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Abstract
Metasurfaces have provided a novel way on modulating the wavefront of electromagnetic (EM) waves, where phase modulating is an important method to control EM waves. Normally, phase can be continuously modulated by changing the size of a meta-atom. For a broadband device, it is essential that phase changes linearly varying against frequency within a wide frequency interval, which is quite difficult to design, especially for the transmissive scheme. In this paper, we propose a 0-1 coding method by using genetic algorithm (GA) to realize broadband linear transmission phase and high transmission amplitude against frequency. To verify the method, a beam bending metasurface is designed based on array of six meta-atoms with step gap of 60°. Simulation and experimental results show that the metasurface deflector achieves perfect beam refraction from 8 to 12 GHz, which is consistent with theoretical calculations. Moreover, the working efficiency is kept at about 75%, with the variation of the frequency, which demonstrates the good stability of the metasurface. This method offers a new insight into the designing of broadband devices.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since N. Yu et. al. in 2011 deduced the generalized Snell’s law [1] and proposed the concept of metasurfaces, metasurfaces have attracted wide attention due to their feasibility, low loss, ease of fabrication [1,2]. It presents many unprecedented electromagnetic properties, such as flexible control of electromagnetic waves, which are not available in natural materials [3]. Research results on optical activities such as negative refractive index and invisible cloaks have appeared [3–9]. Many novel phenomena and applications have been implemented, such as anomalous refraction/reflection [1,10–14], polarization converter [15–22], and so on [23–27]. In addition, more methods such as topology optimization [28–35] are introduced continuously to improve design efficiency and achieve better performance.
The key factor of anomalous refraction (AR) is introducing a plane wave vector by generating a phase gradient through the meta-atoms. For traditional method, meta-atom is designed by changing sizes or rotation directions of the meta-atom to realize 2π phase coverage. Unfortunately, the linearity of phase of each meta-atom varying against frequency is barely satisfactory. The nonconstant group delay makes the metasurfaces’ deflection effect deteriorated, narrows the operating bandwidth and seriously limits their further application. Therefore, it has become an urgent task for researchers to expand the working bandwidth of anomalous refraction metasurfaces effectively by using innovative methods while maintaining high working efficiency. For example, a multi-layer metal pattern is used to generate multiple resonance points to increase the bandwidth [13,27].
In this paper, we propose a new strategy by using 0-1 coding optimization to highly improve the efficient and broad the bandwidth of the metasurface deflector. The introduced genetic algorithms significantly enhance linear phase response of the meta-atoms and keep a very high efficiency polarization conversion for the transmission waves. A microwave sample is fabricated and experimentally measured. Numerical results and simulation results agree well with the experimental ones, which indicates that our metasurface achieves perfect AR from 8 GHz to 12 GHz with the transmission efficiency reaches 75%. Our findings open the door for ultra-broadband anomalous refraction meta-devices, which can lead to many exciting applications.
2. Concept and unit cell design
According to the generalized Snell’s law [1], the anomalous refraction angle could be calculated as:
In this paper, the phase difference between adjacent meta-atom at x-axis direction is 60° within 8-12 GHz as shown in Fig. 1. And then, according to the Eq. (1), we can calculate that the deflection angle ${\theta _t}$ is 30° at 10 GHz.
Fig. 1. Schematic of the anomalous refraction (AR) metasurfaces with GA optimization. For the incident electromagnetic wave, the transmitting cross polarized electromagnetic wave achieves beam deflection from 8 GHz to 12 GHz.
The meta-atom as shown in Fig. 2(b) consists of three metallic layers, separated by two dielectric layers. The top (Fig. 2(a)) and bottom layers of the meta-atom composed of orthogonal grating structures. The middle layer of the preliminary unit is composed of split rings with different opening angles and directions. The total thickness of the meta-atom is approximately 0.17λ0 with respect to the central operating frequency (10 GHz) at free space. The transmission phase and geometric phase of the meta-atom can be changed by changing the size and direction of the opening of the split ring. We characterize the EM property of the meta-atom by using the CST Microwave Studio, and the simulation setup is shown in Fig. 2(d). Figure 2(e) depicts the cross-polarization transmission coefficients and phase when the structure is shined normally along -z axis. We can see that three transmission peaks (blue circle symbols) appear at the particular frequencies due to the multiple Lorentz resonators induced by the mutual interactions of three metallic layers [35]. Moreover, large fluctuations are obviously observed which is the main reason limiting the working bandwidth and working efficiency. Correspondingly, the phase of the cross-polarization transmission wave appears bad degree of linearity. The undesirable phase gradient will affect the deflection of the metasurface to the incident beam. Our main point is to suppress the large transmission fluctuations to enhance the transmission resonances and to improve the linearity of the phase.
