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Abstract
This work presents a theoretical design and experimental demonstration of a transmissive microwave metasurface for generating dual-vector vortex beams (VVBs). The proposed metasurface consists of an array of pixelated dartboard discretization meta-atoms. By rotating the meta-atoms from 0° to 180°, a Pancharatnam-Barry (P-B) phase covering the full 360° range is achieved, with a transmittance exceeding 90% over the frequency range from 9.7 to 10.2 GHz. The measured results demonstrate that when a linearly polarized microwave normally impinges on the metasurface, the transmitted beams correspond to the dual VVBs with different directions. A good agreement among Poincaré sphere theory, full-wave simulation, and experimental measurement is observed. This proposed transmissive microwave metasurface for VVBs may offer promising applications in communications and radar detection.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Structured beams, including vortex beams (VBs) and vector vortex beams (VVBs), have garnered significant attention in recent years due to their unique properties and the additional degree of freedom they offer for manipulating electromagnetic waves [1,2]. Vortex beam generation has been extensively studied, which can carry orbital angular momentum (OAM) and have potential applications in various fields such as communication [3–5]. The VVB are a type of vortex beam that carry both OAM and spin angular momentum (SAM). The VVBs feature an anisotropic polarization distribution in the transverse plane, such as radial polarization. The analysis of VVBs is enriched by the utilization of hybrid-order Poincaré spheres (HyOPS), providing a powerful theoretical tool for understanding the intricate polarization characteristics and interactions of these vector beams [6–8]. As a result, VVBs show high polarization sensitivity and have a significant impact in holography [9,10], quantum communication [11,12], high-resolution lithography [13,14], and encryption [15].
Metasurfaces, a class of structured interfaces with varying profiles of structures, are currently undergoing rapid development for the manipulation of electromagnetic wavefronts in terms of amplitude, phase, and polarization [16–25]. By employing artificially engineered meta-atoms, metasurfaces can locally modify the amplitude, phase, and/or polarization of incident waves, thereby affecting the transmitted or reflected waves. Since their inception, metasurfaces have emerged as a promising platform for manipulating the SAM and OAM of electromagnetic waves, offering a flexible and compact means to generate VVBs [26–43]. However, techniques for generating VVBs using metasurfaces often require complex amplitude modulations or complex multilayer structures [44–54], which could impede practical implementation and widespread adoption. The development of simple microwave metasurfaces capable of generating VVBs is highly anticipated in wireless communications and radar detection areas, where such techniques are rarely reported due to the complexity of inter-element coupling [55].
In this study, we propose a simple single-layer metasurface approach to generate dual VVBs in the microwave frequency range. The meta-atom is composed of pixelated dartboard discretization structures printed on both sides of a substrate layer, ensuring stable amplitude and precise phase modulations [56]. Utilizing the HyOPS theory, the metasurface, consisting of these meta-atoms, generates a specific P-B phase distribution [57–59]. When a normally incident linearly polarized (LP) microwave interacts with the single-layer metasurface, the geometric P-B phase controlled meta-atom can generate dual VVBs with different directions. These VVBs exhibit different polarization orders according to the theoretical description of the Poincaré sphere. Furthermore, a prototype of the metasurface sample is fabricated using the low-cost standard printed circuit board (PCB) technique. The experimental results are in excellent agreement with the theoretical and simulation results, demonstrating the generation of dual VVBs with different directions.
2. Theoretical analysis
Figure 1 shows the schematic of the transmissive microwave metasurface for generating vector vortex beams. When a linearly polarized wave impinges normally on the metasurface along the + z direction, the transmitted Beam1 and Beam2 correspond to the dual VVBs with different directions and the same polarization order. The VVB denoted as $|\psi \rangle$ can be synthesized by the right circularly polarized (RCP) and left circularly polarized (LCP) vortex beams carrying different absolute values of topological charges. This can be expressed as follows:
where $|{\psi_L^n} \rangle $ and $|{\psi_R^m} \rangle $ are complex coefficients corresponding to the initial amplitudes and phase information of LCP and RCP components, respectively. $\{{{L_n},\textrm{ }{R_m}} \}$ forms an orthonormal basis for LCP and RCP, with topological charges n and m, respectively.Figure 2 illustrates the HyOPS and the state-of-polarization corresponding to VVB. On the HyOPS, the north and south poles correspond to the RCP and LCP vortex beams, represented as |R + M› and |L-N›, respectively. The Beam1 is located on the equator of the HyOPS-n, +m, while the Beam2 is on the equator of the HyOPS-m, +n. The VVB located on the equator of the HyOPS can be approximated as follows:
The topological Pancharatnam charge is defined as the number of times the phase changes by 2π when circumventing the phase singularity in the helical misalignment of equiphase surfaces of the Pancharatnam phase, which can be approximated as follows:
From the Eqs. (1) and (2), when $\psi _L^n$ and $\psi _R^m$ are the same and set to 1 for ease of analysis, the VVB can be expressed as follows:
The polarization order p0 and topological Pancharatnam charge lp of the VVB can be obtained as follows:
The principle for generating the dual VVBs is illustrated in Fig. 3. The metasurface deflects the incident LCP and RCP waves towards the left and right directions, respectively, and flips their spin state, generating vortex beams with different topological charges. When the metasurface is illuminated by a LP wave, the LCP and RCP vortex beams superimpose with each other, forming the dual VVBs. The P-B phase distribution of the metasurface is expressed as follows:
Fig. 3. Schematic of the different incident waves imping the metasurface: (a) LCP, (b) RCP, and (c) LP.
