Efficient generation of polarization multiplexed OAM using levitated metasurfaces

Abstract

Dual-polarization (DP) vortex waves (VWs) are widely applied in optical, electromagnetic, and quantum science owing to their ability to simultaneously convey two distinct and non-interfering orbital angular momentums (OAMs). Here, we propose a lightweight levitated meta-atom to achieve 360° phase control with a difference of no more than 1° while maximizing the reflection efficiency. In combination with convergent phase modulation, a OAM metasurface array that facilitates the generation of DP VWs with high mode purity and low divergence angles was designed. The measured DP VW bearing mode l = 1 had only 4° divergence angle and 84% mode purity at 5.8 GHz. Furthermore, DP VWs with integer, fractional (l = 1.5) and higher order (l = 8) modes are discussed based on an OAM purity spectrum analysis. The experimental results were consistent with the simulation results, demonstrating the practicality of the proposed DP OAM metasurface and its potential applications in the field of multithreaded communication systems.

Graphical abstract

Keywords

dual-polarization / orbital angular momentum / low divergence angles / high mode purity

1 Introduction

Orbital angular momentum (OAM), a fundamental physical quantity with extra degrees of freedom [1], has attracted significant research interest. The distinguishing characteristic of vortex waves (VWs) is the wavefront phase factor exp($\mathrm{i}$$l\phi$), where $l$ denotes the OAM mode number, and $\phi$ represents the azimuthal angle. In particular, $l$ can take on arbitrary values, and different modes are reciprocally orthogonal and independent [2-5]. In 2011, a groundbreaking experiment was conducted in the Venetian lagoon, demonstrating the concurrent delivery of two different data within the same frequency band using VWs [6]. This experiment verified that VWs could provide an additional dimension to the capacity of communication systems without increasing the bandwidth [7].

The basic principle for generating VWs bearing OAM is to introduce a continuous current of phase $l\phi$ along the circumference. In the original method of generating OAM beams, designers used spiral phase plates [8-11], spiral paraboloids [12, 13], and holographic phase plates [14] made of all-dielectric materials in order to fulfill the rich phase information of OAM waves. This presented a significant challenge in the construction of ultrathin components. In addition, circular antenna arrays could generate OAM waves through discrete currents, however, they required complex feed networks [15-18]. Metasurfaces have recently become excellent platforms for OAM beam generation, optical imaging and ultra-wideband filters owing to their remarkable ability to manipulate the amplitude, phase, and polarization of reflected waves [19-26]. Based on the robustness of Pancharatnam−Berry (PB) phase atoms, many circularly polarized OAM generators have been proposed [27-33], some of which can generate ultra-wideband OAM [34], OAM with low crosstalk modes [35], OAM with mode spectrum tailoring [36], and dual-band OAM [37]. However, circular polarization (CP) only and single-polarization modulation limit these metasurfaces, ignoring the significant potential of dual-polarization (DP) OAMs in improving channel capacity and the availability of linear polarization (LP) OAMs in wireless communications.

In view of the higher requirements on information transmission rate and channel capacity for 6G wireless communications, DP OAM technology has become a research hotspot. Moreover, in existing wireless communication devices, the base station antenna are typically DP antennas containing x- and y-pol [38, 39]. The DP OAM metasurface transmits different modes of OAM in x- and y-pol, thereby becoming a 6G candidate technology to extend the communication capacity and increase the multithreaded communication rate. Unfortunately, for DP OAM metasurfaces based on the transport phase theory, it is difficult to achieve high purity and low divergence angle because of lack of precise phase control. For example, a hollow cross-shaped metasurface realized multi-beam and DP OAMs at two terahertz frequencies [40]. However, this complex structure generated many crosstalk modes while realizing OAM polarization multiplexing, which degraded the performance of the DP VWs. The four-layer metasurface generated a DP OAM with high mode purity but a large divergence angle at 7.5 GHz [41]. This subjected the DP OAM to severe distortion during wireless transmission, thereby increasing the bit error rate. Thus, considering practical applications, it is essential to minimize the divergence angle while improving the DP OAM performance.

