## Abstract

The orbital angular momentum (OAM) has been widely used in the wireless short-range communication system, but for long-distance communication, the huge difficulty of beam receiving is a great challenge. In this paper, to overcome this challenge, a generation system of radio-frequency rotational orbital angular momentum (RF-ROAM) beams based on an optical-controlled circular antenna array (CAA) is proposed. The ROAM beam is an OAM beam rotating at a certain speed around the beam axis. According to the rotational Doppler effect, the rotation of the OAM beam will induce a frequency shift proportional to the OAM mode and the rotation speed. Thereby, by rotating an OAM beam at a fixed speed scheduled in advance in the transmitting end, the beam can be mode-distinguished by just detecting the frequency shift without receiving the whole wavefront vertical to the beam axis in the receiving end. This provides a partial reception scheme for the OAM-based wireless communication system. The generation system of RF-ROAM beams is proposed and constructed, and the proof-of-concept experiment is performed. In the system, the optical-controlled CAA is constructed to generate the general RF-OAM beam, the optical signal processor (OSP) is employed to control the phase shifts to further control the OAM mode, and the signal with time-varying phase is generated as the rotation factor to control the rotation speed. In the experiment, the RF-ROAM beams with different mode and mode combination are generated and successfully measured by detecting the frequency shift of the signal received in a fixed point.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Orbital angular momentum (OAM) is a fundamental physical quantity of the electromagnetic (EM) [1]. Different from the spin angular momentum (SAM) which shows the EM’s polarization, the OAM represents the spiral spatial distribution of the phase front. This spiral phase term can be expressed as $\textrm{exp} (iL\varphi )$, in which the *L* is the topological charge or the OAM mode (an unbounded integer) and $\varphi$ is the azimuthal angle. Beams carrying OAM are called as OAM beams, and the spiral phase term gives this kind of beam a helical phase structure with a phase singularity in the center and a “doughnut”-shaped intensity as it propagates [1].

The OAM beams have been widely studied in many fields in the last few years, such as optical manipulation [2], micro imaging [3–5], quantum information processing [6], fiber and wireless communication [7–10], and so on. In all these fields, the communication is one of the most promising applications of OAM beams. OAM beams in the same frequency with different mode are orthogonal with each other when propagating together, so the OAM mode division multiplexing (OAM-MDM) can significantly increase the channel capacity and the spectral efficiency for the communication link [7–9]. Besides, the OAM beams are also very useful in the radar system. The unique phase structure can provide more information in the cross-range direction for the radar detecting, and the rotational Doppler effect of the OAM beam can provide a new rotation detection method for the spinning target [11–13].

Whatever the communication or radar system, the OAM-mode measurement is the key to effective beam reception. There are already lots of methods for the OAM-mode measurement, for example, in 1994 M. Harris *et. al.* used interferometer technology to measure the OAM mode of laser [14]. In the radio-frequency (RF) domain, the phase gradient method is usually used, by directly measuring the helical phase front [15–16]. But for these methods, the whole wavefront reception is essential, which is not realistic for long-distance system owing to the OAM-beam diffusion. The rotational OAM (ROAM) beam provides a possibility for partial reception of OAM beams. As the name implies, the ROAM beam is an OAM beam rotating at a certain speed around the beam axis. According to the rotational Doppler effect, when the OAM beam is rotating, there will be a frequency shift which is proportional with the OAM mode and the rotation speed. Thereby, by rotating an OAM beam at a fixed rotation speed scheduled in advance in the transmitter, the OAM mode measurement can be realized by just detecting the spectrum of the signal received in a fixed point without receiving the whole wavefront vertical to the beam axis in the receiver.