Fig. 2. Meta-atom design and the evolution process of the preliminary meta-atom structure. (a) Top layer, (b) perspective view, (c) middle layer, (d) electromagnetic simulation setup. (e) Simulation results of transmission coefficient and the cross-polarization phase of incident wave propagating in -z-direction of the preliminary structure. (f) Schematic diagram of the topologically coding in the middle layer. The composite meta-atom composed of three metallic layers (yellow part) separated by two substrate spacers (blue part). The substrate is made of F4B with relative permittivity of 2.65, a loss tangent of 0.01i and a thickness of h=2.5 mm. Other parameters p=10 mm, a = b=2.5 mm. The metal is made of perfect conductor and the thickness 0.036 mm.
Here we use a new strategy to tune the mutual coupling by using the 0-1 coding optimization based on genetic algorithm. We only optimize the middle layer structure of the meta-atom to tune the mutual coupling interactions. In order to carry out the subsequent optimization smoothly, the middle metal layer is first discretized, and the principle of discretization is shown in Fig. 2(f). The yellow square represents the presence of the metal patch and the green square represents the absence of the metal patch. The side length of each small square is 0.5 mm. The gratings at both sides of the structure are used to transmit one polarization perfectly while blocking the orthogonal polarization, which is perfect and should not be optimized.
Figure 3(a) presents the flowchart of the optimization process. Firstly, it is important to define the fitness function. In this work, the aim is to improve the operating bandwidth and efficiency of the anomalous refraction. The phase and amplitude of tyx should be considered respectively. We set the aim that the working band of the metasurface is 8 GHz to 12 GHz. The fitness function is generally used to distinguish the individual’s superiority and inferiority in the genetic algorithm, which is the base of natural selection. As a principle, the smaller the fitness, the better the individual. Here, the least square method is used to define the individual fitness function. Then the fitness function F is made up of the residuals sum of squares for phase (RSSP) and the residuals sum of squares for amplitude (RSSA). For the part of phase, the improvement in linearity of the phase response among the working band corresponds to the decrease of residual sum of squares. For the part of amplitude, the improved transmission efficiency of cross-polarization waves among the working band corresponds to the decrease of residual sum of squares. The expression of fitness function is as follows:
Fig. 3. Schematic diagram of the topologically coding optimization process. (a) Flowchart of the optimization process. (b) Comparison of the middle layer in the six meta-atoms before and after optimization. The upper row is before optimization and the lower row is after optimization. The cross-polarization transmission tyx (c) amplitude and (d) phase of six meta-atoms before optimization. The cross-polarization transmission tyx (e) amplitude and (f) phase of six meta-atoms after optimization. (g) The genetic curve of the least five meta-atoms.
Pij and Aij represent the transmissive phase and amplitude of cross-polarization respectively. In Eq. (2) to Eq. (4), the superscript i and i-1 represent the serial number of meta-atom within a period (i = 2, 3, 4, 5, 6) and the subscript j represents the jth frequency point among 8-12 GHz. The frequencies are selected by the CST automatically. k presents the number of frequency points. As we assumed the phase difference between adjacent meta-atom at x-axis direction is 60°, in order to reflect the optimization effect more intuitively, 60 is subtracted from the corresponding transmission phase of two adjacent elements in the expression. To better measure the amplitude and phase performance of each “individual”, we multiply the amplitude part of the fitness function by 60, making it the same order of magnitude as the phase part of the fitness function.
