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Abstract
Metasurfaces offer an unprecedented opportunity for flexible manipulation of electromagnetic wave. The azimuth-rotation-independent (ARI) polarization conversion metasurface (PCM) is an ultrathin device, which could convert an arbitrary linearly-polarized incident wave to its cross-polarized state. However, the bandwidth of an ARI PCM with a high cross-polarized transmission is usually limited. Here, a topology optimization method of multi-feature points based on the differential evolution (DE) algorithm is adopted to enhance the bandwidth of the traditional ARI PCM while maintaining a high transmission and polarization conversion ratio. The simulated results of the optimized structure indicate a 2.08 times bandwidth expansion in the cross-polarization conversion compared with the original structure. In addition, the measured results are consistent with the simulated ones and the ARI characteristic is validated. The proposed method provides a promising route for efficient high-performance metasurface designs.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Polarization is one of the most important characteristics of the electromagnetic (EM) waves. The manipulation of the polarization is desirable in analytical chemistry, the biological science, photoelectrons, telecommunications, etc. The thickness of conventional wave plates is challenging for the lower frequency of microwave band. The metasurfaces are two-dimensional (2D) artificial sub-wavelength structures applied for EM manipulation, such as absorption [1,2], asymmetric transmission [3,4], spatial filters [5,6], hologram [7], orbital angular momentum generation [8–10]. The metasurface also provides an opportunity for ultra-thin polarization conversion applications, which is called polarization conversion metasurface (PCM) [11–35].
Generally, the PCM can be classified into two types in terms of the outgoing EM wave propagation direction, namely the reflective type [12–17] and the transmissive [18–35] type. The transmissive PCM could be further divided into two major groups. One is the azimuth sensitive type [18–24], whose polarization conversion performance relies on the polarization azimuth angle of the incident wave. The other is the azimuth-rotation-independent (ARI) type [25–35], which could rotate the polarization of the linearly polarized incident wave by a certain angle (mostly 90°). For the reflective PCM, a high reflectance can be easily obtained due to the metal ground plane and broader polarization conversion bandwidths are usually obtained. In contrast, for the transmissive PCM, both the reflection and transmission occur, leading to a much narrower bandwidth for a high cross-polarized transmission. Typically, it is difficult to broaden the bandwidth of the ARI PCM. Previous reported ARI PCMs with the metal insulator metal (MIM) structure, the bandwidth with cross-polarized transmission coefficient over 0.9 is usually less than 2% [26–35]. Compared with shape and size optimization, the topology optimization enhances the design freedom and could achieve better performances. Most topology optimization method in the metasurface design is the binary coding by dividing one unit cell into square pixels [36–39]. However, the structures of the elements may have sharp boundaries or shapes of small isolated islands with this method, which are challenging to be fabricated.
In this work, we propose a topology optimization method for broadening the bandwidth of the traditionally designed ARI PCM. Firstly, we theoretically analyze the mechanism for designing an ARI PCM, which could usually be satisfied in a very narrow bandwidth. Subsequently, the topology optimization design strategy of multi-feature points is proposed for the performance improvement. Then, a fabrication-friendly ARI PCM element is obtained under this topology optimization with the help of differential evolution (DE) algorithm. The bandwidth of the cross-polarized transmission is significantly increased at the same central frequency for the optimized structure. In comparison to previous reported ARI PCMs with metal-insulator-metal (MIM) structure, our design possesses a much broader cross-polarized bandwidth with a high transmittance. Finally, an ARI PCM sample of the optimized design is fabricated and measured for verification. The measured results are in accordance with the simulated results.