There are already several schemes to generate ROAM beams. In the optical domain, the rotation of the wavefront is usually performed by combining different frequencies of OAM beams to form the dynamic spatiotemporal beams [17–18]. In the RF domain, without the optical equipment, the mechanical rotation or algorithm driving are the common methods. C. Zhang *et. al.* generated RF-ROAM beams by mechanical rotating the spiral phase plate (SPP) in 2016 [19]. In this scheme, the system structure is very simple, but due to the mechanical motion limiting the rotation speed of the OAM beam, the induced frequency shift is only a few *Hz*, which is difficult to detect. In 2017, C. Zhang *et. al.* proposed another scheme by employing the virtual rotational antenna, the RF-ROAM beams with high rotation speed were generated, and corresponding frequency shifts which are big enough were successfully detected [20]. But in this paper, two receiving antennas must be spread over the beam wavefront, increasing the detection complexity. Moreover, the SPP they used limits the OAM-mode flexibility. Therefore, a simple generation scheme of the RF-ROAM beams with the advantage of high rotation speed and high mode flexibility is necessary to be proposed. To generate the RF-ROAM beam, the generation scheme of RF-OAM beam needs to be researched firstly. Except for the SPP, many methods have already been proposed, such as the Q-plate, the geometric method, and the circular antenna array (CAA). In all these methods, the CAA is widely used recently for its OAM mode flexibility. The CAA method was firstly proposed in [21], and its ability to generate the RF-OAM beam has been verified by some numerical experiments. Then in [22], our group has proposed an optical-controlled CAA system to realize the OAM mode-division multiplexing based on the optical-true-time-delay (OTTD) technique. After that, the OTTD lines were replaced by a commercial optical signal processor (OSP) in [23], making the system compact. The usage of the microwave photonics technique provides conveniences in processing the same-frequency radio signal, and advantages of a high OAM-mode flexibility. Moreover, the microwave photonics system also provides the ultra-high frequency, wide bandwidth, and immunity to interference for the system. To make the OAM beam rotating on the premise of employing the CAA, except for the mechanical rotation and algorithm driving, according to the characteristic of the OAM beam’s phase wavefront, driving by a signal with time-varying phase is a more advisable method.

In this paper, we proposed a compact optical-controlled generation scheme of RF-ROAM beams with high rotation speed and high OAM-mode flexibility. In the scheme, the optical-controlled CAA is set up to generate the general RF-OAM beam, and the optical signal processor (OSP) is employed to be as the phase shifters to control the OAM mode, achieving the high OAM-mode flexibility. Furthermore, an RF-signal with time-varying phase is generated as the rotation factor to provide the high rotation speed. The theory and principle of ROAM-beam generation are deduced and the relationship between the signal with time-varying phase and the rotation speed of the ROAM beam is built. Simulations are performed, and results agree well with the theory. Finally, the system is constructed and the proof-of-concept experiment is performed. In the experiment, the +1-mode ROAM beam, the -1-mode ROAM beam, and the multiplexed ROAM beam with mode $L ={\pm} 1$ are generated with the same rotation speed, and the mode measurements are successfully achieved by detecting the corresponding frequency shift of the signal received in a fixed point.

## 2. Principles

To generate an RF-OAM beam, the CAA structure is one of the most popular schemes. As seen in Fig. 1, there are *N* antennas uniformly distributed on a circle with radius of *R*. To generate an OAM beam with the mode *L*, the phase of the signal routed to the ${A_n}$ should be ${\phi _n}\textrm{ = }2\pi nL/N$, and the intensities of all signals should be the same. By changing phases of the feeding signals, the OAM mode can be easily changed, which brings high OAM-mode flexibility. But it still has the mode limitation of $L < |{N / 2}|$ [21].

Supposing the center of the CAA is the origin point, constructing the coordinate system $(r,\theta ,\varphi )$, the coordinate of the ${A_n}$ is ${{\mathbf r}_{\mathbf n}} = ({r_n},{\theta _n},{\varphi _n}) = (R,\pi /2,2\pi n/N)$, the phase of ${A_n}$ is ${\phi _n}\textrm{ = }2\pi nL/N = L{\varphi _n}$, and the *E*-field factor of a random point ${\mathbf r} = (r,\theta ,\varphi )$ should be:

*f*and

*c*are the frequency and the transmission speed in the free space of the beam, respectively. Considering the far-field approximation of the dipole antenna,

