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Abstract
Metasurfaces are ultra-thin artificial structures capable of flexibly manipulating electromagnetic (EM) waves. Among various applications, phase modulation of electromagnetic (EM) waves using metasurfaces holds great significance. The Pancharatnam-Berry (P-B) metasurfaces provides a complete 2π phase modulation by simply rotating the meta-atom. However, the fixed lattice in rotation employed by traditional P-B metasurfaces often results in unstable amplitude and imprecise P-B phase, leading to performance degradation. In this work, we demonstrate transmissive P-B metasurfaces with stable amplitude and precise phase modulation. To ensure stable amplitude and precise P-B phase, we adopt a dartboard discretization configuration with a hexagonal lattice for the meta-atom design. By applying topology optimization to the encoding sequence formed by surface pixels and dimensions, we significantly enhancing the high transmissive bandwidth of the optimized meta-atom. Furthermore, the optimized meta-atom exhibits a stable amplitude and precise P-B phase for each rotation angle. As proof-of-concept demonstrations, two metasurfaces for single and multiplexed vortex beams generating are designed utilizing the optimized meta-atom. Both the simulated and measured results indicate high mode purity of generated vortex beams. The design method can also be readily extended to other high performance metasurfaces with stable amplitude and precise phase manipulations, which can enhance the efficiency and capacity of metasurface-assisted holographic imaging and 6 G wireless communication systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Metamaterials have been widely studied in the past decades for its unique electromagnetic (EM) properties which do not exist in nature. Metasurfaces, which are two-dimensional ultrathin structures of metamaterials, enable flexible manipulation of the amplitude, phase and polarization of EM waves. Metasurfaces have found applications in a wide range of fields, such as absorber [1–3], spatial filters [4], beam steering [5–7], holography [8–11], vortex beam generation [12–15], cloaking [16], and polarization conversion [17–20]. According to the outgoing EM wave propagation direction, the metasurfaces could be classified into two types, namely the reflective type and transmissive type. The occlusion of the feed restricts the applications of reflective metasurfaces. In contrast, the transmissive metasurfaces avoid this drawback and have natural advantages in conformal systems.
Phase modulation is a critical aspect in the design of transmissive metasurfaces. There are two major types of metasurface phase manipulation in terms of the operating principle of the meta-atom. The first type is the Huygens’ metasurface [21–31], where the meta-atoms are meticulously engineered to achieve desired phases through adjustments of structural parameters. The second type is the Pancharatnam-Berry (P-B) metasurface [32–44], which exhibit a non-dispersive phase response achieved by rotating a single meta-atom. In theory, each meta-atom on the phase modulation metasurface should have identical amplitude and precise phase as desired. However, in practical metasurface design, achieving stable amplitude and precise phases is challenging, leading to a deterioration in performance.
In this paper, we demonstrate an ultra-thin and compact transmissive P-B metasurface with stable amplitude and precise phase modulation using dartboard discretization configuration. Firstly, a meta-atom featuring circular metal outline within a hexagonal lattice is designed, ensuring stable amplitude and precise P-B phase characteristics. To enhance the bandwidth of the high transmission, a topology optimization is introduced by discretizing the solid circle into sector annular pixels along the radial and angular directions, resembling a dartboard configuration. This optimization approach is well-suited for meta-atom rotation and ensures improved bandwidth for high transmission. Then, as a practical phase modulation application, the optimized design is applied to the generation of single and multiplexed vortex beams. The simulated results show a high efficiency and high purity of vortex. Compared with the reported research, our work has a stable amplitude, precise P-B phase, and miniaturized meta-atom with a low-cost single substrate configuration. This promising advancement opens up possibilities for high-quality vortex imaging and communication. Finally, the samples of the two metasurfaces are fabricated and measured for the verification, and the results obtained from near-field and far-field measurements demonstrate excellent agreement with simulation results.