2. Theory analysis and optimization design strategy
2.1 Theory analysis
The schematic illustration of the ARI PCM and the polarization state of the incident and the transmitted waves can be seen in Fig. 1. For the polarization conversion application, the Jones matrix is always applied to relate the incident to the transmitted electric field:
In the previous work, we concluded that the ${C_n}(n \ge 3,n \in {N^ + })$ symmetric structure is an ARI structure [35].The Jones matrix for the ARI structure is
andTo obtain a high cross-polarized transmission for the ARI PCM, A is supposed to be 0 and modulus of B is supposed to be close to 1 in Eq. (4) and Eq. (5). Obviously, the eigenpolarizations of the ARI structure are circular states according to Eq. (5). The moduli of ${t_{ +{+} }}$ and ${t_{ -{-} }}$ are both close to 1 with an argument difference of π in the high cross-polarized transmission band. However, the performances along the two eigenpolarizations cannot be individually modulated by adjusting the structure. In other word, ${t_{ +{+} }}$ and ${t_{ -{-} }}$ are both erratically influenced by the structure changes. Therefore, it is difficult to broaden the bandwidth for the ARI PCM just by shape or size optimization.
2.2 Optimization design strategy
Topology optimization is a useful method for constructing the structure with the optimal objective function values and satisfying the constrains through optimizing the structural topology parameters. Figure 2 shows the schematic illustration of the topology optimization design strategy. The ${C_4}$-symmetric elements are selected as the optimization objects in this paper. The basic shape of the original element in [35] is the rotating L-shape, which could be regarded as the structure characterized by two feature points. In our topology optimization design, the structure uses multi-feature points: The starting point ${P_0}$ is fixed at the center of the element; next, a series of feature points are selected and connected to form a polyline; then, a strip with several segments is created from the polyline; finally, the basic shape is established by rotating the conjugated metal strips around ${P_0}$ by 90°, 180°, and 270°.
Fig. 1. (a) The schematic illustration of the azimuth-rotation-independent (ARI) polarization conversion metasurface (PCM); (b) the polarization state of the incident and the transmitted waves.
Fig. 2. Schematic illustration of the topology optimization design strategy. (a) The top view and (b) perspective view of the original design and multi-feature points design.
To demonstrate the effectiveness of the topology optimization, the multi-feature points design is composed with the same material and thickness as the original design. In details, both the original and the proposed design consist of three layers: the F4B substrate (${\varepsilon _\textrm{r}} = 2.65,{\kern 1pt} \,\tan \delta = 0.001$) with a thickness of 1 mm and two conjugated copper patterns (i.e. the patterns in yellow and blue in Fig. 2(b)) with the thickness of 70 ${\mathrm{\mu} \mathrm{m}}$ on both sides of the substrate. Moreover, the width of each segment in these two designs is fixed to be 0.4 mm.
A broader bandwidth with high-efficiency cross polarization conversions can be achieved through the position optimization of the multiple feature points. Theoretically, as the number of the feature points increase, better topology optimization results could be obtained. However, once the feature points reach the enough amount of numbers, each additional feature point will improve few performance, and it will cause time consumption. For this structure, the performance saturation occurs when the number of feature points exceeds three. Therefore, the practical topology optimization is a trade-off between the performance and the cost.
Here, as illustrated in Fig. 2(a), we choose three feature points (${P_1}$, ${P_2}$ and ${P_3}$) in this topology optimization. To reduce computation complexity, all the feature points are selected in the right upper quadrant to avoid the excessive intersections of the segments. Besides the positions of the feature points, we also take the period of the element as an optimization parameter since it has a significant influence on the resonant frequency. Consequently, there are total 7 parameters in the topology optimization.
To realize the topology optimization for high-efficiency cross-polarized transmission, we joint the numerical simulation and differential evolution (DE) algorithm through Python as shown in Fig. 3. Differential evolution is a robust and efficient global optimization technique [40]. In the optimization, the initial population is randomly generated firstly. Then, the new generation is created from the previous evaluated designs through the process of mutation, crossover, and selection. This process repeats over many generations until the termination condition is met.
Fig. 3. Flowchart of the joint simulation for differential evolution algorithm-based topology optimization.