Secondly, we build the optimization matrix to facilitate the joint simulation of MATLAB and CST. As shown in Fig. 3(b), the middle layer structure is divided into M×N particles, with the yellow protruding square covering copper and the green sunken square without copper. In the optimization matrix, the 0-1 code sequence is be substituted for each particle. In order to speed up the optimization process, the split ring in the middle layer in the upper column of Fig. 3(b) is replaced by the split square ring in the lower column of Fig. 3(b). The length of the split square ring is the same as the diameter of the ring, and the opening length is the same as the corresponding ring. After adjustment, the middle layer of square ring is divided to small metal particles instead. In order to reduce the length of the individual, we set the upper triangle of the matrix as an individual by using diagonal symmetry. In this way, it can not only improve the conversion rate of orthogonal polarization [36] but also shortens the optimization time. Finally confirmed the M = N = 20. Each particle exhibits a size of 0.5 mm, and the structure is divided into 400 particles. In the other word, the length of each individual is 210 bits (0-1 code sequence). To accelerate the convergence speed of the genetic algorithm, the initial values of the bits are set according to the structure of the open split square ring. By combining the MATLAB and CST Microwave Studio, we use the phase and amplitude of the transmission coefficient to determine how each “individual” behaves. Here we set the fitness function value of 15, corresponding to that the |tyx| is equal to 0.94 within 8 GHz∼12 GHz and the deviation from the target phase is 1°. When the fitness function value is less than 15, we believe that the objective of optimization has been achieved.
Thirdly, the process of natural selection is carried out to reduce the residual sum of squares of the least square method. Simulation finds that the linearity of the phase response of the split square ring of the encoded has improved compared with the split ring. Next, we process the further optimization to get more perfect phase distribution and higher transmittance in bandwidth. Mean fitness value decreases significantly with iteration, as shown in Fig. 3(g). The number of individuals in each generation is set as 10, which means that 10 simulations are carried out for each generation. The final meta-atom optimization process takes the longest time. It carries out 57 iterative optimizations during the optimization process. The total simulations of this meta-atom are 570, which takes about 11.03 hours. According to the optimization values of the bits, we establish the final structure of the middle layer which is shown in the lower column in Fig. 3(b). On the upper column is the corresponding split ring structure before optimization. Compared with the theoretical analysis [23], which significantly suppresses the transmission fluctuations of the transmissive system by using the ABBA structure and improves the transmission of the co-polarization, the new strategy enhances the performance of the cross-polarization transmission coefficients.
We contrast of the transmission phase and the amplitude of the six meta-atoms before and after optimization, as shown in Figs. 3(c)–3(f). As it can be seen from Fig. 3(f), the linearity of the optimized phase curve is greatly improved compared with the pre-optimized phase (Fig. 3(d)). The step gap in phase among six meta-atoms is 60° from 8 GHz to 12 GHz. Figure 3(c) and Fig. 3(e) compare the transmission amplitude of the six meta-atoms before and after optimization. The results showed that the transmissivity of optimized six meta-atoms is over 0.82.
For a further verification of the design, a 360 mm × 200 mm sample composed of a periodic array of meta-atoms (36×20 meta-atoms) of the PGM was fabricated using the print circuit board (PCB) technique. The upper and middle pictures shown in Figs. 4(a) and 4(b), in which the inset is a zoom view of the 6 × 1 array of the super cells. Two 2.5 mm-thick F4B substrate was used and the metallic pattern surfaces were tinned in order to protect them from being oxidized. The measurement was carried out in an anechoic chamber. In the experiments, the sample is placed between two horn antennas, the transmitter horn antennas with both polarizations are on the side of the vertical grating. Both horn antennas are connected to the vector network analyzer (Agilent E8362C PNA) by coaxial cables so that the scattering coefficients (txx, txy, tyx and tyy) can be detected and recorded. Then, the anomalous transmission power intensity spectra under the x-polarized wave incidence are measured. The simulation, theory and test results of the far field patterns within the operation band are shown in Figs. 4(c). We take the maximum energy radiation direction as the deflection angle. The black pentagram represents the theoretical calculated value of deflection angle, and the pink dot represents the plane wave simulation value of deflection angle. As shown in the Figs. 4(c), the simulation, theory and test results agree well with each other. In the other word, the results prove that the designed metasurface has achieved a relatively perfect beam deflection and the optimization method has achieved the expected effect. In the working band, the energy of the transmitted beam is mainly concentrated in the direction of the anomalous deflection. The diffraction waves of the other undesirable modes are greatly suppressed. There is a small amount of forward reflected electromagnetic wave, which also verifies the good performance of the designed metasurface. The slight difference of the deflection angle is attributed to inevitable fabrication errors and imperfections of the incoming wave fronts generated by the microwave horns. The frequency band for the optimization process is 8 GHz∼12 GHz. As a result, the bandwidth of the abnormal refraction metasurface is about 4 GHz, with relative bandwidth corresponding to 40%, which is better than those of the reported metasurfaces (see the details in Table 1) [37–42]. The relative band of the references listed in Table 1 is calculated on the basis of |tyx| over 0.8.
