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Abstract
We propose a flexible and fast estimation method to calculate the far-field patterns of digital-coding metasurfaces (DCMs) by performing chirp Z-transform (CZT), called the DCM-CZT method. Because of the expression form of convolution, CZT can be accelerated by fast Fourier transform. Compared with the traditional discrete Fourier transform (DFT) method, the DCM-CZT method can accurately estimate the far-field patterns with arbitrary element periods. More importantly, the DCM-CZT method can calculate partial far-field patterns for some specific orientations, instead of the global far-field patterns like DFT does. We show that the DCM-CZT method can efficiently improve the partial space-resolution to avoid the calculation error caused by the fence effect under acceptable computing time. We present six representative examples to demonstrate the capabilities of the proposed method. Results show that the far-field patterns calculated by the DCM-CZT method have good agreements with full-wave simulations and experimental measurements. However, the results of main-lobes calculated by the DFT method have obvious deviations when the element period is about 0.2 wavelengths. We believe that the DCM-CZT method has potential applications in wireless communications and radar detections.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Metamaterials are artificial structures that designed and manufactured to obtain some special properties, such as negative permittivity and permeability, which are not possessed by existing natural materials [1]. The concept of metamaterials was proposed based on the equivalent media model [2], which are known as ‘analog metamaterials’. Compared with large and complex three-dimensional metamaterials [3–5], two-dimensional metasurfaces have attracted more attention in recent years, due to their low-profile and easy-to-process characteristics. Digital coding and programmable metasurfaces were proposed in 2014 [6]. The digital-coding metasurfaces (DCMs) transform the problem from analog domain of equivalent medium parameters to digital domain of coding patterns. DCMs were originally presented to control the electromagnetic (EM) waves in digital way. By arranging the coding particles on a two-dimensional plane with predesigned coding patterns, DCMs can be used to manipulate the EM waves and realize different functions [7–14]. If the coding particles contain pin diodes or capacitors, the metasurfaces can be classified as programmable metasurfaces. It can achieve reprogrammable and real-time controls of the EM waves when combined with field-programmable gate array (FPGA) [13–23]. Some novel directly digital modulation wireless communication systems have been reported [24–27]. These systems transmit the digital information directly via the programmable DCMs, which provides a new architecture for the wireless communications.
The digitization of DCMs connects the physical world and digital information world, and simplifies the design and optimization procedures of metasurfaces [14,15]. DCMs bring a fancy perspective on the connection between the physics and information science. Some concepts and methods of signal processing in information science can be introduced into the physical DCMs to achieve flexible and precise controls of the EM waves [28–30]. For the wireless communication systems based on DCMs, it is increasingly urgent for flexible and fast estimation of the far-field patterns according to different coding patterns. The traditional fast far-field pattern estimation of planar phased array is based on discrete Fourier transform (DFT) method [31–34]. However, the element spacing in the DFT method is defaulted by half wavelength. For DCMs, the element spacing (i.e. the element period) has diverse dimensions other than half wavelength. Therefore, the DFT method will create errors in the far-field pattern estimations, which cannot be neglected. Fortunately, this defect can be remedied by chirp Z-transform (CZT) [35,36].
Here, we propose a DCM-CZT method to modify the traditional DFT method in flexible and fast estimations of the far-field patterns for DCMs, in which the element period is much less than the half wavelength. Compared with the global transformation of DFT, the presented DCM-CZT method can improve the estimation accuracy and calculate partial far-field patterns for specific orientations. To validate the proposed method, we provide six examples of DCMs models for far-field-pattern estimations, which have good agreements with full-wave simulations and experimental measurements.
2. Theory and method
The digital phase-coding metasurfaces can be equivalent to a uniformly-spaced planar phased array, whose element spacing is the element period. We modify the expression of CZT in the form of far-field function of DCMs. Then the element period is directly reflected in parameters of the presented DCM-CZT method. Hence, the DCM-CZT method can accurately estimate the far-field patterns with arbitrary element periods. Furthermore, by controlling the solution path of the DCM-CZT method, we can get the global far-field patterns and partial far-field patterns accordingly. It should be noted that the proposed DCM-CZT method is applicable to any two-dimensional phase matrix, including the bidirectional-coding matrix and random matrix.