Herein, we introduce a lightweight levitated metasurface that combine precise OAM phase modulation with beam convergence phase modulation to produce DP OAM waves with a low divergence angle and high mode purity. The basic unit enables 360° phase coverage while maintaining a high reflectance of not less than 0.99. The feasibility of generating DP VWs through the arrangement of metasurface units with different structural parameters has been theoretically validated. The simulation result showed that a DP OAM wave with a divergence angle of only 4° and 84% mode purity was generated at 5.8 GHz. In addition, DP VWs with integer, fractional ($l$ = 1.5) and higher-order ($l$ = 8) modes are discussed with the OAM purity spectrum. The experiment and simulation results were in excellent agreement, confirming the reliability of the introduced metasurface generated DP OAM. The proposed metasurface has potential applications in multithreaded wireless communication systems.

2 Method and design

Fig.1 shows the schematic diagram of the DP OAM metasurface. The metasurface consists of M × N elements with different structural parameters and is illuminated by a feed antenna at the position vector ${\mathit{r}}_f$.

Fig.1 Schematic of dual-polarization and dual-mode OAM generating levitated metasurface.

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Considering two orthogonal polarizations, the EM fields radiated by the metasurface can be represented as a summation of the radiated fields from elements. With an arbitrary direction $\mathit{u}$, the electric field can be described as

(1)$\begin{array}{rl}{E}_i(\mathit{u})=& \sum _{m=1}^M\sum _{n=1}^{N}F({\mathit{r}}_{mn}\cdot {\mathit{r}}_f)A({\mathit{r}}_{mn}\cdot {\mathit{u}}_0)A({\mathit{u}}_0\cdot \mathit{u})\\ & \times \exp \left\{j\left(k_0|{\mathit{r}}_{mn}-{\mathit{r}}_f|+{\mathit{r}}_{mn}\cdot \mathit{u}+j{\varphi }_{mn}^i\right)\right\},\end{array}$

where $F$ is the feed radiation pattern function, ${\mathit{r}}_{mn}$ is the position vector of the mn-th cell, $k_0$ is the propagation constant in vacuum, ${\mathit{u}}_0$ is the main beam direction, and $A$ is the radiation function of the metasurface element. Based on the OAM phase characteristics, the phase shift required on each cell to generate the desired mode of a DP OAM wave is expressed as follows:

(2)$\begin{array}{rl}{\varphi }_{mn}^i=& l_i{\phi }_{mn}-(k_0|{\mathit{r}}_{mn}-{\mathit{r}}_f|+{\mathit{r}}_{mn}\cdot \mathit{u}),\\ & l_i=0,\pm 1,\pm 2,\dots ,\end{array}$

where $i$ = $x$ or $y$ and the number of OAM modes carried by the x- or y-pol reflected wave is denoted as $l_i$. The azimuthal angle of the n-th cell is ${\phi }_{mn}$.

Fig.2 shows the top and side views of the introduced metasurface unit (metal thickness = 0.035 mm) imprinted on a dielectric layer, F4B (${\epsilon }_r$ = 2.65 and $\tan \theta$ = 0.007). The thickness of dielectric layer F4B was t = 1 mm. The air layer of the cell suppressed the magnetic coupling between the dielectric layer and metal backplane. Thereafter, the thickness of the air layer was parametrically optimized to achieve 360° phase control at high reflectivity, and t₁ = 5 mm. The other structural parameters were w = 2 mm, d = 1 mm, a = 8 mm, and p = 25 mm, and the reflected wave phase was controlled by adjusting L_x and L_y.

Fig.2 (a) Top and (b) side view of proposed levitated meta-atom.

