Generation of Four-Channel Multi-Polarization Bessel Vortex Beams with Equal Divergence Angle Based on Co-Aperture Metasurface

Abstract

This paper proposes a co-aperture reflective metasurface that successfully generates four-channel Bessel vortex beams with equal divergence angle in both Ka and Ku bands. Initially, a frequency-selective surface (FSS) is employed to suppress inter-unit crosstalk. Subsequently, modified cross-dipole metasurface units are implemented using spin-decoupling theory to achieve independent multi-polarization control. Through theoretical calculation-based divergence angle engineering, the dual-concentric-disk structure integrated with multi-polarization control demonstrates enhanced aperture utilization efficiency compared to conventional partitioning strategies, yielding high-purity equal-divergence-angle Bessel vortex beams across multiple modes. Finally, experiments on the metasurface fabricated via printed circuit board (PCB) technology verify that the design simultaneously generates x-polarization +1 mode and y-polarization +2 mode equal divergence angle Bessel vortex beams in the Ku band and ±3 mode beams in the Ka band. Vortex beam divergence angles remain stable at 9° ± 0.5° under diverse polarization states and modes, with modal purity reaching 65–80% at the main radiation direction. This work provides a straightforward implementation method for generating equal-divergence-angle vortex beams applicable to Orbital Angular Momentum (OAM) multimode multiplexing and vortex wave detection.

1. Introduction

Recently, vortex beams have sparked extensive research interest in optical and wireless communications due to their unique helical phase wavefront characteristics [1,2]. Theoretically, orthogonal OAM modes with different topological charges can construct infinitely independent channels at a single frequency, offering a novel approach to enhancing communication system capacity through modal multiplexing [3,4]. As a novel resource, OAM beams enable substantial enhancements in communication capacity and enhance radar resolution [5,6,7]. However, as the absolute value of topological charge increases, the divergence angle of vortex electromagnetic waves significantly enlarges. Moreover, the orthogonality of different OAM modes requires strict alignment between transmitters and receivers to ensure complete beam reception. Otherwise, the carried information cannot be effectively captured, recognized, or utilized. This divergence angle discrepancy between OAM modes complicates multi-modal OAM reception in fixed-aperture systems [8]. More notably, the disordered wavefront structures of hybrid-modal OAM beams [9] make their physical characterization challenging, severely limiting their application potential in long-distance communication scenarios. Therefore, flexible control over the divergence angles of multi-modal vortex beams has become a critical issue to be addressed for high-capacity OAM communication systems.

In the field of OAM generation technologies, existing methods primarily include helical phase plates [10], rotating paraboloids [11], and uniform circular arrays (UCA) [12], but these approaches generally rely on complex feed networks, leading to significant increases in antenna system volume and cost. Additionally, metasurface antennas, owing to their powerful electromagnetic wave manipulation capabilities, find wide applications in areas such as dynamic optical tuning and sensing [13] and infrared absorption devices [14,15,16]. Significantly, by mimicking traditional optical devices through phase engineering, metasurface antennas significantly expand the possibilities for flexible control of OAM beams, enabling their generation and manipulation [17]. Metasurface lenses can provide phase gradients in the phase design, significantly reducing OAM divergence and improving communication range [18,19]. Alternatively, Bessel beams with self-focusing and non-diffracting properties maintain low divergence angles during propagation to significantly extend communication distances in OAM systems. Among existing Higher-Order Bessel Beam (HOBB) generation methods, equivalent axicon phase compensation technology offers notable advantages in structural simplicity [20,21,22]. However, current research primarily focuses on single-mode OAM studies, including improving mode purity, miniaturization, and increasing transmission distance. Dual-polarization metasurface technologies [23,24,25] achieve the simultaneous generation of dual-linear-polarized OAM beams through orthogonal polarization modulation. Ref. [26] realized a single-layer dual-band metasurface vortex beam with variable OAM modes and polarization operating in the 5.2 GHz and 10.5–12 GHz bands. The core challenge faced by existing schemes is as follows: on the one hand, the mode-related divergence radius has a significant impact on receiving sensitivity; more fundamentally, the limited number of multiplexable OAM modes in OAM multiplexing systems restricts their full potential to enhance communication capacity as a new degree of freedom, severely constraining the multiplexing efficiency of OAM. Consequently, the prospects for OAM multiplexing as a novel spatial multiplexing mechanism are being challenged.

