Multi-mode vortex beams generation with single-layer transmissive metasurface

Orbital angular momentum (OAM) is a phenomenon of vortex phase distribution in free space, the infinity of whose modes and the orthogonality between modes allow it to possess great potential in enhancing channel capacity. A transmissive metasurface consisting of a single-layer substrate and a two-layer metallic structure is proposed to enable vortex beam generation based on the concept of the Pancharatnam-Berry (PB) phase. The phase response is achieved by continuous phase modulation under circularly polarised (CP) wave incidence due to the simultaneous rotation of the upper and lower metal structures. Besides, through the introduction of both phase compensation on the metasurface and phase superposition of the beams to the array design, multi-beam generation is expected to be realized. A metasurface sample composed of 30 × 30 elements (450 mm × 450 mm) is fabricated and measured in a microwave darkroom on the basis of the simulation results. Both the simulation and measurement results provide firm demonstration that the designed metasurface is capable of simultaneously generating four beams carrying OAM mode numbers of ±1/±2 and different beam directions at the same aperture. The designed metasurface in this paper utilized for generating multi-mode vortex multi-beams will be of wide application in multi-channel transmission in wireless communication systems.


Introduction
The current communication capacity is exposed to the challenge that can no longer satisfy the existing demand on account of the boomingly developing communication technology and the increasing number of communication devices.Enhancing channel capacity and improving the utilization rate of communication equipment become urgent problems demanding prompt solutions [1][2][3][4].Researchers have focused on optimizing the bandwidth and signal-to-noise ratio to increase the channel capacity of wireless communication systems traditionally.However, limitations in bandwidth and signal-to-noise ratio impose restrictions on the current systems to meet the growing demand for high-speed communications.
Several techniques for improving spectrum utilization have been developed to solve this issue, among which the usage of angular momentum to transmit electromagnetic waves has attracted the attention of scholars and researchers.Electromagnetic waves, with the feature of fluctuation and particle nature, therefore hold angular momentum characteristics in the light of the wave-particle duality principle.Angular momentum can be divided into Spin Angular Momentum (SAM) and Orbit Angular Momentum (OAM) [5,6] which own the property of mutual transformation under the premise of conservation of total momentum, according to the principle of conservation of angular momentum.The spin angular momentum reflects the polarization state of the electromagnetic wave (linearly polarization (LP), right-handed circular polarization (RHCP), or left-handed circular polarization (LHCP), and the reference axis is the wave vector direction.Meanwhile, OAM reflects the spatial phase distribution of the electromagnetic wave, and the reference axis is the antenna aperture plane normal direction.For a vortex wave with OAM mode number l, its isophase plane exhibits a spatial spiral distribution.Attributed to the additional communication degrees of freedom provided by the modal orthogonality between different OAM modes of vortex electromagnetic waves, an infinite number of channels in a limited communication bandwidth can be obtained thus solving the radio band saturation problem [7,8].In addition to its enormous potential in enhancing communication capacity, vortex electromagnetic waves have also proven to possess equally crucial applications in improving radar angular resolution, target electromagnetic imaging and rotating target detection.A variety of methods for generating OAM beams have been proposed previously, such as spiral phase plates [9], circular traveling wave antennas [10], holographic diffraction gratings [11], spiral or twisted reflectors [12] and circular phased radiator arrays [13].However, the existing methods scarcely satisfy the practical requirements of wireless systems on account of the drawbacks of complex structures, large devices, multiple layers, complex feed networks and narrow beams.