2.1 Global far-field patterns for arbitrary element periods
The global far-field function of the digital phase-coding metasurfaces with M × N elements given by the DCM-CZT method for the phase matrix φ(m,n) can be expressed as:
When ignoring the attenuation, we have ${a_1} = {a_2} = {b_1} = {b_2} = 1$. The element period p is directly reflected in the parameters θ01, θ02, ϕ01, and ϕ02. It determines the starting point and ending point of the solution path, and indirectly affects the sampling interval. The uniform sampling of the sine space coordinates u-v for the DCM-CZT method is expressed as
As a reference, we provide the sampling formulas [34] of the sine space coordinates u-v when using the traditional DFT method for the far-field pattern estimation:
As shown in Eq. (3), the sampling area of u-v in the traditional DFT method depends on the element period p, and accurate solution can be performed only when the element period is equal to half wavelength. Whereas the sine space coordinates u-v in the DCM-CZT method are sampled at equal intervals with points K and L, respectively, and are independent of the element period p. Compared with the traditional DFT method, the sampling area of DCM-CZT is fixed for any element period, and there is no need to carry out tedious deleting points or filling points. Finally, the far-field pattern can be obtained by the uv-matrix
It should be noted that the parameters for uniform sampling are u and v. Whereas, the values of θ and φ corresponding to the far-field pattern are derived from the inverse trigonometric function. Hence, the values of θ and φ are nonlinear discretizations.
For the far-field pattern estimation, the global solution path along gradient phase-coding direction of DCM-CZT on the Z-plane is shown in Fig. 1, with the solution path of DFT as a reference. Z-plane is a complex plane, and parameter z is expressed in polar coordinates. The solution path of DFT is unit circle, which starts at point z0 (initial radius a = 1, initial angle θ0 = 0), as shown in Fig. 1(a). For L-point uniform frequency-domain sampling, the sampling interval of DFT is ϕ0 = 2π/L. When ignoring the attenuation (a = b = 1), the solution path of DCM-CZT is a symmetrical arc, as shown in Fig. 1(b). The solution path of DCM-CZT starts at point z0 (initial radius a = 1, initial angle θ0 = 2πp/λ), and the radian of the arc is Ω=2θ0. For K-point uniform sampling, the sampling interval of DCM-CZT is ϕ0=-2θ0/(K-1), and the minus sign indicates that the sampling direction is opposite to the DFT. By contrast, the length of DCM-CZT solution path depends on the element period p. As p increases to half wavelength, the solution path of DCM-CZT changes from a symmetrical arc to a unit circle. We remark that the starting positions of DFT and DCM-CZT are different. For the solution direction, the DFT sampling counterclockwise, whereas the DCM-CZT sampling clockwise. The solution path along gradient phase-coding direction corresponds to the far-field space. As shown in Fig. 1(e), when sampling from point z0 to zK-1 along the solution path Ω, the corresponding pitch angle θ in the far-field space ranges from 90° to −90°, and the corresponding sine space coordinate u ranges from 1 to −1.
Fig. 1. The corresponding relation between the solution path and the far-field space. (a),(b) The global solution path (solid black line) of DFT and DCM-CZT, respectively. (c) The partial solution path (solid green line) of DCM-CZT. (d) The correspondence between the solution path (black line), frequency domain (orange line), and far-field space (yellow line) for DFT. (e),(f) The correspondence between the solution path (black line), sine space coordinates u (blue line), and far-field space (yellow line) for DCM-CZT.