*i.e.*, $|{{\mathbf r} - {{\mathbf r}_{\mathbf n}}} |\approx r$ for the intensity and $|{{\mathbf r} - {{\mathbf r}_{\mathbf n}}} |\approx r - \widehat {\mathbf r} \cdot {{\mathbf r}_{\mathbf n}}$ for the phase, wherein the $\widehat {\mathbf r}$ is the unit vector of the ${\mathbf r}$, the

*E*-field factor can be rewritten as:

*L*-order Bessel function of the first kind. Supposing the CAA is rotating with the speed of $\Omega rad/s$ to make the OAM beam rotating, the azimuth of the ${A_n}$ can be rewritten as ${\varphi _n}(t) = {\varphi _n} + \Omega t$, as referred in the [24]. Substituting ${\varphi _n}(t)$ into (2), the E-field factor of the ROAM beam can be expressed as:

*L*and rotation speed of $\Omega $. Therefore, on the premise that the rotation speed $\Omega $ of the ROAM beam is known, by detecting the spectrum of the beam to obtain the frequency shift $\Delta f$, the OAM mode can be calculated as $L = 2\pi \cdot \Delta f/\Omega $. In this method, the spectrum can be detected by a single antenna, so the whole wavefront reception is not necessary, which may provide a partial reception scheme for the OAM-beam long-distance transmission.

To generate the ROAM beam, the phase front of the OAM beam is analyzed. The phase term $\textrm{exp} (iL\varphi )$ in (1) shows that on the phase front of the OAM beam with the mode *L*, when the azimuthal angle $\varphi $ changes once from $- \pi $ to $\pi $, the phase changes *L* times from $- \pi $ to $\pi $. That is, the phase arrangement in the CAA determines the distribution of the phase in the wavefront of the OAM beam. Assuming that phases routed to the CAA change with time from $- \pi $ to $\pi $, periodically and linearly, the phase in the wavefront will change with it. In a period, the time-varying phase can be expressed as $\phi (t) = \phi + kt = L(\varphi + \Omega t)$, wherein the $k = L\Omega $ is the slope. It can be concluded that the linear change of phases routed to the CAA will lead to the change $\varphi (t) = \varphi + \Omega t$ of the azimuth, which signs the OAM beams’ rotation with the speed of $\Omega rad/s$. And as seen in (2), the frequency shift $\Delta f = L\Omega /2\pi$ will be produced.

Taking an OAM beam with $L ={-} 1$ generated by a *4*-antenna CAA as an example, as seen in Fig. 2(a), a signal $S(t)$ with a time-varying phase $\phi (t)$ which is shown in Fig. 2(b) is power-divided equally into four signals. After each signal is added the corresponding phase by the phase shifter and routed to the corresponding antenna in the CAA, in theory, the OAM beam can be generated and is rotating around the beam axis with a certain rotation speed. To verify the beam’s rotation, the 2-D phase patterns of the generated OAM beams in different time are simulated.

In the simulation, the frequency of the $S(t)$ is set at 20*GHz*, the period of the time-varying phase is set as 2*ns*, and the number of the antenna in the CAA is 32. The phase relative to the beam’s frequency of 20*GHz* is calculated, and the 2-D phase patterns of the simulated OAM beams received in 0*ns*, 0.5*ns*, 1*ns*, 1.5*ns* and 2*ns* are shown in Fig. 3(a)-(e), and each phase pattern is rotated 90° clockwise from the previous one (the rotation in this direction is defined as positive). The phase pattern in Fig. 3(e) is the same with that in Fig. 3(a), which means the OAM beam has rotating one full circle, so the rotation speed can be calculated as $\Omega \textrm{ = }\pi \times {10^9}/s$. The received signal in one point is also simulated, and the receiving point is set at about 60 times the wavelength (about 1 meter) away from the transmitter. The simulation results are shown in Fig. 4. Figure 4(a) shows the local phase of the signal, which agrees well with the Fig. 2(b). Figure 4(b) is the spectrum of the signal, and the -0.5GHz frequency shift is also consistent with the theoretical value calculated from the rotational Doppler effect, which is $\Delta f = L\Omega /2\pi ={-} 0.5GHz$[11].