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In the simulation, the full-wave simulation software, CST Studio Suite, was used to analyze the behavior of the structure. The boundary conditions were defined as a unit cell in the x- and y-directions; the z-direction was defined as an open boundary, and the LP wave was illuminated along the −z-direction. By adjusting the structural parameters (L_x and L_y) of the DP OAM unit, the full phase coverage from 0° to 360° of the x- and y-polarized reflected waves was achieved, respectively, with a reflectance above 0.99. Fig.3 shows the reflectance and reflection phase of the DP OAM unit. In addition, we simulated the reflectance and phase of the y-polarized wave with different structural parameters of L_x. Owing to the periodicity of the phase, the phase difference at L_y = 16.5 mm was extremely small. Thus, the reflectance and phase curves of the y-polarized reflected wave for different L_x values practically coincided, indicating that the designed device had excellent polarization isolation.

Fig.3 The reflectance and phase under the LP incident waves.

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For phase modulation, 12 structural parameters were selected based on the precise phase control of the levitated structure, with a phase gradient of 30° (difference ≤1°) in the x- and y-pol directions. Tab.1 lists the parameters. Owing to the isolation of the DP OAM cell in both axes, the phase modulation of the y-pol coincided with that of the x-pol. Taking the phase modulation in the x-pol direction as an example, the superposition of the OAM and the convergence phases reflected the converging OAM wave. The converging phase satisfied

Tab.1 Structural parameters with a phase gradient of 30°.

Phase in $\mathit{x}$-pol
EP (°)	−30°	−60°	−90°	−120°	−150°	−180°	150°	120°	90°	60°	30°	0°
Lx (mm)	24.5	8.03	9.2	11.5	15.04	16.6	17.7	18.44	19.2	20.04	21.2	22.8
Phase (°)	−30°	−60°	−90°	−120°	−150°	−180°	150°	121°	91°	61°	31°	0°
PD (°)	0°	0°	0°	0°	0°	0°	0°	1°	1°	1°	1°	0°
Phase in $\mathit{y}$-pol
EP (°)	−30°	−60°	−90°	−120°	−150°	−180°	150°	120°	90°	60°	30°	0°
Ly (mm)	24.6	8.11	9.32	11.6	15.2	16.62	17.66	18.4	19.3	20.02	21.2	22.8
Phase (°)	−30°	−61°	−90°	−120°	−150°	−180°	150°	119°	90°	60°	30°	0°
PD (°)	0°	1°	0°	0°	0°	0°	0°	1°	0°	0°	0°	0°

1) EP: Expected phase;2) PD: Phase difference.

(3)${\varphi }_1(m,n)={\displaystyle \frac{2\pi f_c}{c}}(\sqrt{(mp{)}^2+(np{)}^2+{z_f}^2}-z_f)+{\varphi }_0,$

where $f_c$ is the operating frequency, $z_f$ is the distance of the feed antenna to the metasurface, c is the light velocity in free space, ${\varphi }_0$ is the phase at the center of the metasurface, and $(m,n)$ is the coordinate position of the metasurface unit relative to the origin.

The OAM phase satisfied Eq. (3). The proposed design contains a square array with dimensions of g = 475 mm × 475 mm (19 × 19 unit cells). The phase distributions of the convergence phase [Fig.4(a)], the OAM phase for $l$ = 1 [Fig.4(b)] and overall phase compensation [Fig.4(c)] were computed. Fig.4(d) shows the distribution of the introduced cells with a gradient of 30° phase in the metasurface.

Fig.4 Phase distribution. (a) Focusing phase compensation. (b) OAM phase factor. (c) Phase distribution of converging vortex waves. (d) Distribution of units.

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The presence of loss and array patch coupling in the generation of OAM waves indicates that the resulting wave comprised a combination of multiple modes rather than a single mode. It was imperative to conduct a quantitative analysis to determine the proportion of each mode present. The periodic nature of the azimuth angle $\phi$ suggests a Fourier transform between the OAM mode $l$ and the azimuth angle $\phi$, as evidenced by a comparison with the defining equation of the Fourier transform. Consequently, the OAM spectrogram was obtained by decomposing the individual modes in the OAM wave according to the Fourier transform. Taking the x-pol OAM wave as an example, the relation can be expressed as