In details, the population size is set to 50 and the minimum step of each parameter is set to be 0.1 mm. The numerical simulation is performed with CST Microwave Studio Suite software controlled by home-made Python code. The objective values are used to evaluate the designs in every generation. The optimization objective is set to be the bandwidth with the cross-polarized transmission coefficient higher than 0.9 at center frequency of 9.8 GHz. The objective function is set to be as follows:
In this differential mutation step, the mutant population is obtained by the DE/best/1 operator, in which the best individual in the current population is selected as the base vector, and the scaling factor F is set to 0.5. Exponential crossover is selected in the crossover step under the crossover rate Cr =0.5. In the selection step, the better of the offspring and its parent is chosen for the next generation, which is as known as the greediness mechanism. The termination condition is whether the iterative times reach the number 150. It is worth noting that the full-wave numerical simulation is the most time-consuming step in the optimization process. Thus, previously-simulated structures are checked to prevent repeated tasks.
3. Optimization results
Figure 4 depicts the best and mean bandwidth value of the whole DE iteration. Overall, the mean value and the best value increase with the number of generation. The best value reaches the optimal value at the 89th generation, and the mean value meet the best value at 100th generation. The elements evolved over the course of the optimization are displayed below the curves and the optimized design is shown as the element E. The period of the optimized design is 8.6 mm and the coordinates of the three feature points are ${P_1}({4.1\,\textrm{mm},0.8\,\textrm{mm}} )$, ${P_2}({4.1\,\textrm{mm},3.2\,\textrm{mm}} )$ and ${P_3}({0\,\textrm{mm},0\,\textrm{mm}} )$.
Fig. 4. The best and mean bandwidth value of the whole differential evolution (DE) iteration. The patterns below the curves indicates how the best element design evolved for the structures.
The simulated co- and cross-polarized transmission coefficients of the original and the optimized structures are depicted in Fig. 5. Compared with the original design, a higher cross-polarized transmission can be observed in the whole X-band. The cross-polarized transmission coefficients are over 0.96 at 9.8 GHz for the optimized structure, indicating a high-efficiency polarization conversion. Moreover, the bandwidth of the cross-polarized transmission coefficient over 0.9 is 2.45% (9.69-9.93 GHz) for the optimized designs, which is the corresponding optimized objective values in Fig. 4. Therefore, the bandwidth is expanded to 2.08 times the original structure (1.18%) for the optimized structure. The maximal co-polarized transmission coefficients of the original and the optimized structures are 0.32 and 0.25 in the X-band, respectively. Both the low co-polarized transmissions are observed in the operating band for the two structures.
Fig. 5. The simulated co- and cross-polarized transmission coefficients of the original and the optimized structures.
Figure 6 plots the polarization conversion ratio (PCR) of the original and the optimized structure. The PCR characterizes the polarization conversion efficiency and it can be calculated by Eq. (7),
To further validate the performance of the optimized ARI PCMs, the electric field distributions are illustrated for the optimized structure at 9.8 GHz, as illustrated in Fig. 7. The incident wave is y-polarized and propagates along the –z direction. The electric field drastically varies near the PCM, indicating a strong resonance at the operating frequency. The transmitted wave is completely converted into x-polarized with an extremely close distance.
Fig. 7. The electric field distributions of the optimized structure at 9.8 GHz under y-polarized EM wave incidence.
The performances of the ARI PCMs reported recently with the optimized structure in this work are listed in Table 1. All the selected structures are ${C_4}$-symmetric and composed of two conjugated metal patterns and one substrate layer, namely the MIM structure. The bandwidths of this work with the cross-polarized transmission coefficient (Tcro-pol.) over 0.9 and 0.8 are much wider than the other listed works. Moreover, the maximum cross-polarized transmission is still competitive among the mentioned works. This also proves that the topology optimization method proposed in this paper has an obvious effect on broadening the bandwidth of the ARI PCMs.
Table 1. Comparisons between the reported works and this work.