It’s worth noting that to make RF-ROAM beams with different mode rotating at the same rotation speed, the driving signals $S(t)$ should be different. As mentioned above, the slope *k* of $S(t)$ and the rotation speed $\Omega $ of the OAM beam should satisfy $k = L\Omega $. To verified it, another simulation of a 20-*GHz* RF-ROAM beam with ${L_2} = 3$ is performed. The phase-time curve of the driving signal ${S_2}(t)$ is shown in Fig. 5. The phase relative to the 20-*GHz* frequency is calculated, and the 2-D phase patterns of the generated ROAM beams received in 0*ns*, 0.5*ns*, 1*ns*, 1.5*ns* and 2*ns* are shown in Fig. 6(a)-(e). The OAM beam with ${L_2} = 3$ rotating one circle in 10*ns*, and the rotation speed can be calculated as ${\Omega _2}\textrm{ = }\pi \times {10^9}/s$, which is the same with the rotation speed of the OAM beam with $L = 1$. In the receiving end, the frequency shift $\Delta {f_2} = {L_2}\Omega /2\pi = 1.5GHz$ can be received, so these two OAM beams can be mode-distinguished by just detecting the frequency shift of the signal received in a single point.

Overall, the OAM beams generated by the time-varying-phase signal through the CAA are rotating, *i.e.*, the ROAM beam, and the rotation speed is related to the time-varying period of the phase and the OAM mode. Consequently, in the transmitter, different signals are used to drive OAM beams with different mode to generate ROAM beams with the same rotation speed. In the receiver, different frequency shifts which are proportional to the mode value can be received, so the OAM modes can be distinguished by receiving the spectrum of the signal in a single point. With the OAM mode, the demultiplexing and demodulation of OAM beams can be easily performed, and the beams-carried data can be recovered.

Furthermore, compared with other existed RF-ROAM beam-generation schemes, this scheme has its unique advantages. In this scheme, the CAA provides the advantage of high OAM-mode flexibility, and ROAM beams with any mode if $L < |{N / 2}|$ can be generated by a same CAA. The time-varying period of the phase in the signal can control the rotation speed, and then control the frequency shift. As seen in the simulation, the period can be *ns*-level, and the induced frequency shift can be *GHz*-level, which is big enough to detect.

## 3. System structure

Figure 7 shows the schematic diagram of the generation system of RF-ROAM beams. In the system, *N* optical carriers with different wavelength radiated from the TLSs are combined by a wavelength division multiplexer (WDM). The combined optical signal can be written as:

*E*is the same intensity of all the optical carriers, and ${\omega _n}$ is the angular frequency of the ${n^{th}}$ carrier in the combined optical signal.

_{o}The signal generated from the AWG can be written as $S(t) = A\textrm{exp} [j\phi (t)]$, in which *A* is the amplitude, and $\phi (t)$ is the time-varying phase, and this phase can be coded in the AWG. Signals sent out from the I-port and Q-port are $I(t) = Re [S(t)] = A\cos [\phi (t)]$ and $Q(t) = {\mathop{\rm Im}\nolimits} [S(t)] = A\sin [\phi (t)]$. The VNA-generated RF signal is ${E_{RF}}(t) = {E_{RF}}\cos ({\omega _{RF}}t)$, wherein the ${E_{RF}}$ is the intensity and the ${\omega _{RF}}$ is the angular frequency of the signal. The generated time-varying-phase RF signal can be calculated as:

Then the RF signal is double-sideband (DSB) modulated into the combined optical signal by the MZM, and the modulated optical signal can be expressed as:

After power-amplified by the EDFA, the optical signal is then phase-shifted in the programmable OSP. The OSP can be seen as the combination of many groups of BPF and PS, as seen in the inset of Fig. 7. The optical signal will be divided into *N* single-frequency optical signals by the wavelength, and be phase shifted simultaneously and separately. According to the principle of the CAA-generating OAM beams mentioned above, the equal-difference phases are shifted to the corresponding optical signals. For clarity, taking the condition of generating the OAM beam with mode $L ={-} 1$ as an example, Fig. 8(a) shows intensities of optical signals routed to the OSP and corresponding shifted phases ${\phi _n}$ added by the OSP. Blue arrows respect intensities of optical carriers and green arrows respect intensities of their sidebands. Red short lines sign corresponding shifted phases: ${\phi _1}$, ${\phi _2}$, ${\phi _3}$ and ${\phi _4}$, and according to the beat frequency principle of the DSB modulation, these shifted phases should be added in the up-sideband (the left sideband in Fig. 8(a)) and subtracted in the down-sideband (the right sideband in Fig. 8(a)) at the same time, just as depicted by Fig. 8(a).