To address this, we introduce an approach combining a multi-polarization control metasurface unit with a dual-disk partitioning strategy. By adjusting the aperture size of each partition, we control the divergence angles of vortex beams across different modes. This enables the generation of four-channel vortex beams with distinct polarization states and equal divergence angle. Furthermore, employing an equivalent axicon method to generate Bessel vortex beams increases the transmission distance of the vortex beams. Figure 1 demonstrates a dual-band co-aperture metasurface generating four coaxial vortex beams with equal divergence angle. This design reduces receiver antenna design complexity and signal processing requirements. Experimental validation of the PCB-fabricated metasurface demonstrates its capability to generate high-purity Bessel vortex beams with equal divergence angle. This not only significantly enhances communication efficiency but also reduces the practical implementation complexity of OAM transceivers, offering a potential solution for mode and polarization separation in microwave-band OAM transmission.

2. Theoretical Formulation and Metasurface Unit Design

2.1. Spin-Decoupling Theory

The Jones matrix is commonly used in optics to describe the characteristics and polarization states of electromagnetic waves. When the rotation angle is α, it can be expressed as $R(\alpha )=S^{-1}(\alpha )\dot{R}S(\alpha )$, where $S(\alpha )$ is a rotation matrix. The broadband spin-decoupled metasurface-enabled reflection Jones matrix is calculated by converting its LP (linear polarization) form into the corresponding CP (circular polarization) form [27]:

$$R^{\mathrm{cir}}(\alpha )={\Lambda }^{-1}R(\alpha )\Lambda =\left(\begin{array}{cc}{r}_{++}& r_{+-}\\ r_{-+}& r_{--}\end{array}\right)$$

(1)

In this context, the ‘+’ and ‘−’ signs denote right-hand circularly polarized (RHCP) and left-hand circularly polarized (LHCP) wave excitations. The first subscript indicates the reflected wave, while the second corresponds to the incident wave. Specifically, the cross-polarization reflection coefficients are

$$r_{-+}=r_{xy}{e}^{j({\Phi }_{xy}+2\alpha -\pi /2)}$$

(2)

$$r_{+-}=r_{xy}{e}^{j({\Phi }_{xy}-2\alpha +\pi /2)}$$

(3)

where $r_{xy}$ is the cross-polarization reflection coefficient under LP illumination, and ${\Phi }_{xy}$ refers to the phase shift between x and y polarization states. The generation efficiency of CP is equivalent to the LP polarization conversion rate. By setting two demand phases ${\Phi }_L$ and ${\Phi }_R$ under LHCP and RHCP, respectively:

$${\Phi }_{xy}+2\alpha -{\displaystyle \frac{\pi }{2}}={\Phi }_R$$

(4)

$${\Phi }_{xy}-2\alpha +{\displaystyle \frac{\pi }{2}}={\Phi }_L$$

(5)

Through simple equation transformation, we can obtain

$${\Phi }_{xy}={\displaystyle \frac{{\Phi }_R+{\Phi }_L}{2}}$$

(6)

$$\alpha ={\displaystyle \frac{({\Phi }_R-{\Phi }_L+\pi )}{4}}$$

(7)

As shown in (6) and (7), the two phases are decoupled, and they can be independently modulated by the rotation angle and phase difference. Therefore, metasurface units featuring independent polarization tunability for orthogonal x- and y-polarizations are essential to implement spin-decoupling theory and achieve multi-polarization regulation.

