{n-1}(\gamma )|+|{J}_{n+1}(\gamma )|}|$$
(7)

where \(\gamma ={k}_{0}{a}_{e}\,\sin \,\theta ={X}_{nm}\,\sin \,\theta /\sqrt{{\varepsilon }_{r}}\). The calculated AR is shown in Fig. 2. For different integer orders with \({\varepsilon }_{r}=3.66\), the AR value is less than 0.6 indicating that the CP radiation is obtained. For the case of the four-feed antenna working in the TM21 mode, the AR value decreases with the increase in permittivity of the substrate as shown in Fig. 2(b). Because \(|{E}_{Ln}^{t}| < |{E}_{Rn}^{t}|\), the RHCP component is the co-polarized and the LHCP component is the cross-polarized. Equation (6) shows that the RHCP component of the electric field has the helical phase wavefront of \({e}^{-jn\varphi }\), which indicates the generation of the RHCP vortex waves carrying an OAM mode of −n. Similarly, the phase wavefront of \({e}^{jn\varphi }\) can be obtained for LHCP, indicating the four-feed antenna can obtain an OAM mode with a positive number.

Figure 2
figure 2

The calculated AR (a) for integer orders \(n=1,2,3,\ldots ,7\) and (b) for \(n=2\) with different permittivities.

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To verify the formula of the electric field, the phase and amplitude distributions under ideal conditions operating in the 6 GHz are analysed using MATLAB software, as shown in Fig. 3(a). The square observation window lies at \(z=2{\lambda }_{0}\) with a side length of 12λ0. The phase change along a concentric circle is 2πl for l = 2, 3, and 4. An amplitude null is observed in the center of the beam, similar to that of the optical vortex beam carrying OAM. Based on the model shown in Fig. 1(b), the theory is further verified by using HFSS software. Figure 3(b,c) shows the simulated results for different OAM modes for different sizes of the metal ground at 6 GHz. The calculated and simulated results for the metal ground of the \(10{\lambda }_{0}\times 10{\lambda }_{0}\) square are in good agreement. However, Figure 3(c) reveals a difference in the center of the electric field for the OAM mode of +4 for the metal ground of the \(1.6{\lambda }_{0}\times 1.6{\lambda }_{0}\) square, which is mainly due to the effect of the finite metal ground. Both the size of the patch antenna and the divergence angle of the radiation start to rise with an increase in the order of the OAM mode. Therefore, the beam peak of the higher mode antenna is at a lower angle. The finite metal ground cannot maintain the horizontal electric field to propagate well for low-angle radiation, which deteriorates the electric field distribution of the TM41 mode. In conclusion, these features with the helical phase and the donut-shaped amplitude validate that the four-feed circular patch antenna can generate ±n OAM states for the CP TMn1 mode.

Figure 3
figure 3

Calculated and simulated phase and amplitude distributions of CP component of the electric field for l = +2, l = +3, and l = +4 OAM modes at 6 GHz. (a) Calculated results for the infinite metal ground. (b) Simulated results for the metal ground (\(10{\lambda }_{0}\times 10{\lambda }_{0}\) square). (b) Simulated results for the metal ground (\(1.6{\lambda }_{0}\times 1.6{\lambda }_{0}\) square).

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Fabricated structure of the antenna working in a CP TM21 mode