2. Design strategy and the topology optimization method
2.1 Design strategy and theoretical analysis
Figure 1 depicts the topology optimization process of the meta-atom with dartboard discretization configuration for the high-efficiency transmissive metasurface. As the schematic diagram shown in Fig. 1(a), the hexagonal meta-atom consists of two metallic patterns printed on both sides of a substrate. According to the P-B phase principle, the meta-atom acts as a half-wave plate, and the Jones matrix can be expressed as Eq. (1) with the fast axis along the x direction.
When the meta-atom rotates about the center along z direction with an angle of $\varphi $, the new Jones matrix ${J_\varphi }$ is calculated by the rotation matrix ${R_\varphi }$ [45].
The Jones matrix can also be expressed under the circular base by substituting Eq. (3),
Fig. 1. The design process of the topology optimized meta-atom with dartboard discretization configuration for high efficiency transmissive metasurface. (a) The schematic diagram of the meta-atom with dartboard discretization. The region between the central circle and the outermost annulus is discretized into sector annular pixels, forming a surface pixel matrix by concentric annuli. (b) The encoding sequence composed of the surface pixel matrix (topology) and the dimension of the meta-atom. (c) The topology optimized design and the metasurfaces for the (d) single vortex and (e) multiplexed vortex generating.
However, in the practical design, the structure cannot entirely rotate because of the fixed lattice. In this condition, the coupling between the adjacent meta-atoms varies depending on the relative positions of the patterns as the rotation angle changes, which causes unstable amplitudes and imprecise P-B phases For the square and hexagonal lattices, the amplitude A and the P-B phase ${\Phi ^{\textrm{P - B}}}$ at a given rotation angle ${\varphi _0}$ can be described by the following relationships:
As shown in Fig. 1(c), an optimized meta-atom is obtained for the high-efficiency transmissive metasurface through the topology optimization of the encoding sequence. The optimized design is capable of various phase-only modulation application. As proof-of-concept demonstrations, we utilize the optimized meta-atom to generate a single vortex beam and a multiplexed vortex beam as illustrated in Fig. 1(d) and (e). These examples illustrate the versatility and effectiveness of the optimized meta-atom for generating desired beam patterns.
2.2 Topology optimization method
The binary encoding sequence can be optimized through various algorithms. In this work, the topology optimization method employs the differential evolution (DE) algorithm, and the flowchart is shown in Fig. 2(a). Firstly, an initial population with a size of 200 is randomly generated. Then, the individuals with R < p/2 are constructed and simulated in the CST Microwave Studio Suite software. The materials of the substrate and metal are F4B (${\varepsilon _r} = 2.65$, $\tan \delta = 0.001$) and copper, respectively. Subsequently, the simulated results are converted into the objective values for evaluating the designs in each generation. Here, the optimization objective is defined as the relative bandwidth with the cross-polarized transmission coefficient higher than 0.9 at the center frequency of 9.8 GHz under a circular polarized incidence. The objective function g is calculated as follows:
Fig. 2. (a) The flowchart of the differential evolution algorithm-based topology optimization and (b) the best and mean relative bandwidth value of the whole iteration. The optimized structure is displayed beside the graph.
The following process of mutation, crossover, and selection creates new structures from the best previous designs. In the mutation step, the scaling factor F of 0.5 is chosen. In the crossover step, the exponential crossover is selected with a crossover rate of 0.5. In the selection step, greediness mechanism is applied to create the new generation from the parents and the offspring. Finally, the process repeats until no better individual is produced for 50 consecutive generations, which means the termination condition is met.
The left part of Fig. 2(b) shows the best and mean bandwidth values throughout the whole DE iteration. As the generation number increases, both the best and mean values exhibit significant improvements. The difference between the two curves indicates that population diversity is maintained during the search for the best individual in each generation. This suggests that the solution obtained through our optimization method is close to the global optimal solution. The topology optimization end at the 333rd generation, achieving a best relative bandwidth of 5.18%. The optimized design is displayed in the right part of Fig. 2(b). The pixels filled with copper is colored in yellow as “1”, and the optimized dimensional parameters: p = 7.2 mm, R = 3.4 mm and t = 2 mm.