2.2. Theory of Multimode OAM Reflective Metasurface with Equal Divergence Angle

To mitigate the inherent divergence of vortex waves, researchers have explored various methods [28,29]. These methods primarily control the divergence angle by adjusting the feeding scheme of coaxial annular structures. The study [25] examined how antenna aperture radius affects OAM beam radiation patterns. Under non-focusing conditions, the radiation distributions in the central area can be described as

$$|F_D(\theta ,\phi )|=\sqrt{1-{(\sin \phi \sin \theta )}^2}|{\sin }^l\theta {\times }_1{F}_2(\theta ,R_1)|$$

(8)

$${}_{1}F_2(\theta ,R)={}_{1}F_2\left(\frac{l}{2}+1;\frac{l}{2}+2,l+1;-{\left(\frac{\mathit{kR}\sin \theta }{2}\right)}^2\right)$$

(9)

In Formula (8), $\theta$ and $\phi$ respectively show the elevation angle and azimuth angle, and $l$ is the OAM modes generated in the inner area and the outer area, respectively. ${}_{1}F_2(\theta ,R)$ denotes the hypergeometric function, while $R$ is the radius of the disk.

To achieve uniform divergence angles for OAM radiation patterns across different modes, this work employs an optimization method based on hypergeometric function zero-matching. The core concept adjusts the radius ratios $R_l$ for different modes $l$, ensuring that all radiation patterns $|F_D(\theta ,\phi )|$ reach their first null at the same observation angle $\theta$. This is established by solving the condition ${}_{1}F_2(\theta ,R)=0$, with $z_l={\left({\displaystyle \frac{k{R}_l\sin \theta }{2}}\right)}^2$. Under the equal divergence constraint, the first positive root $z_l$ of the hypergeometric function depends solely on the mode order $l$. This zero-matching condition decouples the size parameter:

$$R_l={\displaystyle \frac{2}{k\sin \theta }}\sqrt{z_l}$$

(10)

Following proportional coefficient calculation and normalization, we obtain

$$R_1:R_2:R_3=\sqrt{z_1}:\sqrt{z_2}:\sqrt{z_3}\approx 144:170:190$$

(11)

As illustrated in Figure 2, the array layout employing multi-mode partitioned radii ($R_1$, $R_2$, $R_3$) calculated via Equation (11) achieves four-channel vortex beams with equal divergence angle across distinct modes. Notably, every divided section needs to maintain a width of at least one basic period, and the radius of the metasurface determines the highest number of OAM modes achievable [8].

2.3. Multi-Polarization Element Design

As illustrated in Figure 3, the proposed metasurface unit cell comprises five metallic layers and three dielectric layers. The dielectric layers utilize Rogers 4350b material (${\epsilon }_r=3.66$, $\tan \delta =0.004$), with a metal ground plane integrated on the metasurface backside. The periods of the phase units for the Ku band and Ka band are set as $D_{\mathrm{ku}}=10\mathrm{mm}$ and $D_{\mathrm{ka}}=5\mathrm{mm}$, respectively. An interlayer FSS suppresses cross-band interference. Optimized geometric parameters are listed in

Table 1. Geometrical parameters of the meta-atom.

Parameters	${\mathit{t}}_1$	${\mathit{t}}_2$	${\mathit{t}}_3$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{w}}_{\mathit{u}}$
Values (mm)	0.508	0.762	0.762	2	6	1.5	0.5
Parameters	$L_1$	$L_2$	$L_3$	$d_1$	$d_2$	$d_3$	$w_a$
Values (mm)	4.5	2.8	1.5	0.35	0.2	0.45	0.3

.

This paper implements a stepwise design strategy for constructing multi-polarization manipulation units. Utilizing the CST Microwave Studio 2023 commercial platform, three-dimensional full-wave electromagnetic simulations were conducted with the frequency-domain solver. To validate the operational mechanism of the optimized FSS, Figure 4 presents its transmission/reflection amplitude responses at 0° and 30° incidence angles. The results confirm the effective realization of frequency-selective characteristics featuring a Ku-band passband and Ka-band stopband.