The configuration of the proposed antenna consists of a parasitic circular patch, a four-feed circular patch, a metal ground, and a feeding network, as shown in Fig. 4. The parasitic circular patch is located on the top layer of Substrate 1. The four-feed circular patch which works as a driven patch is located on the Substrate 2. There is an air gap between Substrate 1 and Substrate 2 in order to achieve the wideband property of CP radiation. The metal ground with four feeding slots is situated on the top layer of Substrate 3. A four-way feeding network with 90° phase differences is printed on the bottom layer of Substrate 3. An inexpensive FR4 substrate (\({h}_{1}=0.8\,{\rm{mm}}\), \({\varepsilon }_{r1}=4.4\), loss tangent = 0.02) is used as the top substrate. The middle and bottom substrates are Rogers 4350B (\({h}_{2}=1.524\,{\rm{mm}}\), \({h}_{3}=0.508\,{\rm{mm}}\) \({\varepsilon }_{r2}=3.66\), loss tangent = 0.004). The detailed dimensions of the proposed antenna are set as follows (unit: mm): \({a}_{1}=16\), \({a}_{2}=12.5\), \({l}_{1}=60\), \({l}_{2}=70\), \({h}_{a}=4\), \({l}_{c1}=14.48\), \({l}_{c2}=11.08\), \({w}_{1}=1.78\), \({w}_{2}=1.72\), \({l}_{t}=7.2\), and \({l}_{f}=9\).

Figure 4
figure 4

Configuration of the proposed four-feed patch antenna. (a) Exploded view. (b) Side view. (c) Top view without the parasitic patch. (d) Bottom view.

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Figure 4(d) shows the phase-shifting feeding network consisting of a 3-dB three-branch coupler, two T-junction power dividers, and two quarter-wavelength impedance transformers. The 3-dB three-branch coupler is considered as a cascade connection of two conventional couplers28, introducing another resonance point, thus providing a broadband 90° phase shift. We designed the feeding network with six ports: Ports 1, 2 are input ports, while other ports are output ports used to feed the patch. When Port 1 is excited and Port 2 is terminated with a 50-Ω load, Ports 3, 4, 5, and 6 have equal amplitude and a phase shift of \({0}^{\circ },-\,{90}^{\circ },{0}^{\circ },-\,{90}^{\circ }\) respectively. The proposed antenna can generate the OAM mode l = −2 for RHCP. Whereas when Port 2 is excited and Port 1 is matched, Ports 3, 4, 5, and 6 have equal amplitude and a phase shift of \({0}^{\circ },{90}^{\circ },{0}^{\circ },{90}^{\circ }\) respectively and the OAM mode l = +2 is generated for LHCP. The simulated S-parameters of the broadband feeding network are shown in Fig. 5. The simulated \(|{S}_{11}|\), \(|{S}_{22}|\), \(|{S}_{12}|\) and \(|{S}_{21}|\) are less than −20 dB from 5.5 to 6.5 GHz. For the OAM mode l = −2, the variations of output powers and phases at Ports 3, 4, 5, and 6 are less than 0.9 dB and 0.9° from 5.5 to 6.5 GHz, respectively. For the OAM l = +2 mode, the variations of output powers and phases are less than 0.8 dB and 0.5°, respectively. Therefore, the four-way phase-shifting feeding network can maintain a stable phase and amplitude response over a wide frequency band, which makes it possible for the proposed antenna with broadband performance.

Figure 5
figure 5

Simulated results of the four-way phase-shifting feeding network. (a) l = −2 mode. (b) l = +2 mode.

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Simulated and measuredt results