(4)$\{\begin{array}{l}{E}_{l_x}={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_0^{2\pi }\varphi (\phi ){\mathrm{e}}^{-\mathrm{j}{l}_x\phi }\mathrm{d}\phi ,}\\ \varphi (\phi )={\displaystyle \sum _{l_x}{E}_{l_x}{\mathrm{e}}^{\mathrm{j}{l}_x\phi },}\end{array}$

where $\varphi (\phi )$ is the sampled field function along the z-axis. The energy weights of the OAM modes ($l$ = −4 to $l$ = 4) in the observation plane perpendicular to the OAM wave radiation direction were defined as follows:

(5)$\text{Energy}\text{weight}={\displaystyle \frac{E_{l_x}}{\sum _{l=-4}^4{E}_{l_x}}}.$

3 Simulation and discussion

Here, two DP OAM metasurfaces of identical sizes were compared. The first metasurface comprised DP OAM with convergent phases, whereas the second metasurface only had OAM phases. Both metasurfaces exhibited the same expected OAM modes, with the x- and y-pol bearing the OAM modes $l$ = 1 and $l$ = −1, respectively. The metasurfaces were simulated using the CST software with time-domain solution, and the horn antenna was placed 300 mm from the metasurface as an excitation for transmitting LP waves. The boundary conditions were defined as open boundaries. Fig.5 shows the near-field simulation results for the two metasurfaces.

Fig.5 Near-field observation and OAM purity spectrum of vortex beam with $l$ = 1 in x-pol and $l$ = −1 in y-pol.

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The dimensions of the near-field sampling surface were 480 mm × 480 mm, with a distance of 300 mm from the metasurface. In the simulation results lacking superimposed convergent phase modulation, the x-pol component exhibited an OAM mode number of 1 with a clockwise phase shift direction ranging from 0 to $2\pi$. However, the y-pol component had an OAM mode number of −1 with an opposite phase shift direction. The anticipated OAM was evident in the donut-shaped amplitude distribution. Contrarily, the simulation results demonstrated that the incorporation of convergent phase modulation led to a reduction in the amplitude diffusion angle of the resulting DP OAM and the emergence of a vortex-arm-like phase pattern.

The OAM spectra were analyzed through the Fourier spectral analysis of the primary lobe of the resulting OAM beam. The OAM purity of the convergent vortex wave (CVW) generation from the DP OAM metasurface with a superimposed convergent phase was 84%, which was 5% lower than that of the non-converged OAM generation from the DP OAM metasurface with only an OAM phase (89%).

Two primary factors contributed to the decrease in the purity of the CVW mode. First, the converging phase amplified the intensity of the pencil-shaped plane wave in the reflected wave, leading to an increase in the component of mode $l$ = 0 in the purity spectrum. Second, CVWs are more significantly impacted by the blocking effect of the source, resulting in a greater loss of the electric field of the reflected wave.

Next, we examined the comparison between CVWs and VWs through far-field simulations. The radiation patterns of waves at 5.8 GHz (Fig.6) showed that the CVW exhibited a hollow pencil-shaped wave with enhanced directionality while preserving its OAM characteristics. The main lobe gain of the CVW increased from 13 to 22 dBi compared with the VW. In addition, the main lobe divergence angle of the CVW was 4°, which was considerably smaller than that of the VW. This indicated that the convergent phase modulation exhibited high gain and a small divergence angle, demonstrating good transmission performance.

Fig.6 Far-field radiation patterns of the converging vortex waves (CVWs) and vortex waves (VWs). (a) Three-dimensional radiation patterns of the VWs and (b) CVWs. (c) One-dimensional radiation pattern.

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The polarization multiplexing of fractional modal OAMs has not been previously investigated. Thus, we conducted simulations of dual-polarized VWs for modes $l$ = 1.5 and $l$ = 3. Fig.7(a) shows the x-polarized near-field electromagnetic (EM) performance for the OAM mode number of 1.5. The amplitude zero point was positioned at the center of the beam and extended in the −x-direction. The phase exhibited a pattern of 1.5 spiral arms. Fig.7(b) shows the y-pol near-field EM performance, at an OAM mode number of 3; the near-field characteristics were consistent with integer OAM modes.