Furthermore, Equations (6) and (7) show that the core of the spin-decoupled strategy is the independent phase control of orthogonal linear polarizations. This is achieved by independently controlling the reflection phase and rotation angle for x- and y-polarized waves. While existing designs achieve this through parametric scanning of unit cells [30,31], the required multi-objective optimization significantly increases design complexity. Consequently, this work employs both single-layer and multi-layer cross-dipoles to streamline the design process. To comprehensively evaluate inter-band crosstalk, the influence of Ka-band unit parameters, including variable patch length $a_x$ and rotation angle ${\theta }_a$, on Ku-band phase response was analyzed. Results demonstrate negligible crosstalk as illustrated in Figure 5a,b. Similarly, variations in Ku-band parameters such as patch length $u_x$ and rotation angle ${\theta }_u$ exhibit minimal crosstalk effects on Ka-band units. To validate the suppression of crosstalk, the same tests were performed on units without the FSS structure. As shown in Figure 5c,f, larger upper-layer unit sizes cause greater phase crosstalk on lower-layer units. Correspondingly, smaller lower-layer unit sizes cause greater crosstalk on upper-layer units, significantly affecting phase design. Notably, the rotational symmetry of the cross-dipoles ensures consistent y-polarization characteristics. In summary, units incorporating the FSS structure demonstrate excellent polarization and band independence, thereby enabling the generation of four-channel vortex beams.

Finally, generating high-purity vortex beams with equal divergence angle requires full 360° phase coverage from unit cells. This work employs a double-layer cross-dipole configuration, extending the phase range beyond conventional single-layer designs through interlayer coupling resonance. As shown in Figure 6a,b, the Ku-band unit achieves complete phase coverage by tuning patch sizes continuously from 1.0 mm to 3.5 mm, while the Ka-band unit attains equivalent coverage via size adjustments from 2.0 mm to 10.0 mm; both bands maintain reflection efficiencies exceeding 95%.

In summary, the implemented metasurface unit cells effectively suppress crosstalk between the Ku- and Ka-bands. Leveraging spin-decoupling theory enables independent phase manipulation of multi-polarization incident waves. This configuration simultaneously delivers excellent reflection characteristics and full 360° phase coverage. Moreover, these units can be adapted to other dual-band applications by adjusting FSS structural parameters.

3. Reflective Metasurface Design and Empirical Validation

Building upon multi-polarization independent control units and a divergence control method based on hypergeometric function zero matching, we designed a dual-layer co-aperture metasurface. This structure generates four-channel vortex beams with equal divergence angle to validate the theoretical framework. The array features a 190 × 190 mm² aperture integrating 19 × 19 Ku-band and 38 × 38 Ka-band unit cells. For simulations, 10 dB gain linearly, and circularly polarized horn antennas were positioned 150 mm from the metasurface center as feeds.

To extend the transmission distance of vortex beams, HOBBs carrying OAM are generated. An axicon phase term is incorporated into the phase design, and the required compensation phase ${\psi }_{mn}$ for the unit cell $\left(x_{mn},y_{mn}\right)$ is calculated as

$${\psi }_{mn}={\displaystyle \frac{2\pi }{{\lambda }_0}}\left(\sqrt{r_{mn}^2+F^2}-F\right)+l\arctan \left({\displaystyle \frac{y_{mn}}{x_{mn}}}\right)+k_0{r}_{mn}\sin ({\theta }_0)$$

(12)

where ${\lambda }_0$ is the wavelength corresponding to the operating frequency, F is defined as the vertical distance of the feed’s phase center, $r_{mn}$ represents the distance from the center of the $\left(x_{mn},y_{mn}\right)$ unit to the center of the metasurface, and ${\theta }_0$ is the numerical aperture (NA) of an axicon.