2.2 Partial pattern for specific orientation
Fourier transform is a kind of global transformation. If we are only interested in the scattered waves in some specific directions, the DFT method will waste a lot of computational time on the orientations we are not interested in. The DCM-CZT method is a competitive candidate for specific-orientation far-field calculation by its unique solution path. The green part in Fig. 1(f) describes the analytical correspondence between the solution path and far-field. We assume a target orientation $\theta _1^\ast{=} {\theta _s} - {\theta _e}\; (\varphi = 0^\circ )$ in the far-field space, then we locate the position of $\theta _1^\ast $ in the sine space coordinates u as $u_1^\ast $=sin$\theta _1^\ast $, and the solution path $\Omega _1^\ast{=} sin\theta _1^\ast{\cdot} 2\pi \textrm{p}/\lambda $. Then, the partial far-field function of DCMs given by the DCM-CZT method for the phase matrix $\varphi $(m,n) can be modified as:
We use the same number of sampling points in the global solution for the partial solution, hence improve the space-resolution under the same consuming time. It should be noted that the space-resolution here refers to the resolution of sine space u-v, since uniform sampling actually occurs in the u-v space. Since the far-field space θ-φ is the result of the transformation of the sine space u-v, the space-resolution can reflect the solution accuracy in the far-field space θ-φ.
When the specific orientation we are interested in is relatively narrow, the presented DCM-CZT method requires fewer sampling points to obtain the same space-resolution. Furthermore, CZT can be expressed in convolution form [35], so that the computation can be accelerated by fast Fourier transform (FFT). According to the convolution theorem, a CZT is implemented by three times of FFT. Therefore, when the number of array elements is large or the space-resolution is high, the DCM-CZT method has advantages in calculation speed.
3. Design verifications
The DCMs models provided in this paper are phase-coding metasurfaces, and the gradient phase-coding is only along the horizontal direction, whereas maintain the same in the vertical direction. We model the DCMs using a kind of square-shaped coding particle, as shown in Fig. 2(c). The coding particle working at X-band consists of two square-shaped metallic sheets separated by a dielectric substrate. The phase response of the coding particle is controlled by the size of the upper metallic sheet wn, with the subscript representing the value of coding digits. In 1-bit coding, the digits “0” and “1” correspond to two different coding particles of w0 and w1, whose reflection phase difference is 180°. In 2-bits coding, the digits “00”, “01”, “10”, and “11” correspond to four different coding particles of w0, w1, w2, and w3, whose reflection phase difference is 90°. For simplicity, the 2-bits digits “0”, “1”, “2”, and “3” stand for “00”, “01”, “10”, and “11”, respectively. The parameters of different coding particles mentioned in the paper are shown in Table 1.
Fig. 2. The X-band DCMs models. (a) Schematic illustration of multi-beam generation function of DCMs under the normal illumination. (b) Phase-coding pattern configuration of DCM. (c) Structure of the square-shaped coding particles. (d)-(f) Reflection phases of the coding particles cell-1, cell-2, and cell-3, respectively.
Table 1. Parameters of Coding Particles
3.1 Element period p = 1/2 λ
To validate the effect of the proposed DCM-CZT method and traditional DFT method on far-field pattern estimations, we present two models of DCMs, with element period p = 1/2 λ. The DCM based on the periodic coding sequence “0011223300112233…” is denoted as S1, whereas the DCM based on the periodic coding sequence “1111333311113333…” is denoted as S2. The coding pattern configuration is 48 ${\times}$ 48. Both S1 and S2 are arranged by coding particle cell-1 shown in Table 1. The reflection phases of the four cells based on cell-1 are illustrated in Fig. 2(d). The far-field pattern results for S1 and S2 using the presented DCM-CZT method and traditional DFT method are demonstrated in Fig. 3, with the reference results of full-wave simulations by CST. The calculated results are consistent with the full-wave simulations very well. The main-lobe of S1 is located at θ=-14.4° (φ=0°), and the main-lobe of S2 is located at θ=±14.3° (φ=0°). When the element period is p = 1/2 λ, the far-field patterns of DCM can be accurately estimated by both DCM-CZT and DFT methods.