To verify the effectiveness of the theory and the design, a prototype of the four-feed antenna is fabricated and measured, as shown in Fig. 6. The simulated and measured S-parameters of the four-feed antenna are shown in Fig. 7(a). The measured 10-dB return loss bandwidth is 45% from 4.75 to 7.45 GHz for Port 1 and 44.2% from 4.75 to 7.40 GHz for Port 2. The measured transmission coefficient \(|{S}_{12}|\) is lower than −10 dB from 5.50 to 6.65 GHz. The simulated and measured gains and AR values of the four-feed antenna versus frequency are depicted in Fig. 7(b). The measured 1-dB gain bandwidth of the antenna is 16.67% from 5.50 to 6.65 GHz for both ports. The measured 3-dB AR bandwidth is 28.33% from 5.00 to 6.70 GHz for both ports. The simulated and measured radiation patterns in xoz plane at 6 GHz are shown in Fig. 8(a,b). For the OAM with l = −2 mode, the angle of the maximum gain is around \(\theta =38^\circ \). The measured peak gain is 5.81 dBic at 6 GHz. The half-power bandwidth (HPBW) ranges from −46° to −12° and from 20° to 60°. For the OAM with l = +2 mode, the angle of the maximum gain is also around \(\theta =38^\circ \) with the peak gain of 6.23 dBic at 6 GHz. The HPBW ranges from −44° to −20° and from 24° to 56°. The measured cross-polarization levels are substantially lower than −16 dB over the HPBW. Figure 8(c,d) shows that the measured AR is substantially below than 3 dB over the HPBW. Figure 9 represents the simulated and measured near-field distributions of the electric field at different frequencies, which highly coincide with each other in a wide band. The observation window lies in the plane at \(z=150\,{\rm{mm}}\) with dimensions of 300 mm × 300 mm with a step of 20 mm. The measured and simulated results are in good agreement. The purity of different OAM modes is quantitatively calculated based on the Fourier transform. Since the azimuth angle \(\varphi \) is a periodic function, its Fourier conjugate is the OAM spectrum. The Fourier relationship between the OAM spectrum Al and the sampling phase \(\psi (\varphi )\) is given by29

$${A}_{l}=\frac{1}{2\pi }\,{\int }_{0}^{2\pi }\,\psi (\varphi )d\varphi \,\exp (\,-\,il\varphi ).$$
(8)
Figure 6
figure 6

Fabricated antenna. (a) Top view. (b) Bottom view. (c) The near-field test scenario in the anechoic chamber.

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Figure 7
figure 7

Simulated and measured results of the four-feed antenna. (a) S-parameters. (b) Gains and AR values versus frequency in xoz plane.

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Figure 8
figure 8

Simulated and measured radiation patterns and AR in xoz plane at 6 GHz for (a,c) l = −2 mode, (b,d) l = +2 mode.

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Figure 9
figure 9

Simulated and measured near-field phase and amplitude distributions of the CP component of the electric field for l = −2 and l = +2 OAM modes at z = 150 mm at different frequencies. (a,c) Simulated results. (b,d) Measured results.

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Figure 10 shows the OAM spectrum at maximum gain angle at different frequencies. We can see that the OAM purity of l = ±2 is greater than 90% from 5.5 to 6.5 GHz.

Figure 10
figure 10

OAM spectrum generated by the four-feed antenna at (a) 5.5 GHz, (b) 6 GHz, (c) 6.5 GHz.

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Discussion

In this paper, a four-feed circular patch antenna is proposed for the first time to generate high-purity OAM beams. In principle, arbitrary OAM modes can be obtained by changing the patch radius together with the feed angular spacing. The fabricated prototype of the proposed TM21-mode antenna has an ability to generate the OAM mode of l = +2 with LHCP and the OAM mode of l = −2 with RHCP. Compared with previously reported OAM antennas shown in Table 2, the proposed antenna has a broader bandwidth of about 44.2% and a higher mode purity above 90%. Unlike the designs with single polarization10,14,15 and dual LP property18,19, the proposed antenna has the benefit of providing dual CP property. Moreover, in comparison with the patch antenna13 and array antennas18,19,20,21, the proposed antenna has a higher order of the OAM mode. All of these attractive features illustrate that the proposed four-feed circular patch antenna offers a tremendous potential in high-capacity wireless communications and radar applications of detection and imaging.

Table 2 Comparison Between the Proposed and Reported OAM Antennas.
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Methods

The theoretical calculation results are processed by MATLAB software. The proposed four-feed circular patch antenna is designed by ANSYS High Frequency Structure Simulator (HFSS), and manufactured on the Rogers 4350B and FR4 substrates using the PCB technology.

The S-parameters are measured by the Keysight N5247A vector network analyzer and the field distributions are obtained by near-field scanning with an open-ended waveguide probe. The radiation pattern and near-field scanning are measured in a microwave anechoic chamber.