4. Experimental results
An ARI PCM sample of the optimized structure is fabricated through the printed circuit board (PCB) fabrication technology for the experimental verification as shown in Fig. 8(a). The copper patterns are printed on a circular F4B substrate with a diameter of 440 mm. Vias are punched every 30° along the edge of the substrate. The measurement environment is depicted in Fig. 8(b). Two horn antennas are selected as the transmitter and the receiver, and the PCM sample is placed coaxially between the two horns. Absorbing materials are fixed around the PCM for reducing the electromagnetic wave diffraction. The co- and cross-polarized components can be measured by placing the receiver parallel and perpendicular to the transmitter, respectively. To measure the results of the incident wave with different polarization azimuth angles φ, the PCM sample could be relatively rotated and fixed by the vias.
Fig. 8. (a) The fabricated PCM sample of the optimized structure and (b) the measurement environment.
The simulated and measured transmission coefficients of the fabricated PCM sample are plotted in the Fig. 9. The measured results agree very well with the simulated ones. The ARI characteristic can also be noticed since both the simulated and measured results show imperceptible changes with different linearly polarized azimuth angles φ. The co-polarized transmissions coefficients show slight difference in the measurement and simulation, which mainly due to the electromagnetic wave diffracting from the absorbing materials in the measurement.
Fig. 9. Simulated and measured transmission coefficients for the ARI PCM of the optimized structure with different linearly polarized azimuths under normal incidence: (a) φ=0°, (b) φ=30°, and (c) φ=60°.
5. Conclusion
In summary, we demonstrate a DE algorithm-based topology optimization method for enhancing the bandwidth of the ARI PCM, which is an enormously challenging task for empirical designing or conventional optimization methods. Through multi-feature points based topology design strategy and DE algorithm, the fabrication-friendly topology optimized ARI PCM element can be obtained. The bandwidth with the cross-polarized transmission coefficient over 0.9 is 2.45% for the optimized structure, which has been expanded to 2.08 times compared with the original structure at the same central frequency (9.8 GHz). A PCR over 90% is also achieved in the high cross-polarized transmission band. Particularly, the bandwidth of the optimized structure is also superior to the works in recent years. The measured results are consistent with the simulated results, and the ARI characteristic is easily observed. The proposed topology optimization method has the potential to be extended to many metasurface designs for its effectiveness.
Funding
National Key Research and Development Program of China (Grant no. 2022YFF0604801); National Natural Science Foundation of China (Grant no. 61527805, Grant no. 62171186, Grant no. 62271056); Basic Research Foundation of Beijing Institute of Technology (Grants no. BITBLR2020014); the 111 Project of China (Grant no. 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.
References
1. Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014). [CrossRef]
2. L. Khosravi Khorashad and C. Argyropoulos, “Unraveling the temperature dynamics and hot electron generation in tunable gap-plasmon metasurface absorbers,” Nanophotonics 11(17), 4037–4052 (2022). [CrossRef]
3. B. Han, S. Li, Z. Li, G. Huang, J. Tian, and X. Cao, “Asymmetric transmission for dual-circularly and linearly polarized waves based on a chiral metasurface,” Opt. Express 29(13), 19643–19654 (2021). [CrossRef]