Figure 7a–d present the phase compensation profiles of the Ku-band metasurface, including the vortex phase, Bessel beam focusing phase, feed phase compensation, and superimposed total phase. Figure 8a,b display the dual circular polarization phase distributions for the Ka-band, while Figure 8c–e further provide the unit orientation angle distributions and x/y-linear polarization phase response curves, all calculated using Equations (6) and (7). Utilizing our previously designed dual-band polarization-multiplexed metasurface unit combined with the spin-decoupling principle, independent control of multi-polarization phases can be achieved. Simultaneously, we established the correspondence between unit dimensions and their corresponding reflection phases. First, the lower-layer array exhibits excellent linear polarization independence. When illuminated by an x-polarized incident wave, the reflection phase distribution of the array is adjusted by tuning the unit dimensions along the x-axis direction to generate an x-polarized vortex beam with mode 2. Correspondingly, adjusting the dimensions along the y-axis direction generates a y-polarized vortex beam with mode 1. The specific phase distribution is shown in Figure 7d. Second, for the upper-layer array, applying the spin-decoupling principle enables independent phase control of dual circularly polarized waves. As shown in Figure 8a,b, these depict the intended phase distributions for LCP and RCP waves, respectively. Figure 8c,d, and e display the unit rotation angles along with the required phase responses for x-polarization and y-polarization. By leveraging the phase-dimension correspondence, we tune the unit dimensions and rotation angles. The detailed design methodology is described in Section 2.1. Finally, we successfully generated an LCP vortex beam with +3 mode and an RCP vortex beam with −3 mode. Crucially, according to Equation (12), configuring the array radius with sector partitioning enables equal divergence angle for all vortex modes.

Performance validation was conducted using the time-domain solver in CST Microwave Studio 2023, with a 130 × 130 mm² observation plane at 600 mm from the metasurface. Figure 9 shows simulated transverse electric field distributions along propagation: vortex beam phase distributions exhibit $\pi$, $2\pi$, and $3\pi$, while amplitude distributions display characteristic annular null structures, matching theoretical predictions. Figure 9e–h confirm consistent ring radii for +1, +2, and ±3 modes, validating the equal divergence angle design. Notably, the null radius scales with topological charge, aligning with OAM beam properties.

As shown in Figure 10, under the same structural parameter configuration, the free-space electric field distribution characteristics of traditional vortex beams and Bessel beams (l = 1, f = 12.45 GHz) are compared via full-wave simulation. Results indicate that HOBBs possess higher focused energy distribution and improved transmission efficiency over long distances. Figure 11 exhibits the normalized radiation pattern of the reflective metasurface, demonstrating the successful generation of Bessel vortex beams with an equal divergence angle of 9° across multiple modes.

To assess the purity of the vortex beams, e-field data within the main radiation region was extracted, as displayed in Figure 9e–h. Additionally, to quantitatively assess the beam purity, the OAM spectral distribution of each mode is analyzed [32]. The electric field can be modeled as a combination of multiple helical harmonic components, expressed in the following form:

$$E(\phi )={\displaystyle \sum _{l=-\infty }^{+\infty }{A}_l}{e}^{-jl\phi }$$

(13)

Among them, $A_l$ corresponds to the amplitude coefficient of helical harmonics, which can be described as

$$A_l={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_0^{2\pi }E}(\phi )e^{-jl\phi }d\phi$$

(14)

The extracted electric field data is subjected to Fourier transform and normalization processing [33]. Dominant mode purity is defined as the ratio of dominant OAM energy to total OAM energy, expressed as

$$S_{\mathrm{n}}={\displaystyle \frac{|A_{l=\mathrm{n}}{|}^2}{{\displaystyle \sum _{l=-\infty }^{l=+\infty }|}{A}_l{|}^2}}$$

(15)

Histogram plots show dominant OAM mode purities for Ku and Ka bands from simulations. Analysis results reveal that all four OAM beam modes exhibit high purity levels, as depicted in Figure 12. Specifically,

To verify the optimized design and proposed strategy, a reflective metasurface array was fabricated via PCB technology. To facilitate fixed assembly, a 10 mm margin was left on each side, three holes were drilled, and nylon screws were used for interlayer fixation. Infrared alignment was employed to ensure proper alignment between the metasurface and the feed sources.

Subsequently, experimental verification of the fabricated metasurface was conducted within a microwave absorber-lined anechoic chamber. During the experiment, the test system was simplified by alternately positioning the feed horns along the z-axis to guarantee the independence of dual-band feeding, as presented in Figure 13a,b. Furthermore, a receiving prode was attached to a scanning frame to obtain the farfield radiation patterns.