The phase-shifted signals are then routed to corresponding PDs, and the RF signals can be recovered. The ${n^{th}}$ RF signal can be expressed as:

The recovered RF-signals with time-varying phase ${\phi _{RFn}}(t)$ are then sent to the corresponding antennas in the CAA, and an RF-ROAM beam can be generated. A receiving antenna is arranged after the CAA to receive the beam, and send it to a OSC to get the waveform and to a OSA to record the spectrum.

With this system, the OAM-MDM is easy to be performed. Let’s taking the condition that multiplexing a +1-mode and a -1-mode OAM beams as the example. The VNA and another microwave source are employed together to generate two incoherent RF signals. After multiplexing with corresponding time-varying-phase signals, these two signals are then modulated onto two groups of optical carriers by two MZMs, and then wavelength division multiplexed together. After power-amplified by the EDFA, these two groups of optical signals are added with two groups of phase shifts corresponding to $L ={\pm} 1$ OAM mode by the OSP, as depictured by Fig. 8. In Fig. 8, these two light-modulated incoherent same-frequency RF signals are respected by green and yellow arrows, respectively. Optical signals with the same background color are routed out from the same output port of the OSP, and will be sent to the same antenna of the CAA after recovered by the PD. Then, the OAM-MDM of $L ={\pm} 1$ can be achieved.

## 4. Experiments and results

A proof-of-concept experiment is performed as depicted by Fig. 9. In the experiment, the 20*GHz* RF-signal is generated by the VNA (Agilent, 8722ES). The designed signal $S(t)$ is generated by the AWG (M9505A). The number of the antenna in the CAA and optical carrier are set as $N = 4$. The DSB modulation is performed by a MZM (Optilab, IM-1550-20), whose half-wave voltage is ${V_\pi } = 6.4v$ and the direct-current (DC) bias voltage is set in the orthogonal offset point.

The power is amplified in the EDFA (Conquer, KG-EDFA-P), and signals are sent to the OSP (WaveShaper 4000s) to be phase shifted. Then signals from the 4 out-ports of the OSP are further routed to corresponding PDs (Optilab, LR-30), where the RF signals are recovered and sent to the CAA. Finally, the RF-ROAM beam is generated and transmitted by the CAA.

To generate the -1-mode ROAM beam, the laser (Coherent Solutions, Mtp 1000&LaserBlade) is employed to radiate optical carriers with wavelengths of 1550 nm, 1550.5 nm, 1551 nm, and 1551.5 nm, respectively. The phase-time curve of the AWG-generating signal ${S_1}(t)$ is shown in Fig. 10(a), which is recorded by the OSC. The phases are shifted as 270°, 180°, 90°, and 0°, respectively. To generate the +1-mode ROAM beam, optical carriers with the wavelengths of 1552 nm, 1552.5 nm, 1553 nm and 1553.5 nm are radiated. The phase-time curve of the signal ${S_2}(t)$ generated by the AWG is shown in Fig. 10(b), which is also recorded by the OSC. The phases are shifted as 0°, 90°, 180°, and 270°, respectively. From the simulation mentioned above, the rotation speed of these two beams can be calculated as $\Omega \textrm{ = }\pi \times {10^9}/s$.