Fig. 3. The far-field patterns for 2-bits DCMs with the element period p = 1/2 λ. (a) DCM S1. (b) DCM S2.
3.2 Element period p < 1/2 λ
To further validate the effect of the DCM-CZT and DFT methods, we consider the case when the element period is less than half wavelength. We present four different DCMs with the element period p < 1/5 λ. The DCM with the periodic coding sequence “0011223300112233…” is denoted as S3, whereas the DCM with the periodic coding sequence “1111333311113333…” is denoted as S4. The coding pattern configuration is 48${\times} $48. Both S3 and S4 are arranged by coding particle cell-2 shown in Table 1. The reflection phases of the four cells based on cell-2 are presented in Fig. 2(e). The far-field pattern results for S3 and S4 using the DCM-CZT and DFT methods are illustrated in Fig. 4(a) and Fig. 4(b), with the reference results of full-wave simulations by CST. The calculated results of presented DCM-CZT method are consistent with the full-wave simulations. The main-lobe of S3 is located at θ=-48.6° (φ=0°), and the main-lobes of S4 are located at θ=±48.1° (φ=0°). The DCMs based on the aperiodic coding sequences “001011” and “0010100” are denoted as S5 and S6, and their phase matrix configurations are $42 \times 42$ and $49 \times 49$, respectively. Both S5 and S6 are arranged by coding particle cell-3 shown in Table 1. The reflection phases of the two cells based on cell-3 are demonstrated in Fig. 2(f). By comparing Fig. 2(d), Fig. 2(e), and Fig. 2(f), we can see that the effective bandwidth of the 1-bit coding particle is significantly wider than that of the 2-bits coding particle. The far-field pattern results for S5 and S6 using the DCM-CZT and DFT methods are shown in Fig. 4(c) and Fig. 4(d), with the reference results of full-wave simulations by CST. The calculated results of the DCM-CZT method are consistent with the full-wave simulations. The main-lobes of S5 are located at θ=±4.5°, ± 14.8° (φ=0°), and the main-lobes of S6 are located at θ=0°, ± 7.6°, ± 16.5° (φ=0°).
Fig. 4. The far-field results for DCMs with the element period p < 1/5 λ. (a),(b) The global and partial far-field patterns of 2-bits periodic DCMs S3 and S4, respectively. (c),(d) The global and partial far-field patterns of 1-bit aperiodic DCMs S5 and S6, respectively.
When the element period is p < 1/5 λ, the far-field patterns of both periodic and aperiodic phase-coding metasurfaces can be accurately estimated by the DCM-CZT method. However, the results of the traditional DFT method (not perform deleting points or filling points) show that the main-lobe of S3 is located at θ=-14.5° (φ=0°), the main-lobes of S4 are located at θ=±14.3° (φ=0°), the main-lobes of S5 are located at θ=±1.8°, ± 5.5° (φ=0°), and the main-lobes of the S6 are located at θ=0°, ± 3.0°, ± 6.2° (φ=0°). The deviation between the results of the traditional DFT method and full-wave simulation is obvious and unacceptable.
3.3 Partial far-field patterns for specific orientations
We assume two target orientations $\theta _1^\ast $ and $\theta _2^\ast $ for partial scattering pattern calculations. Here, $\theta _1^\ast $= −60∼−30° (φ=0°) is set for S3 and S4, whereas $\theta _2^\ast $= −30∼30° (φ=0°) is set for S5 and S6. The partial scattering results for S3-S6 using the presented DCM-CZT method are shown in Fig. 4, marked with blue triangles. It can be seen that the results of partial scattering patterns agree very well with the full-wave simulations. Hence the DCM-CZT method can improve the space- resolution by partial pattern calculations, with the same or more sampling points than the global pattern calculations, to effectively avoid the calculation error caused by the fence effect.