3. Simulation results
3.1 Topology optimized meta-atom
To evaluate the performance of the optimized meta-atom, the cross-polarized transmissions at different rotation angles are simulated under an LCP incidence propagating along the + z axis. As shown in Fig. 3(a), the magnitudes of the cross-polarized transmission coefficients for different rotation angles $\varphi $ ranging from 0° to 60° overlap with each other. The cross-polarized transmission coefficients remain above 0.9 from 9.72 to 10.23 GHz, with a maximum value of 0.93 in this band. Additionally, Fig. 3(b) depicts the constant shift in transmission phases. To provide further insight into the relationship between the phase shift and the rotation angle, Fig. 3(c) displays the P-B phases versus frequency for different rotation angles. Obviously, the P-B phase precisely corresponds to twice the rotation angle $\varphi $ across the entire frequency band, validating the suitability of the optimized structure for phase-only modulation.
Fig. 3. (a) The magnitude and (b) the phase of the cross-polarized transmission coefficients for topology optimized meta-atom with different rotation angle $\varphi $. (c) The P-B phase of each rotation angle compared to $\varphi \textrm{ = 0}^\circ $.
3.2 Metasurface for single vortex generating
Firstly, the optimized meta-atom is arranged to generate a single vortex beam. The rotation angle of each meta-atom at position (x, y) is calculated as follows:
where l is the topological charge of the vortex, i.e. the number of OAM mode. The arrangement of meta-atoms is confined within a circular area with a radius of 80 mm on the metasurface.In the simulation, an incident LCP plane wave propagates along the + z axis, and the transmitted electric fields are recorded at the x-y plane, which is located 300 mm away from the metasurface. Figure 4 shows the simulated results of three typical frequencies. Figure 4(a)-(c) and (d)-(f) depict the electric field amplitude and phase distributions on the recording plane at 9.72, 9.8, and 10.23 GHz, respectively. At all the three frequencies, a doughnut-shape amplitude pattern with a central null is observed, indicating the intrinsic characteristic of a vortex beam. Moreover, the spiral gradient phase shift from 0 to ${2{\mathrm{\pi}}}$ in a counter-clockwise direction indicates a topological charge of +1. To evaluate the quality of the generated vortex beams, the OAM mode purity can be calculated from Eqs. (10)–(12) [46].
Fig. 4. The simulated electric field (a)-(c) amplitude, (d)-(f) phase distribution and (g)-(i) corresponding mode purity of a single vortex beam with topological charge l=+1 on the x-y plane at z = 300 mm for 9.72 GHz, 9.8 GHz and 10.23 GHz, respectively. (j)-(l) The radiation pattern of each frequency.
To emphasize the advancement of the precise P-B phase manipulation, we compare this work with the reported researches on transmissive P-B metasurfaces for vortex generation in Table 1. The stability of amplitude and the precision of P-B phase are evaluated using the root mean square error (RMSE). The RMSE-A, which measures the stability of amplitude, is calculated by:
where ${A_0}$ is the cross-polarized transmission coefficient of rotation angle $\varphi = 0$ at the center frequency, and ${A_i}$ is each corresponding cross-polarized transmission coefficient of n rotation angles at the same frequency. Similarly, for the P-B phase, the $\textrm{RMSE} - {\Phi ^{\textrm{P - B}}}$ is achieved by:
Table 1. Comparison of the reported transmissive P-B metasurfaces for vortex generation and this work.
3.3 Metasurface for multiplexed vortex generating
As mentioned above, the precise P-B phase modulation becomes even more significant for the cases with more complex phase distributions. Here, we design a metasurface to generate a multiplexed vortex beam with topological charges of l=+1 and l=+2, with a 1:1 power ratio, to verify the effectiveness of the optimized meta-atom.