Metasurface, the two-dimensional form of metamaterials, has intrigued researchers greatly due to its strength of excellent wavefront modulation capability.Metasurfaces consist of periodic or nonperiodic arrangements of cells and are characterized by low loss, low profile and high integration in practical applications [14][15][16] with an excellent performance in manipulating electromagnetic waves in multiple forms and functions.Diverse usages such as polarization conversion [17,18], multi-beam generation [19], beam modulation [20,21] and enhancement of spatial beam shift [22,23] have been implemented by transmitting and reflecting metasurfaces.In addition, metasurfaces offer superior performance in vortex beam modulation, where OAM beam control is achieved by introducing phase mutations.This OAM beam control technique, usually exhibiting the superiorities in its simple design and low cost, is versatile in phase modulation and is capable of furnishing equivalent control efficiencies as focal lenses, axial prisms and parabolic surfaces through phase compensation [24].The OAM beam generated by the transmissive metasurface which is less susceptible to the influence of the feed source than the reflective metasurface, thus holds a higher purity.How to introduce an azimuth-dependent spiral phase in the optical field proved to be the only one design principle for generating vortex beams based on the metasurface, although the design of the metasurface differs in materials and structures.Therefore, one of the two existed types is designed to be independent of the polarization state of the incident beam.The resonant frequency of the emitting unit is adjusted by converting the geometry of the metasurface unit, leading to the variation of the phase shift of the emitting unit in accordance with the alteration of frequency.The other design is dependent on the polarization state of the incident beam on the basis of the coupling transition between SAM and OAM, which is the Pancharatnam-Berry phase metasurface [25].The geometric phase-based metasurface with simple design produces a vortex beam without dispersion.
Nowadays, with the continuous development of metasurface technology, it is necessary to investigate multibeam and multimode vortex wave generation to further improve the utilization of hardware and communication capabilities.To date, there have been numerous studies on multimode vortex beam generation.Literature [26] achieved vortex multibeam generation by rotating a metal structure to realize the transmission phase shift.However, the disparity of vortex beam gain in different modes is more obvious and is not conducive to the simultaneous and stable transmission of multi-channel information.Literature [27] realized symmetric dual-beam and four-beam vortex waves by adjusting the dimensional parameters of the metal structure to achieve phase shifting.However, the metasurface in this study only produces symmetric beams.Literature [28] successfully generated a single-mode vortex multibeam by realizing a single-mode vortex multibeam on the surface of a single-layer substrate element and encoding the array using the PB phase principle.The above study enables to generate a multimode vortex multibeam, but it is not able to control the beam direction and the number of OAM modes of each beam uniformly, stably and independently.Therefore, the simultaneous generation of stable multimode vortex multibeam at the same aperture can further improve the hardware utilization and transmission capability.Meanwhile, realizing arbitrary modulation of OAM modes and beam directions will provide more versatility and flexibility for the metasurface.
In this paper, we propose a metasurface of a single-layer substrate that enables to realize four vortex beams in four modes through applying the PB phase principle.Under circularly polarized wave incidence conditions, phase coverage of 360 degrees is achieved by simultaneously rotating the angles of the upper and lower metal structures while the transmission amplitude remains almost constant.The incident circularly polarized wave passed through the metasurface guarantee the accomplishment of the polarization conversion of LHCP and RHCP waves in virtue of the special symmetry of the metallic structure.The phase superposition method is introduced in the design to rearrange the array phases to achieve wavefront modulation.Arrays that can generate two-mode vortex dual-beam and four-mode vortex quad-beam are designed on the ground of adopting the proposed metasurface element in this study.Simulation results provide validation data for the feasibility of the designed metasurfaces.The metasurface samples are fabricated to generate  1 and  2 four-mode vortex quad beams.The test results agree well with the simulation ones, and the beam directions and OAM mode numbers of the generated beams are in excellent consistency with the preset contents.The designed vortex wave multiplexing metasurface effectively reduces the profile of the equipment, and the unloaded active devices can decrease the energy loss.It possesses stable performance in the field of multi-channel information transmission and holds a promising application prospect in the demand of short distance and multiple fixed target communication.