Fig. 4. Sample picture and the experimental results. (a) Top layer and (b) middle layer of the fabricated sample. (c) Far-field patterns of theory, simulation and test results within the operation band. The black hollow star represents the theoretical calculated value of deflection angle, and the pink dot represents the plane wave simulation value of deflection angle. (d) Simulation and experimental efficiency of the metasurface.
Table 1. The comparison between references and this work.
Then we measure the working efficiency of the sample, as shown in Fig. 4(d). A metal plate with same size of the metasurface was made as a reference plane. The total reflected wave is received, integrated and defined as Ptot. And then, the reference metal plane is replaced by the metasurface and the crossed-polarized transmitted wave is received. The main lobe of anomalous refraction is integrated, defined as Pano. The absolute efficiency of beam deflection is:
Based on this method, the efficiency of the tested and simulated metasurface is calculated at the corresponding frequency points every 0.5 GHz in the range of 8 GHz∼12 GHz. The efficiency curve is obtained as shown in Fig. 4(d). As it can be seen, the experimental and simulation results agree well with each other and fluctuate around 75%, which verifies the high efficiency of the metasurface. The physical mechanism of the broadband transmission attributes to the enhancing of transmission resonance. The messy distribution of the middle layer provides more chances to achieve high intensity resonance. The large transmission fluctuations has been suppressed to improve the working efficiency. For the mechanism of polarization conversion, it can be understanded that the antiparallel current excites magnetic dipoles, which results from the antiparallel current [28].
Finally, in order to further visually demonstrate the contribution of discrete metal patches with disordered distribution after optimization to improving transmission efficiency and expanding bandwidth, we present the current distribution diagram of three metal layers before and after optimization of the fifth meta-atom at 8 GHz, 9 GHz, 10 GHz, 11 GHz and 12 GHz. From the current distributions in Fig. 5, it can be found that after optimization, the added metal patches have a strong current distribution, indicating that the newly added metal patches have certain help to improve the performance. Moreover, the current density at the bottom layer and top layer both are significantly enhanced after optimization compared with the initial ones, which further indicated that the stray arrangement of the middle layer made the resonance intensity greatly improved. As we can see in Fig. 3 (e), the resonance peak appears at 8 GHz of the optimized fifth meta-atom, and the transmittance amplitude is significantly improved compared with the initial design.
Fig. 5. Schematic diagram of current distributions on surface of the initial and optimal for fifth meta-atom.
3. Conclusion
In this paper, we proposed a novel method which uses the 0-1 coding to realize broadband anomalous deflection from 8 GHz to 12 GHz while keeping the transmissivity over 0.8 for x- and y- polarized incidence respectively at both sides. The relative bandwidth reaches 40%, which is broadened extremely compared with the reported achievements. In addition, we fabricated a sample and the experiment results are in good agreement with the simulation ones. The efficiency of the designed metasurface is about 75% in the working bandwidth, and the highest can reach 80%. Our method has more potential application prospect in meta-devices with other functionalities or at other frequencies.
Funding
National Natural Science Foundation of China (61701572, 61871394, 61901512); Key Projects of Aviation Foundation (201918037002); Natural Science Foundation of Shaanxi Province (2019JQ-013).
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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