Fig.7 Near-field observation and OAM purity spectrum of fractional order vortex wave. (a) $l$ = 1.5 in x-pol. (b) $l$ = 3 in y-pol.

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Fig.7(a) shows the results of the Fourier spectral analysis of the fractional modes, where the energy was fully distributed in the fractional modes, indicating that the fractional-order OAM was also orthogonal. As shown in Fig.7(b), the energy was mainly distributed in the desired mode $l$ = 3, and the other energies were primarily concentrated in mode $l$ = 0, consistent with the properties of CVWs. This further supported the suitability of fractional modes for polarization multiplexing in OAM applications.

The sampling surface was shifted to a distance of 1000 mm, and the x-pol component bearing the OAM mode $l$ = 1.5 shown in Fig.8(a) was subjected to Fourier analysis. Contrary to the near-field results, the OAM beam for mode $l$ = 1.5 was observed to exhibit significant energy spreading and a blurred phase distribution. In addition, the purity of the mode reduced by 27%. Conversely, as shown in Fig.8(b), the y-pol component of mode $l$ = 3 demonstrated significantly less energy spread with an increase in distance, and the phase distributions of the three vortex arms remained discernible. Noteworthily, the x-pol component bearing mode $l$ = 3 maintained the highest amplitude with a decay of only 8%. The fractional-order mode exhibited a greater diffusion effect while maintaining orthogonality with the DP OAM.

Fig.8 Far-field observation and OAM purity spectrum of fractional order vortex beam. (a) $l$ = 1.5 in x-pol. (b) $l$ = 3 in y-pol.

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Furthermore, a DP VW simulation was conducted for higher-order OAM modes with $l$ = 8 and integer-order OAM modes with $l$ = 3. The near-field spectral analysis of the y-polarized component of the $l$ = 8 mode shown in Fig.9 revealed that, although mode $l$ = 8 retained the largest energy weight, it was significantly attenuated compared with mode $l$ = 3 in the x-polarized component. The x- and y-polarized components exhibited distinct vortex arms in the phase distribution. However, the amplitude analysis revealed that the spreading effect of the OAM wave was evident in the higher mode. The significant divergence angle of higher-order VWs led to increased spectral crosstalk during transmission, resulting in reduced purity and energy dispersion. The generation of fractional-order OAMs and higher-order OAM modes that align with theoretical predictions demonstrated the effectiveness and adaptability of the metasurfaces for multithreaded communication.

Fig.9 Near-field observation and OAM purity spectrum of a high-order vortex beam.

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4 Experimental verification

For the experiment, the near-field characteristics of the DP vortex beam were measured in a microwave darkroom with a sample comprising a grid of 19 × 19 cells with a size of 475 mm × 475 mm. Fig.10 shows the setup including the test environment. The LP wave excitation was produced using a horn antenna positioned 300 mm from the sample. The DP vortex wave was received using a near-field scanning system, and the observation surface was swept by a probe connected to a sweep frame located 400 mm from the metasurface. The size of the observation plane was 600 mm × 600 mm, and the step size of the scan was 5 mm. As shown in Fig.11(a), the energy of the hollow donut shape expanded with an increase in the modes, which are characteristics of OAM waves. Thereafter,the region with the largest energy weight was scanned with a step size of 1 mm (the x- and y-polarized reflected wave observation plane had dimensions of 2R₂ × 2R₂ and 2R₁ × 2R₁, respectively). In Fig.11(a), the phase of the region with the largest energy weight was analyzed, where there were four vortex arms for the x-pol component of $l$ = 4 and one arm for the y-pol component of $l$ = 1. The Fourier analysis of the experiment result [Fig.11(b)] showed that OAM mode $l$ = 4 dominated in the energy distribution of the x-pol OAM wave (approximately 67% energy weight), whereas mode $l$ = 1 dominated in the y-pol component (approximately 74% energy weight). The experimental results were consistent with the simulation results, confirming the multifunctionality and flexibility of our introduced meta-atom for polarization multiplexing in OAM.

Fig.10 (a) OAM metasurface sample. (b) Test environment.