The single receiving antenna is located in a fixed point which is about 2*m* away from the CAA, and sends the received signal to the OSC (Agilent DSO-X 93204A) to record the amplitude and phase and to the OSA (Advantest, Q8384) to record the spectrum. The signal received from the -1-mode RF-ROAM beam is shown in the Fig. 11, and that from the +1-mode RF-ROAM beam is shown in the Fig. 12. The time-varying phases in Fig. 11(a) and Fig. 12(a) are relative to the beams’ frequency of 20*GHz*, and agree well with that in Fig. 10(a) and (b), respectively. The frequency shift shown in Fig. 11(b) and Fig. 12(b) agree well with the theoretical value $\Delta {f_1} = {L_1}\Omega /2\pi ={-} 0.5GHz$ and $\Delta {f_2} = {L_2}\Omega /2\pi = 0.5GHz$. Therefore, the mode measurement of +1-mode RF-OAM beam and -1-mode RF-OAM beam is realized by the generation of the RF-ROAM beams with the same rotation speed and corresponding mode.

For the multiplexed RF-OAM beams, these two RF-ROAM beams generated in the last experiment are multiplexed and transmitted together, and the phase and spectrum of the received signal in the receiving end are shown in Fig. 13. As seen in Fig. 13(b), the 20.5-*GHz* spectrum component respects the +1-mode OAM beams, and the 19.5-*GHz* spectrum component respects the -1-mode OAM beams, proving the successfully measurement of these two multiplexed RF-ROAM beams with mode $L ={\pm} 1$.

The detecting distance is 2 meters in the experiment, which is limited by the size of laboratory and the power of the signal. With the help of electrical power amplification, the distance can easily reach hundreds of meters.

All in all, by setting the rotation speed of the RF-ROAM beams in advance, in the receiving end, the single mode or the multiplexed mode combination can be easily measured by just detecting the frequency shift of the signal received in a fixed point. This proposed system provides the practical reception scheme for OAM-based long-distance wireless communication, and has promising prospects for use in the future wireless communication system.

## 5. Summary

In conclusion, an optical-controlled system for the generation of RF-ROAM beams is proposed, and by this system, the mode measurement of RF-OAM beams can be realized by just detecting the frequency shift of the signal received in a fixed point. This provides a partial reception scheme for the long-distance RF-OAM beam transmission. The theory and simulation of CAA generating ROAM beams and signal with time-varying phase controlling the rotation speed of the ROAM beam are analyzed and performed. Demonstration experiments are also conducted. In the experiment, the +1-mode 20*GHz* ROAM beam and the -1-mode 20*GHz* ROAM beam with the same rotation speed are generated, and their modes are successfully measured by detecting the spectrum of the received signal in the fixed point. Then these two RF-ROAM beams with mode $L ={\pm} 1$ are multiplexed and transmitted together, and the multiplexed modes’ measurement is also realized by detecting the spectrum of the received signal in the fixed point. This provides a partial reception scheme for the long-distance RF-OAM beam transmission, and has promising prospects for use in the future OAM-based wireless communication system.

## Funding

National Natural Science Foundation of China (61622102, 61690195, 61801038, 61821001).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## References

**1. **L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef]

**2. **A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics **3**(2), 161–204 (2011). [CrossRef]

**3. **J. Wang, K. Liu, Y. Cheng, and H. Wang, “Three-Dimensional Target Imaging Based on Vortex Strip map SAR,” IEEE Sens. J. **19**(4), 1338–1345 (2019). [CrossRef]

**4. **L. Li and F. Li, “Beating the Rayleigh limit: Orbital-angular-momentum-based super-resolution diffraction tomography,” Phys. Rev. E **88**(3), 033205 (2013). [CrossRef]

**5. **T. Yuan, H. Wang, Y. Cheng, and Y. Qin, “Electromagnetic vortex-based radar imaging using a single receiving antenna: theory and experimental results,” Sensors **17**(3), 630 (2017). [CrossRef]

**6. **S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser & Photon. Rev. **2**(4), 299–313 (2008). [CrossRef]

**7. **J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics **6**(7), 488–496 (2012). [CrossRef]

**8. **Z. Wang, N. Zhang, and X.C. Yuan, “High-volume optical vortex multiplexing and demultiplexing for free-space optical communication,” Opt. Express **19**(2), 482 (2011). [CrossRef]

**9. **Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, and Z. Zhao, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat Commun **5**(1), 4876 (2014). [CrossRef]