Element design and analysis
To elaborate on the operating mechanism of the proposed metasurface, we first illustrate the principle of the PB phase to implement continuous phase shift.Under the assumption that the RHCP wave is incident to the metasurface element along the z-direction, the relationship between the incident and transmitted waves can be expressed as: The upper and lower metal structures are rotated along the y-axis as shown in figure 1(c) and the coordinate transfor-mation is defined below: x y x y The incident and transmitted waves are represented in the new coordinate system as: e j e x y 5 Since it is a circularly polarized wave incidence, the trend of transmission coefficient and phase change before and after coordinate transformation is the same, which is T T T T , , x x y y = ¢ = ¢ the transmitted waves can be represented as: x y cos sin sin cos x j y e e e x j y 2 7 Incident RHCP waves are converted to LHCP waves through transmission grounded on the derivation and analysis of equation (1) to equation (8).In addition, the transmitted LHCP wave produces a phase shift of 2j as the metallic structure rotates j along the y-axis when the RHCP incident wave is irradiated to the metasurface.Suppressing the transmission of co-polarized waves as much as possible becomes a requisite to meet the formation conditions of PB phase.When the values of t yx and t xy are low enough to be negligible, then T T , A full-wave simulation of the element is performed by utilizing CST Microwave Studio to verify whether the proposed metasurface element is capable of satisfying the PB phase shift.Figure 2(a) illustrates the result that the proposed metasurface unit enables to accomplish equal amplitudes of t xx and t yy and a phase shift of 180 degrees at about 7.8 GHz under the line polarization wave incidence condition, which implies the conclusion that the proposed metasurface element satisfies the condition of PB phase shift based on the above analysis and equation (8).Subsequently, we adjusted the incident wave to a circularly polarized wave to demonstrate whether the phase shift reaches 360 degrees as the metal structure rotates.The amplitude and phase response of the transmitted wave at the incidence of the LHCP wave and RHCP wave to the metasurface are displayed respectively in figures 2(b) and (c) representing the trend of the phase shift of the transmitted wave with the rotation angle more distinctly, where 0, 1, 2, 3, 4, 5, 6, 7 denote the eight modes set, meaning that the rotation angles j are 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, respectively.It can be observed from the figure that along with the increasing rotation angle , j the amplitude variation of the transmitted wave remains within a small range, while the phase changes constantly with a trend of 2 .j It is worth noting that the incident wave reaches a bicircular polarization transition after transmission through the metasurface element.As the simulation results revealed, the proposed metasurface satisfies the condition of PB phase shift and brings about a phase shift of 360 degrees and meanwhile makes the amplitude keep at the same level.The magnitude-phase response of the metasurface element at 7.8 GHz under bicircular polarization wave incidence is indicated more clearly in figure 2(d).What draws attention is the opposite trend of the phase change of the transmitted waves resulted by respective LHCP and RHCP incident waves.It can be deduced from equations (1) and (5) that the phase shift of the transmitted wave calculated in equation ( 8) is 2j and 2 , j respectively, due to the different expressions of the incident waves LHCP and RHCP.The LHCP wave is applied as the incident wave in the subsequent simulations and measurements in this paper.
To elaborately illustrate the working principle of the proposed metasurface, simulations are performed for the electric field, as shown in figure 3. What depicted in figure 3(a) is the generation of a toroidal induced current on the upper metal surface as the LHCP incident wave irradiates on the upper surface of the metasurface unit.After radiation, an induced current is produced as well on the surface of the lower metal structure, as presented in figure 3(b).However, the toroidal induced currents on the surface of the metal structure of the upper and lower layers are in opposite directions, which is easily observed in figure 3. Thus, the outcome that the transmitted wave electric field and the incident wave electric field are in opposite motion provides further verification on the bicircular polarization conversion results mentioned in the above analysis.