As depicted in Fig. 3(a), the metasurface is illuminated by LCP incident wave that is decomposed into two RCP waves with different topological charge. The vortex wave |R + M› has a topological charge of + M and is oriented in the direction of the polar angle ϑ and azimuthal angle −180°. Similarly, the vortex wave |R + N› has a topological charge of + N and is oriented in the direction of the polar angle ϑ and azimuthal angle 180°. When RCP plane wave is incident, compared with the LCP, the metasurface generates two LCP vortex beams with opposite sign P-B phases and same amplitudes, as illustrated in Fig. 3(b). Hence, when the metasurface are illuminated by a LP incident wave, the dual VVBs are generated with opposite sign topological Pancharatnam charges and same sign polarization orders in two directions with mirror symmetry, as depicted in Fig. 3(c). The dual VVBs, characterized by polar angle and azimuth angle (ϑ, φ), can be represented as follows:
3. Results and discussion
To demonstrate the proposed methodology for VVB generation, the metasurface is designed to generate the VVBs with polarization order of −2 where M = 3, N = 1, operating in the microwave range. The transmission metasurfaces are constructed and simulated in the CST Microwave Studio Suite software. The excitation source is a Gaussian beam with x-direction polarization.
Figure 4(a) and (b) depict a meta-atom employed for generating dual-VVBs. The top and bottom metallic layers exhibit the same patterns, where the pixels filled with copper are colored in brass as “1” and those without filling are colored dark cyan as “0”. The metallic layer consists of an external metal ring and an internal metal pattern. The external metal ring is designed to reduce variations in coupling between the meta-atoms. The annular region between the outer ring and the central metal circle is uniformly segmented radially and angularly from the center of the meta-atom, resembling shapes seen on a dartboard. Subsequently, a differential evolution algorithm is employed to obtain a topological structure with the widest bandwidth and stable P-B phase. The radius of the topological structure is 4.4 mm. The middle substrate, comprised of F4B (ɛr = 2. 65, tan δ = 0. 001) is arranged in a hexagonal period 7.2 mm with thickness of 2 mm.
Fig. 4. (a) The schematic diagram of the meta-atom with (b) pixelated dartboard discretization. (c) Transmission magnitude and (d) P-B phase of the meta-atom versus rotation angle and frequency.
The performance of the meta-atom at different rotation angles γ along the center is evaluated, as shown in Fig. 4(c) and (d). The meta-atom exhibits a transmittance exceeding 90% over the frequency range from 9.7 to 10.2 GHz. The magnitude of the meta-atom remains constant as the rotation angle varies from 0° to 180°. Simultaneously, the P-B phase changes cover the full 360° range, which is twice the rotation angle. Overall, the meta-atom demonstrates high transmission and precise 360° P-B phase control across a wider microwave frequency range.
Figure 5(a) illustrates that the P-B phase distribution ϕ of the metasurface. When M = 3, N = 1, the dual VVBs with polarization order of −2 are obtained in the (38°, 180°) and (38°, 0°) directions at 9.8 GHz. The Beam1 is located on the equator of the HyOPS−1, + 3, while the Beam2 is on the equator of the HyOPS−3, + 1. The meta-atom rotation angle corresponding to each position is γ=ϕ/2. The metasurface consists of 1140 meta-atoms. Right-handed vortex beams with topological charges of +3 and +1 are also obtained in the (38°, 180°) and (38°, 0°) directions, respectively. Meanwhile, left-handed vortex waves with topological charges of −1 and +3 can be obtained under same rotation angle distribution.