4. Y. Ling, L. Huang, W. Hong, T. Liu, Y. Sun, J. Luan, and G. Yuan, “Asymmetric optical transmission based on unidirectional excitation of surface plasmon polaritons in gradient metasurface,” Opt. Express 25(12), 13648–13658 (2017). [CrossRef]
5. J. Zhang, S. Zhang, and G. F. Pedersen, “Dual-band structure reused antenna based on quasi-elliptic bandpass frequency selective surface for 5 g application,” IEEE Trans. Antennas Propag. 68(11), 7612–7617 (2020). [CrossRef]
6. K. Sarabandi and N. Behdad, “A frequency selective surface with miniaturized elements,” IEEE Trans. Antennas Propag. 55(5), 1239–1245 (2007). [CrossRef]
7. L. Li, T. Jun Cui, W. Ji, S. Liu, J. Ding, X. Wan, Y. Bo Li, M. Jiang, C.-W. Qiu, and S. Zhang, “Electromagnetic reprogrammable coding-metasurface holograms,” Nat. Commun. 8(1), 197 (2017). [CrossRef]
8. Q. Feng, X. Kong, M. Shan, Y. Lin, L. Li, and T. J. Cui, “Multi-orbital-angular-momentum-mode vortex wave multiplexing and demultiplexing with shared-aperture reflective metasurfaces,” Phys. Rev. Appl. 17(3), 034017 (2022). [CrossRef]
9. R. C. Devlin, A. Ambrosio, N. A. Rubin, J. P. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358(6365), 896–901 (2017). [CrossRef]
10. M. Xin, R. Xie, G. Zhai, J. Gao, D. Zhang, X. Wang, S. An, B. Zheng, H. Zhang, and J. Ding, “Full control of dual-band vortex beams using a high-efficiency single-layer bi-spectral 2-bit coding metasurface,” Opt. Express 28(12), 17374–17383 (2020). [CrossRef]
11. N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science 340(6138), 1304–1307 (2013). [CrossRef]
12. Y. Sun, Y. Liu, T. Wu, J. Wu, Y. Wang, J. Li, and H. Ye, “Broadband anomalous reflective metasurface for complementary conversion of arbitrary incident polarization angles,” Opt. Express 29(23), 38404–38414 (2021). [CrossRef]
13. M. Barkabian, N. Sharifi, and N. Granpayeh, “Multi-functional high-efficiency reflective polarization converter based on an ultra-thin graphene metasurface in the THz band,” Opt. Express 29(13), 20160–20174 (2021). [CrossRef]
14. J. Liu, J.-Y. Li, and Z. N. Chen, “Broadband polarization conversion metasurface for antenna RCS reduction,” IEEE Trans. Antennas Propag. 70(5), 3834–3839 (2022). [CrossRef]
15. X. Wang and G.-M. Yang, “Linear-polarization metasurface converter with an arbitrary polarization rotating angle,” Opt. Express 29(19), 30579–30589 (2021). [CrossRef]
16. M. Karamirad, C. Ghobadi, and J. Nourinia, “Metasurfaces for wideband and efficient polarization rotation,” IEEE Trans. Antennas Propag. 69(3), 1799–1804 (2021). [CrossRef]
17. J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99(6), 063908 (2007). [CrossRef]
18. C.-J. Gao, H.-Q. Dong, L. Zeng, and H.-F. Zhang, “Theoretical proposal of electromagnetically induced transparency with a transmissive polarization conversion based on metamaterials,” Phys. Scr. 97(2), 025505 (2022). [CrossRef]
19. Z. Li, X. Cai, L. Huang, H. Xu, Y. Wei, and N. Dai, “Controllable polarization rotator with broadband high transmission using all-dielectric metasurfaces,” Adv. Theory Simul. 2(9), 1900086 (2019). [CrossRef]
20. Y. Zhang, J. Zhu, C. Huang, and S. Ma, “Wide-band and high-efficiency 90° polarization rotator based on tri-layered perforated metal films,” J. Lightwave Technol. 35(21), 4817–4823 (2017). [CrossRef]
21. W. Liu, S. Chen, Z. Li, H. Cheng, P. Yu, J. Li, and J. Tian, “Realization of broadband cross-polarization conversion in transmission mode in the terahertz region using a single-layer metasurface,” Opt. Lett. 40(13), 3185–3188 (2015). [CrossRef]
22. L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013). [CrossRef]