Design of multi-mode vortex beams metasurface
The capability of the proposed metasurface unit with continuous phase shift of 360 degrees has been testified in the previous analysis.The following section will concentrate on the design process and simulation results of the multimode vortex multibeam metasurface based on this element.The principle of the array phase design will be expounded first.This paper adopts the phase superposition method for the design of the metasurface array on the basis of the design principle of multibeam.Assuming that the number of beams is k, the phase of the element (m, n) can be expressed as: is the wave vetor of free space.r mn  and r f  represent the position vector of the element (m, n) and phase center of the feed, respectively.l i ( ) denotes the topological charge of the i-th vortex beam, which is the mode number of the OAM wave.mn j means azimuth angle.b i is the beam direction of the i-th beam which can be expressed as b sin cos sin sin .
A metasurface array consisting of 30 ´30 elements (450 mm ´450 mm) is designed in this paper for the purpose of validating the performance of the designed multimode vortex multibeam metasurface.Two arrays of l 1 =  dual-mode dual-beam and / l 1 2 =   four-mode four-beam are conceived for carrying out simulation experiments.For the dual-mode vortex dual-beam metasurface, the beam directions , ( ) q j are set to (20°, 90°) and (20°, −90°) for mode numbers l = 1 and l = −1, respectively.Similarly, for the four-mode vortex four-beam metasurface, the beam directions , ( ) q j for mode numbers l = 1, l = −1, l = 2 and l = −2 are set to (40°, 45°), (40°, −135°), (30°, −45°) and (30°, 135°), separately.The number of modes and beam directions of the beams expected to be generated by the two metasurfaces are displayed in detail in table 1.The array phase is calculated using equation ( 9) and (10).The desired phase of each element corresponds to the rotation angle of the metal structure in the element.The final resulting array phase distribution and element rotation angle distribution are exhibited in figure 4. The full-wave simulation is performed in CST Microwave Studio for the two metasurface at 7.8 GHz, utilizing a standard gain horn antenna serving as the circular polarization feed source and excited by the waveguide port method.The boundary in the x, y and z directions is set to open (add space).The distance from the phase center of the waveguide antenna to the center of the metasurface aperture is fixed at F = 351 mm and F/D is 0.78.
The feed source in the experiments of this paper is a horn antenna capable of radiating circularly polarized waves, as expressed in figures 5(a) and (b).The butterfly-type dielectric plate is loaded in the conical horn antenna, and the dielectric plate is able to radiate LHCP and RHCP waves by rotating 45 and 135 degrees along the y-axis, respectively.Figure 5(c) demonstrates the simulated and measured two-dimensional gain patterns at 7.8 GHz.The two-dimensional directional maps of the E-plane and H-plane are in good consistency with each other.The reflection coefficients of the waveguide antenna are shown in figure 5(d).Although some differences exist between the simulated and measured reflection coefficients, they are both below −10 dB and can be matched.Figures 5(e) and (f) depict the simulated and measured results of the gain variation of the waveguide antenna in the bandwidth of 5-9 GHz and the axial ratio at 7.8 GHz are shown respectively.
The simulation results are presented in figure 6, which demonstrates the 3D radiation direction map, 2D radiation direction map in polar coordinate system and the and phase distribution of each beam, respectively.First, the 3D radiation direction maps of the two metasurfaces are depicted in the uv-plane (u v sin cos , sin sin reveals sharp distinction of four beams with OAM modes of  1 and  2 in beam focusing characteristics, where the more well-focused beams are  1 in mode number and the other two beams are  2 in mode number.Secondly, figures 6(c) and (d), which depict the far-field directional maps of M 1 and M 2 in polar coordinate system respectively, clearly manifest the direction of the generated beam.The two beams that can be noticed in figure 6(c) point at 150°and 210°, separately.The beams are shifted by 30°and 30°, respectively, with respect to the -z-direction, which is consistent with the pre-defined beam direction.Figure 6(d) illustrates the directional maps in the polar coordinate system for azimuths phi of 45°and 135°, respectively, in which both curves contain two beams pointing at 200°and 150°, offsetting by 40°and 30°with respect to the -z-direction, which concurs with the preset four-beam beam directions as well.Finally, we extracted the phase distribution and amplitude distribution of each beam to firmly attest that the number of modes in each beam is same as the preset content.The number of OAM modes represented can be determined from the phase rotation characteristics in the phase distribution.What can be concluded from the figure is that the simulation results are in good agreement with the pre-defined parameters.The simulation results elaborated above have preliminarily substantiated that the proposed metasurface may possess the capability of generating multi-mode vortex multi-beam.A metasurface object capable of producing four-mode, four-beam vortex waves and consisting of 30 ´30 elements (450 mm ´450 mm) is fabricated to provide further verification on the application value of the metasurface by means of using the PCB printing technique, as observed in figure 7(a).The measurements are conducted through adopting a free-space test method and all experiments are in a microwave darkroom.The experimental equipment included a vector network analyzer, two standard gain horn antennas and two coaxial cables.The feed antenna for transmitting circularly polarised incident waves is shown in figure 7(b).The physical sample is placed on a turntable, and the transmitting antenna is placed at the center of the sample with a focal-diameter ratio of 0.78, while the receiving antenna is positioned on the other side of the sample to reach far-field conditions, as depicted in figure 7(c). are both centered at Theta = 220°and deflected by 40°relative to the -z-direction, while the two beams with OAM mode 2  are both centered at Theta = 150°and deflected by 30°relative to the -z-direction.
This experimental result supports the conclusion that the actual measured beam directions is in good conformity with both the preset beam directions and the simulation results.Figures 8(c) and (d) illustrate the axial ratios at various azimuth angles, respectively, and it can be summarized from the 3 dB horizontal line that the generated beams are all circularly polarized waves.We have sampled the planes in each beam direction with a metal probe and obtained the corresponding phase and amplitude distributions for each beam, and the results of which are displayed in figures 8(c)-(f) and (i)-(l).The number of OAM modes carried by each beam can be obtained from the sampling results, which more firmly verifies that the beams generated by the samples in the actual measurements are consistent with the preset parameters.The comparison of the measured and simulated results of beam gain, beam direction and other data are listed in table 2.