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Fig.11 Near-field sampling results. (a) Reflectance and phase of x- and y-pol reflection vortex beam. (b) Spectral analyses of sampling results.

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For the far-field experiments, the feed antenna was fixed 300 mm from the metasurface on the rotating stage for the excitation of the incident LP waves, and the metasurface was fixed along the direction perpendicular to the ground on the rotating stage. The center of radiation of the feed antenna and the center of the metasurface were aligned. The distance between the receiving antenna and sample was 1000 mm. The radiation patterns of the OAM metasurface were measured using the rotating antenna technique. Double-ridged horn antennas operating within the range of 0.75−18 GHz with voltage standing wave ratio < 2.5 were used as the feed and receive antennas.

Fig.12 shows the radiation patterns of the simulations and measurements of the sample in x- and y-pol. The y-polarization component bearing mode $l$ = 1 exhibited a main lobe divergence angle of only 5°. Contrarily, the x-pol component, bearing OAM mode $l$ = 4, exhibited a beam main lobe divergence angle of 36° and a peak gain of 9 dBi. The measured and simulated radiation patterns exhibited typical characteristics of VWs, indicating that the designed metasurface could generate DP OAM waves with high gain and minimal divergence angles.

Fig.12 Normalized radiation patterns of simulations and measurements. (a) Reflected vortex beam under y-polarized incident waves. (b) Reflected vortex beam under x-polarized incident waves.

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A slight discrepancy was observed between the simulation and measurement of the radiation direction map. This error may have been caused by the small deformation of the dielectric layer during the assembly of the metasurface samples, which rendered the thickness of the air layer between the dielectric layer and bottom layer nonuniform and not accurate to 5 mm. Furthermore, manufacturing tolerances and experimental environment can lead to certain unavoidable errors.

Finally, Tab.2 presents a comparison with previous studies. The proposed design, which enables the independent modulation of the OAM in two polarization channels, exhibited superior characteristics in terms of OAM purity and divergence angles, compared with existing OAM metasurfaces featuring single-polarization or multi-polarization multiplexing channels.

Tab.2 Comparison of OAM generation metasurfaces between the proposed metasurface and reported metasurfaces.

Ref.	Type	Number of polarizations	OAM purity	Divergence angle (l = 1)	Layer number
[42]	TMS	2(LP)	N/A	3.5°	1
[43]	TMS	1(LP)	N/A	5°	5
[44]	TMS	1(LP)	N/A	9°	3
[34]	RMS	1(CP)	70%	N/A	1
[45]	RMS	4(CP & LP)	96%	N/A	3
[46]	RMS	2(CP)	62.7%	20°	1
[47]	RMS	2(CP)	96%	5.75°	3
[48]	RMS	2(LP)	N/A	3°	1
[49]	TMS	2(LP)	82%	6.5°	2
[50]	TMS	2(CP)	72%	N/A	2
[51]	RMS	1(CP)	80%	3°	2
This work	RMS	2(LP)	84%	4°	1

1) TMS: Transmissive metasurface;2) RMS: Reflective metasurface;3) LP: Linear polarization;4) CP: Circular polarization.

5 Conclusion

Here, we introduced a highly reflective levitated metasurface element that can precisely manipulate a 0°−360° phase shift (differences ≤1°). Using this element, we developed a DP OAM metasurface, facilitating the simultaneous transmission of two OAM signals with distinct mode numbers at a frequency of 5.8 GHz. By incorporating convergent phase modulation, we successfully minimized the divergence angle of the DP OAM waves. The simulation results demonstrated that the generated DP VW of the OAM mode $l$ = 1 had only a 4° divergence angle, and the mode purity reached 84%. Furthermore, the analysis of DP VWs with integer, fractional, and higher-order modes, along with the examination of the OAM purity spectra, clearly depicted the transmission properties of the DP OAM. This holds significant potential in OAM polarization multiplexing toward enhancing multithreaded communication.

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Declarations

The authors declare no competing interests and no conflicts.

Acknowledgements

This work was supported by the Key Research and Development Program of Shaanxi Province (No. 2022GD-TSLD-64).

RIGHTS & PERMISSIONS

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