23. P. Fei, G. A. E. Vandenbosch, W. Guo, X. Wen, D. Xiong, W. Hu, Q. Zheng, and X. Chen, “Versatile cross-polarization conversion chiral metasurface for linear and circular polarizations,” Adv. Opt. Mater. 8(13), 2000194 (2020). [CrossRef]
24. S. Wu, S. Xu, T. L. Zinenko, V. V. Yachin, S. L. Prosvirnin, and V. R. Tuz, “3D-printed chiral metasurface as a dichroic dual-band polarization converter,” Opt. Lett. 44(4), 1056–1059 (2019). [CrossRef]
25. S.-Y. Wang, J.-D. Bi, W. Liu, W. Geyi, and S. Gao, “Polarization-insensitive cross-polarization converter,” IEEE Trans. Antennas Propag. 69(8), 4670–4680 (2021). [CrossRef]
26. L. Wang, F. Lv, Z. Xiao, and X. Ma, “Theoretical and experimental verification of 90° polarization converter based on chiral metamaterials,” Plasmonics 16(1), 199–204 (2021). [CrossRef]
27. Y. Cheng, W. Li, and X. Mao, “Triple-band polarization angle independent 90° polarization rotator based on fermat’s spiral structure planar chiral metamaterial,” Prog. Electromagn. Res. 165, 35–45 (2019). [CrossRef]
28. J. Zhao and Y. Cheng, “Ultrathin dual-band polarization angle independent 90° polarization rotator with giant optical activity based on planar chiral metamaterial,” Appl. Phys. B 124(9), 185 (2018). [CrossRef]
29. H. Shi, A. Zhang, S. Zheng, J. Li, and Y. Jiang, “Dual-band polarization angle independent 90° polarization rotator using twisted electric-field-coupled resonators,” Appl. Phys. Lett. 104(3), 034102 (2014). [CrossRef]
30. R. Rajkumar, N. Yogesh, and V. Subramanian, “Cross polarization converter formed by rotated-arm-square chiral metamaterial,” J. Appl. Phys. 114(22), 224506 (2013). [CrossRef]
31. C. Xi, “Terahertz angle-insensitive 90° polarization rotator using chiral metamaterial,” Phys. B 422, 83–86 (2013). [CrossRef]
32. K. Song, Y. Liu, Q. Fu, X. Zhao, C. Luo, and W. Zhu, “90° polarization rotator with rotation angle independent of substrate permittivity and incident angles using a composite chiral metamaterial,” Opt. Express 21(6), 7439–7446 (2013). [CrossRef]
33. M. Mutlu and E. Ozbay, “A transparent 90° polarization rotator by combining chirality and electromagnetic wave tunneling,” Appl. Phys. Lett. 100(5), 051909 (2012). [CrossRef]
34. Y. Ye and S. He, “90° polarization rotator using a bilayered chiral metamaterial with giant optical activity,” Appl. Phys. Lett. 96(20), 203501 (2010). [CrossRef]
35. G. Cheng, L. Si, P. Tang, Q. Zhang, and X. Lv, “Study of symmetries of chiral metasurfaces for azimuth-rotation-independent cross polarization conversion,” Opt. Express 30(4), 5722–5730 (2022). [CrossRef]
36. L. L. Cong, X. Y. Cao, H. Yang, J. Gao, and T. Song, “Coding anisotropic metasurface with integrated broadband tunable radiation and low-scattering performance,” Nanoscale Res. Lett. 14(1), 113 (2019). [CrossRef]
37. S. Sui, H. Ma, J. Wang, M. Feng, Y. Pang, S. Xia, Z. Xu, and S. Qu, “Symmetry-based coding method and synthesis topology optimization design of ultra-wideband polarization conversion metasurfaces,” Appl. Phys. Lett. 109(1), 014104 (2016). [CrossRef]
38. M. Borgese, F. Costa, S. Genovesi, A. Monorchio, and G. Manara, “Optimal design of miniaturized reflecting metasurfaces for ultra-wideband and angularly stable polarization conversion,” Sci. Rep. 8(1), 7651 (2018). [CrossRef]
39. J. A. Bossard, L. Lin, S. Yun, L. Liu, D. H. Werner, and T. S. Mayer, “Near-ideal optical metamaterial absorbers with super-octave bandwidth,” ACS Nano 8(2), 1517–1524 (2014). [CrossRef]
40. K. R. Opara and J. Arabas, “Differential evolution: a survey of theoretical analyses,” Swarm Evol. Comput. 44, 546–558 (2019). [CrossRef]