Conclusion
In summary, this paper puts forward a transmissive metasurface on a single-layer substrate.The continuous phase adjustment is achieved on the basis of the PB phase principle, and the metasurface array is designed with the integration of the metasurface phase compensation principle and phase superposition theory, which enables the simultaneous generation of four vortex beams carrying four OAM modes in the same aperture at around 7.8 GHz at CP wave incidence.Furthermore, the metasurface samples are fabricated and measured in a microwave darkroom as well.Both near-field spectra and far-field results validate the practical performance of the metasurface capable of generating beams carrying the OAM modes / 1 2.   The proposed transmission metasurface is important for modern wireless communications in terms of channel capacity enhancement and multi-channel communication.Meanwhile, it holds a wide range of applications in scenarios of short-range communication with multiple stationary targets.

Figure 1 (
Figure 1(a) depicts a schematic diagram of the multimode vortex multibeam generated by the proposed TM, which produce a four-beam four-mode vortex beam.The material of the dielectric substrate is F4B ( 2.65, tan 0.009 r e d = = ) with a thickness of 3.6 mm, and the metal structure is fabricated by copper with a thickness of 0.035 mm.Other parameters displayed in figure 1(b) are optimized with the values of: p 12mm, =

Figure 1 .
Figure 1.Schematic of the proposed TM.(a) Visualisation of the generation of four-mode, four-beam vortex waves.(b) Schematic diagram of TM element structure.(c) Coordinate transformation diagram.(d) Schematic diagram of the eight rotation angles of the TM element.

Figure 2 .
Figure 2. Simulation results of metasurface element.(a) Magnitude-phase response of the element under LP wave incidence.(b) Magnitude-phase response of eight modes under LHCP wave incidence.(c) Magnitude-phase response of eight modes under LHCP wave incidence.(d) Magnitude-phase response of the element with rotation angle j at 7.8 GHz.

Figure 3 .
Figure 3. Surface current of metal layer in LHCP wave incidence.(a) Upper layer.(b) Lower layer.

Figure 4 .
Figure 4. Array phase distribution and rotation angle distribution of the elementary metal structure in the array.(a) Phase distribution and (b) elementary metal structure rotation angle distribution of dual-mode dual-beam metasurface.(c) Phase distribution and (d) elementary metal structure rotation angle distribution of a four-mode, four-beam metasurface.
as shown in figures 6(a), (b) so as to display the beam distribution generated by the metasurface from a global perspective.The beam directions of the double and quadruple beams colored in red which can be apparently observed in the figure both exhibit distinct vortex beam characteristics.Particular attention is directed towards the modes of the two OAM beams in figure 6(a), which are 1 and −1, so two beams with similar beam features but different beam directions are shown in the figure.And figure 6(b)

Figure 5 .
Figure 5.The performances of the source feed.(a) Side view of the source feed.(b) Perspective view of the source feed.(c) Simulated and measured realized gain at 7.8 GHz.(d) Simulated and measured reflection coefficient S .11 | | (e) Simulated and measured axis ratio at 7.8 GHz.(f) Simulated and measured realized gain in 5-9 GHz.

Figure 6 .
Figure 6.Simulation results.(a) Radiation far-field diagram of M1.(b) Radiation far-field diagram of M2.(c) Far-field diagram of M1 in the polar coordinate system.(d) The amplitude and phase distribution of the two beams generated by M1.(e) Far-field diagram of M2 at Phi = 45°in the polar coordinate system.(f) Amplitude and phase distribution of the two beams generated by M2 at Phi = 45°.(g) Far-field diagram of M2 at Phi = 135°in the polar coordinate system.(h) Amplitude and phase distribution of the two beams generated by M2 at Phi = 135°.

Figures 8 (
Figures 8(a) and (b) demonstrate the far-field measurements at azimuth angles of 45°and 135°respectively, which indicate that among the four beams generated by the metasurface sample, the two beams with OAM mode number 1 are both centered at Theta = 220°and deflected by 40°relative to the -z-direction, while the two

Figure 8 .
Figure 8. Measurement results.(a) Far-field radiation map and (b) axial ratio when Phi = 45°.(c) Phase distribution and (d) amplitude distribution of the beam with mode l = 1.(e) Phase distribution and (f) amplitude distribution of the beam with mode l = 2. (g) Farfield radiation diagram and (h) axial ratio at Phi = 135°.(i) Phase distribution and (j) amplitude distribution of the beam with mode l = −1.(k) Phase distribution and (l) amplitude distribution of the beam with mode l = −2.

Table 2 .
Comparison of simulation and measurement results.