Fig. 5. (a) P-B phase distribution of the metasurface, (b) the fabricated sample of microwave VVB metasurface, and (c) the measurement environment.
By controlling rotation angle distribution of the geometric phase meta-atoms, the dual VVBs can be obtained. The VVB metasurface was manufactured using printed circuit board (PCB) technology, and the resulting sample is illustrated in Fig. 5(b). The distance from the horn to the array is 700 mm, and the distance from the array to the probe is 130 mm. To assess the VVB metasurface's performance, a near-field testing system was utilized, as shown in Fig. 5(c). A linearly polarized horn antenna served as the transmitting antenna. Following the scanning process using the receiving probe, the system was capable of directly generating far-field directional patterns for x/y linear polarization as well as left/right circular polarization.
Figure 6 illustrates the simulation and measured results of the VVB metasurface. The amplitude and phase distributions of the Beam1 and Beam 2 under different polarizations received in two different directions have been obtained. The polarization states of the two beams exhibit a counterclockwise rotational variation completing two cycles within one 2π period, which indicate polarization orders p0 = −2 at 9.8 GHz frequency. The far-field results clearly show the characteristic intensity patterns of the x-polarization and y-polarization components. It is evident that the generated dual VVBs exhibit a four petals shape with hollow characteristics. In the case of x-polarization, the four petals are distributed along the ±45° directions, whereas in the y-polarization scenario, the four petals are distributed along the 0° and 90° directions, which conform to polarization order of −2 characteristic.
Fig. 6. Simulated and measured results of Beam1 and Beam2. The amplitude distributions of x-polarization and y-polarization for 9.8 GHz. The amplitude distributions and phase distributions of RCP and LCP for 9.8 GHz.
The two generated vector vortex beams are synthesized from vortex beams under left-handed and right-handed circular polarizations. It is crucial to observe both their left-handed and right-handed components to determine their constituent elements. For RCP reception, the amplitude distributions of the two beams show larger and smaller ring-shaped patterns, corresponding to vortex waves with topological charges of absolute values 3 and 1, respectively. In terms of RCP phase distributions, counterclockwise rotations complete three and one cycles of 2π changes, indicating the presence of components |R + 3› and |R + 1›. Conversely, for LCP reception, the amplitude distributions of the two beams are reversed, completing one and three cycles of 2π changes with clockwise phase increments, indicating the presence of |L−1› and |L−3› components. The amplitude, phase distributions of simulated and measured results affirm the consistency of the dual VVBs topological charges with theoretical values. The far-field results further indicate that Beam1 contains component |R + 3› and |L−1›, while Beam2 contains component |R + 1› and |L−3›. The experimental results are in excellent agreement with the theoretical and simulation results. The slight differences between the amplitude distribution of measured results and simulation results occur because a horn antenna was used as the transmitter, and its directivity is slightly inferior to the Gaussian source used in the simulations, resulting in larger diffraction effects.
For evaluating the quality of vortex beams, the purity of OAM modes can be calculated according to Eqs. (10–12).
Fig. 7. (a) The mode purity results of Beam1 and Beam2 under RCP and LCP for 9.8 GHz. (b) The power ratio results of Beam1 and Beam2 at 9.7 GHz, 9.8 GHz and 10.2 GHz frequencies
In Table 1, the performance of the VVB generation is compared with previous works in the field. As it can be seen, the method can generate four modes of high-purity vortex waves under two circular polarization. Dual-VVBs with high transmission efficiency can be obtained in the microwave frequency range.
Table 1. Comparison of the reported metasurfaces for vortex generation and this work.
4. Conclusion
In conclusion, we have experimental demonstrated a simple transmissive metasurface to achieve microwave dual VVBs. Based on a rigorous theoretical description of Poincaré sphere, when a linearly polarized microwave impinges on the metasurface, the transmitted beams correspond to the dual-VVBs with different directions. Utilizing the meta-atom of the pixelated dartboard discretization enables precise control of the P-B phase. Subsequently, employing the designed metasurface achieves the synthesis of left and right circularly polarized vortex waves in the same direction, resulting in the formation of dual VVBs. The experimental results are in qualitative agreement with full wave simulations. The metasurface design not only simplifies the fabrication process but also enhances work efficiency. We believe that microwave structured VVBs could expand the communication capacities and detection accuracy, opening up potential applications in wireless communications and radar detections.
Funding
National Key Research and Development Program of China (2022YFF0604801); National Natural Science Foundation of China (62071291, 62171186, 62201037, 62271056).
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.
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