Figure 14a displays normalized measured radiation patterns, confirming linearly polarized vortex beams in Ku-band and dual-circularly polarized vortex beams of multiple modes in Ka-band, all achieving near 9° divergence angles. Corresponding mode purity is shown in Figure 14b. Results demonstrate strong agreement with simulations, consistently yielding approximately 9° divergence angles. Minor discrepancies exist between measured and simulated results, primarily attributable to metasurface manufacturing tolerances, non-ideal absorption performance of the microwave anechoic chamber, and alignment errors introduced during manual assembly. Note that environmental factors (atmospheric attenuation, multipath interference) and alignment errors cause signal ambiguity in practical applications, significantly reducing communication efficiency. This technology is currently optimal for fixed satellite communication systems with controlled environments—their stable platforms effectively ensure precise alignment, fully leveraging the transmission advantages of multi-mode OAM beams.

Finally,

Table 2. Comparison of Proposed and Reported OAM Metasurfaces.

Ref.	Frequency	Band Width	Aperture Size	Polarization	OAM Mode	Mode Purity (Simulation)	Divergence Angle	Angle of Modes
[36]	12.5 GHz 13.0 GHz 13.6 GHz	0.9 GHz	12 $\lambda$ × 12 $\lambda$	LP	1, 2, 3	60% 50% 45%	12.5	equal
[34]	29.0 GHz	Central Frequency	15 $\lambda$ × 15 $\lambda$	DLP	1, 2	-	3	equal
[29]	5.8 GHz	Central Frequency	8 $\lambda$ × 8 $\lambda$	LP	1, 2, 3	-	12	equal
[37]	9 GHz	Central Frequency	6 $\lambda$ × 6 $\lambda$	CP	0, ±1, 2	-	16.4	equal
[38]	60 GHz	Central Frequency	10 $\lambda$ × 10 $\lambda$	LP	±1	95%	10	Unequal
[26]	5.2 GHz 10.5–12 GHz	1.5 GHz	8.5 $\lambda$ × 8.5 $\lambda$ 19 $\lambda$ × 19 $\lambda$	LP CP	1, 2	65%, 83%	30, 12	Unequal
[39]	11.2–12.9 GHz 28.3–29.6 GHz	1.7 GHz 1.3 GHz	8.8 $\lambda$ × 8.8 $\lambda$ 21 $\lambda$ × 21 $\lambda$	DCP	±1, ±2	96%, 93%	12, 9	Unequal
this work	11.75–13.15 GHz 29.5–30.5 GHz	1.4 GHz 1 GHz	8 $\lambda$ × 8 $\lambda$ 19 $\lambda$ × 19 $\lambda$	DLP, DCP	1, 2, ±3	80%, 79%, 70%, 65%	9	equal

presents a comparison between our work and relevant domestic and international studies. This design generates OAM modes (+1, +2, ±3) surpassing prior antennas. Compared to traditional annular or disk partitioning strategies [34], by integrating a partitioned dual-disk structure with zero-matching-based divergence control, we enhance aperture efficiency, mode purity, and OAM channel number. Unlike [34,35,36], our spin-decoupling approach enables independent multi-polarization control. Versus a uniform circular array antenna [29,37], the dual-layer co-aperture reflective metasurface eliminates complex feeding networks. Furthermore, HOBBs via equivalent axicon effects achieve longer ranges than same-mode vortices in [26,38]. The metasurface ultimately delivers four-channel multi-polarized Bessel vortex beams with equal divergence angle across Ku/Ka bands and extended transmission.

.

4. Conclusions

In summary, this paper innovatively employs polarization-multiplexed units, improving the partitioning strategy by using the inner disk to generate x-polarized vortex beams while the entire disk simultaneously generates y-polarized vortex beams. Furthermore, frequency multiplexing is added via spin-decoupling principles in the upper-layer array. This enables the generation of more OAM modes within the same aperture. Finally, based on a divergence angle control method that adjusts the array surface size, vortex beams of different modes are controlled to maintain equal divergence angle. For a given aperture size, our design achieves smaller divergence angles and higher mode purity. Finally, the fabricated reflective metasurface was experimentally characterized, with results verifying the successful generation of Bessel vortex beams carrying +1, +2, and ±3 OAM modes with equal divergence angle. These research results provide a promising solution for spatially overlapping multiplexed OAM in wireless communication systems, and have application potential in OAM communication and radar detection.