In the case of generating multiplexed vortex beam, the phase distribution on the metasurface is determined by:
Substituting l=+1 and l=+2 with a 1:1 power ratio into Eqs. (15) and (16), the theoretical mode purities can be calculated. The metasurface is then arranged according to Eq. (15), and the simulated results are shown in Fig. 5. The simulated amplitude distributions of this design exhibit petal-shaped patterns as depicted in Fig. 5(a)-(c), which are caused by interference of different modes. Intensity nulls at the center are still observed at the three typical frequencies. The phase distributions in Fig. 5(d)-(f) show an interaction of the spiral phases of the two modes. The mode purities are also calculated through Eqs. (10)–(12) and shown in Fig. 5(g)-(i). To evaluate the difference between the actual and desired power ratio at n target modes, the RMSE-M is introduced by:
Fig. 5. The simulated electric field (a)-(c) amplitude, (d)-(f) phase distribution and (g)-(i) corresponding mode purity for the multiplexed vortex beam with topological charges of l=+1 and l=+2 with a 1:1 power ratio on the x-y plane at z = 300 mm for 9.72 GHz, 9.8 GHz and 10.23 GHz, respectively. (j)-(l) The radiation pattern of each frequency.
Table 2. The theoretical and simulated behavior of the multiplexed vortex.
4. Experimental verification
The fabrication of the metasurface prototypes for generating the vortex beams was performed using the standard print circuit board (PCB) technique. The photos of the fabricated metasurfaces for the single vortex and multiplexed vortex generations are shown in Fig. 6(a) and (b), respectively.
Fig. 6. The fabricated sample for (a) single and (b) multiplexed vortex metasurface generating, and (c) the measurement environment.
To measure the performance of the fabricated metasurfaces, a near-field scanning system was employed, as depicted in Fig. 6(c). The system employed a circularly polarized horn antenna operating in the X-band, positioned 700 mm away from the metasurface sample. This horn antenna generated a left circularly polarized (LCP) plane wave, which served as the transmitter for the measurement setup. A field probe, located 300 mm away from the sample, was used to scan the electric field on the recording plane. The field probe scans twice, once with the probe polarized in the x-direction and then in the y-direction, allowing for the acquisition of the x- and y-polarized electric fields. The horn antenna and the field probe are connected to a vector network analyzer, and they are coaxial with the center of the metasurface samples. This configuration ensured accurate alignment and measurement of the electric field distribution on the recording plane. To obtain the cross-polarized electric field on the recording plane, the x- and y-components of the electric field obtained from the two scans with the field probe are synthesized. This provided a comprehensive representation of the electric field distribution. The far-field results are analyzed using the software that accompanied the measurement system.
The near field measurement results of the single vortex generation are shown in Fig. 7. In Fig. 7(a)-(c), the measured amplitude distributions on the record plane present clear vortex beam characteristics with a topological charge of +1 for all three frequencies. The presence of a central null in the intensity confirms the vortex nature of the beams. Additionally, the phase distributions in Fig. 7(d)-(f) display the expected spiral phase structure, further supporting the generation of vortex beams. To evaluate the quality of the generated vortex beams, the mode purities are calculated and shown in Fig. 7(g)-(i). The mode purities for the frequencies 9.72 GHz, 9.8 GHz, and 10.23 GHz are 0.941, 0.965, and 0.963, respectively. These high mode purities indicate that the majority of the energy is concentrated in the desired mode, confirming the successful generation of high-quality vortex beams at these frequencies.
Fig. 7. The measured electric field (a)-(c) amplitude, (d)-(f) phase distribution and (g)-(i) corresponding mode purity of the single vortex generating metasurface on the x-y plane at z = 300 mm for 9.72 GHz, 9.8 GHz and 10.23 GHz, respectively.
The far-field results of the single vortex generation are shown in Fig. 8. Both the 3D and 2D radiation patterns clearly depict the characteristic amplitude nulls along the z-axis, which are indicative of vortex beams. The measured valley bottoms are -14.4 dB, -21.3 dB, and -16.2 dB at 9.72 GHz, 9.8 GHz, and 10.23 GHz, respectively. These valley bottoms indicate the presence of the central nulls in the vortex beams and are in agreement with the expected behavior. Moreover, the divergence angles of the vortex beams, which represent the spread of the beams, are measured to be 8.8°, 8.2°, and 8° at frequencies 9.72 GHz, 9.8 GHz, and 10.23 GHz, respectively. These measured divergence angles match well with the simulated values, indicating the accuracy of the design and the successful generation of vortex beams with the desired characteristics.