**10. **F. Tamburini, E. Mari, A. Sponselli, F. Romanato, T. Bo, A. Bianchini, L. Palmieri, and C. G. Someda, “Encoding many channels in the same frequency through radio vorticity: first experimental test,” New J. Phys. **14**(11), 78001–78004 (2011). [CrossRef]

**11. **M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light's orbital angular momentum,” Science **341**(6145), 537–540 (2013). [CrossRef]

**12. **M. Zhao, X. Gao, M. Xie, W. Zhai, W. Xu, S. Huang, and W. Gu, “Measurement of the rotational Doppler frequency shift of a spinning object using a radio frequency orbital angular momentum beam,” Opt. Lett. **41**(11), 2549 (2016). [CrossRef]

**13. **K. Liu, Y. Cheng, Z. Yang, H. Wang, Y. Qin, and X. Li, “Orbital-angular-momentum-based electromagnetic vortex imaging,” Antennas Wirel. Propag. Lett. **14**, 711–714 (2015). [CrossRef]

**14. **M. Harris, C. A. Hill, and J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. **106**(4-6), 161–166 (1994). [CrossRef]

**15. **S. M. Mohammadi, L. K. S. Daldorff, K. Forozesh, B. Thidé, J. E. S. Bergman, B. Isham, R. Karlsson, and T. D. Carozzi, “Orbital angular momentum in radio: Measurement methods,” Radio Sci. **45**(4), 1–14 (2010). [CrossRef]

**16. **M. Xie, X. Gao, M. Zhao, W. Zhai, W. Xu, J. Qian, M. Lei, and S. Huang, “Mode measurement of a dual-mode radio frequency orbital angular momentum beam by circular phase gradient method,” Antennas Wirel. Propag. Lett. **16**, 1143–1146 (2017). [CrossRef]

**17. **G. Pariente and F. Quéré, “Spatio-temporal light springs: extended encoding of orbital angular momentum in ultrashort pulses,” Opt. Lett. **40**(9), 2037–2040 (2015). [CrossRef]

**18. **Z. Zhao, H. Song, R. Zhang, K. Pang, C. Liu, H. Song, A. Almaiman, K. Manukyan, H. Zhou, B. Lynn, R. W. Boyd, M. Tur, and A. E. Willner, “Dynamic spatiotemporal beams that combine two independent and controllable orbital-angular-momenta using multiple optical-frequency-comb lines,” Nat Commun **11**(1), 4099 (2020). [CrossRef]

**19. **C. Zhang and L. Ma, “Millimetre wave with rotational orbital angular momentum,” Sci Rep **6**(1), 31921 (2016). [CrossRef]

**20. **C. Zhang and L. Ma, “Detecting the orbital angular momentum using virtual rotational antenna,” Sci Rep **7**(1), 4585 (2017). [CrossRef]

**21. **S. M. Mohammadi, L. K. S. Daldorff, J. E. S. Bergman, R. L. Karlsson, T. Bo, K. Forozesh, and T. D. Carozzi, “Orbital Angular Momentum in Radio—A System Study,” IEEE Trans. Antennas Propag. **58**(2), 565–572 (2010). [CrossRef]

**22. **X. Gao, S. Huang, J. Zhou, Y. Wei, C. Gao, X. Zhang, and W. Gu, “Generating, multiplexing/demultiplexing and receiving the orbital angular momentum of radio frequency signals using an optical true time delay unit,” J. Opt. **15**, 5401 (2013). [CrossRef]

**23. **X. Gao, S. Huang, Y. Song, S. Li, Y. Wei, J. Zhou, X. Zheng, H. Zhang, and W. Gu, “Generating the orbital angular momentum of radio frequency signals using optical-true-time-delay unit based on optical spectrum processor,” Opt. Lett. **39**(9), 2652 (2014). [CrossRef]

**24. **Z. Zhou, Y. Cheng, K. Liu, H. Wang, and Y. Qin, “Rotational Doppler Resolution of Spinning Target Detection Based on OAM Beams,” IEEE Sens. Lett. **3**(3), 1–4 (2019). [CrossRef]