Fig. 8. (a)-(c) The measured 3-D radiation patterns of the single vortex generating metasurface for 9.72 GHz, 9.8 GHz and 10.23 GHz, respectively. (d)-(f) The simulated and measured normalized radiation patterns on x-z plane of the three frequencies.
The near field measurement results of the multiplexed vortex generation are shown in Fig. 9. The measured electric fields on the record plane have the same distribution tendency as the simulated ones displayed in Fig. 5. This correspondence between the measured and simulated electric fields validates the effectiveness of the design in generating the multiplexed vortex beams. The mode purities of the three frequencies, as depicted in Fig. 9(g)-(i) and listed in Table 3, provide further insight into the performance of the multiplexed vortex generation. The total power ratios of the target modes, l=+1 and l=+2, are found to be above 0.8 at all three frequencies, indicating a satisfactory power distribution among the desired modes. Additionally, the calculated RMSE values are very small, suggesting a close agreement between the actual power ratios and the desired ratios. These results affirm the successful generation of the multiplexed vortex beams with accurate power distribution.
Fig. 9. The measured electric field (a)-(c) amplitude, (d)-(f) phase distribution and (g)-(i) corresponding mode purity of the multiplexed vortex generating metasurface on the x-y plane at z = 300 mm for 9.72 GHz, 9.8 GHz and 10.23 GHz, respectively.
Table 3. The measured behavior of the multiplexed vortex.
The measured far-field results of the multiplexed vortex generation are shown in the Fig. 10. Besides the 3-D radiation patterns, the patterns on x-z and y-z planes are also plotted to provide a comprehensive view of the beam characteristics as presented in Fig. 10(d)-(f) and (g)-(i), respectively. As observed in the Fig. 10, the amplitude null of the multiplexed vortex beams is 3° deviating from the z axis towards –y direction. Overall, the measured far-field results align well with the simulated ones, confirming the validity of the design and the successful generation of the multiplexed vortex beams. The slight differences between the simulated and measured results may be caused by the measurement errors and imperfections in the plane wave emitted by the horn antenna.
Fig. 10. (a)-(c) The measured 3-D radiation patterns of the multiplexed vortex generating metasurface for 9.72 GHz, 9.8 GHz and 10.23 GHz, respectively. The simulated and measured normalized radiation patterns on (d)-(f) x-z plane and (g)-(i) y-z plane of the three frequencies.
5. Conclusion
In summary, we have developed a design method for high-efficiency transmissive metasurfaces that enable stable amplitude and precise P-B phase modulation using dartboard discretization configuration. The meta-atom is based on a hexagonal lattice with a single-layer substrate, where metallic layers on both sides are dartboard discretized into sector annular pixels, and binary coding indicates the presence of metal for topology optimization. The optimized meta-atom exhibits stable amplitude and precise P-B phase modulations within the operating band, while also significantly broadening the bandwidth with high cross-polarized transmission coefficients. These features make it an excellent candidate for various phase-only modulation applications. we designed and fabricated two types of metasurfaces for generating single and multiplexed vortex beams using the optimized meta-atom. The metasurface for single vortex generation achieved a remarkable maximum mode purity above 0.99, successfully demonstrating the effectiveness of precise P-B phase modulation. Additionally, we designed a more complex phase distribution for generating multiplexed vortex beams with topological charges of l=+1 and l=+2 in a 1:1 power ratio. The metasurface achieved the desired power distribution of target modes at the concerned frequencies. Both simulated and measured results validated the excellent performance of the generated vortex beams, which closely matched the predicted outcomes. Our design strategy of achieving stable amplitude and precise P-B phase modulation holds significant potential for various EM wavefront modulation applications, such as anomalous refraction, beam focusing, and holography, and opens up new possibilities for advanced electromagnetic wave manipulation devices and systems.
Funding
National Key Research and Development Program of China ( 2022YFF0604801); National Natural Science Foundation of China (62271056, 62201037, 62171186); Beijing Municipal Natural Science Foundation of China-Haidian Original Innovation Joint Fund ( L222042); 111 Project (B14010).
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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