4. Experiment and error analysis
We fabricate four samples of DCMs, S3-S6, as exhibited in Fig. 5 to verify the effect of the DCM-CZT method. The test environment is shown in Fig. 5(e) and Fig. 5(f), and the relevant experimental results are illustrated in Fig. 6, with the results of full-wave simulations and the DCM-CZT method as references. Figure 6(a) and Fig. 6(b) show the scattering patterns of S3 and S4 at the frequency of 10 GHz, and φ=0° on the azimuth plane. Figure 6(c) and Fig. 6(d) show the scattering patterns of S5 and S6 at the frequency of 11.4 GHz, and φ=0° on the azimuth plane. The differences between the measured and simulated results are mainly due to the manufacturing tolerances and shrinkage, and some errors could be caused by the measurement setup. The elevation of the side-lobes is related to the disturbance of the test environment, and the beam broadening of the outermost two beams shown in Fig. 6(d) could be caused by the high side-lobes nearby. Despite of such differences, the measured and simulation results are in acceptable agreements.
Fig. 5. Fabricated sample and experimental setup. (a)-(d) Photographs of the fabricated DCM samples of S3, S4, S5, and S6. (e),(f) Photographs of the experimental setups for the measurements of far-field patterns.
Fig. 6. Experimental results of the four fabricated samples. (a),(b) The measured far-field patterns of the 2-bits periodic DCMs S3 and S4. (c),(d) The measured far-field patterns of the 1-bit aperiodic DCMs S5 and S6.
5. Conclusion
We proposed a DCM-CZT method to modify the traditional DFT for far-field pattern estimation. The DCM-CZT method can flexibly and rapidly deal with DCMs with arbitrary element periods. The computation of DCM-CZT can be accelerated by FFT, and the flexible solution path brings the DCM-CZT method various unique capabilities. The DCM-CZT method can also be used to calculate partial far-field patterns for specific orientations. This method can efficiently improve the partial spatial resolution to avoid the calculation error caused by the fence effect, and further increase the speed of calculation. It is helpful for the programmable DCMs to estimate partial far-field patterns in specific orientations rapidly and effectively, which has potential application values in radar detection and wireless communications.
We choose 1024-point DCM-CZT sampling for six DCMs models in this paper. It should be noted that the accuracy of the calculation result can be guaranteed when the sampling points are smaller than 1024-point. The acceptable number of minimum sampling points depends on the designed direction of the main-lobes in the far-field pattern. Because of the nonlinear relationship between the sinusoidal space and the far-field space, the farther the deviation of designed beam direction from the normal direction is, the more sampling points are needed to ensure the accuracy of the calculation results.
The results from six DCMs models validate the performance of the DCM-CZT method. We show that both DCM-CZT and DFT methods can accurately estimate the far-field patterns when the element period is equal to half wavelength, and the consuming time of DCM-CZT is on the same order of magnitude as that of DFT. When the element period is less than half wavelength, the DCM-CZT method is still fast and accurate, whereas the error of traditional DFT method cannot be ignored. Compared with the traditional DFT method whose solution path is unit circle, the solution path of DCM-CZT can be extended from unit circle to a segment of helix. This provides a higher degree of freedom for the design of super-large-scale DCMs.
Funding
National Natural Science Foundation of China (92167202, 62288101); National Key Research and Development Program of China (2022YFA1404903, 2017YFA0700201, 2017YFA0700202, 2017YFA0700203); Major Project of Natural Science Foundation of Jiangsu Province (BK20212002); State Key Laboratory of Millimeter Waves, Southeast University, China (K201924); Fundamental Research Funds for the Central Universities (2242018R30001, 2242022R20017); 111 Project (111-2-05); China Postdoctoral Science Foundation (2022T150112, 2021M700761).
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.
Supplemental document
See Supplement 1 for supporting content.
References
1. T. J. Cui, D. R. Smith, and R. P. Liu, Metamaterials: Theory, Design, and Applications, 1st ed. (Springer, 2009).
2. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]
3. R. A. Shelby, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef]
4. H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1(1), 124 (2010). [CrossRef]
5. B. Li and Z. X. Shen, “Wideband 3D frequency selective rasorber,” IEEE Trans. Antennas Propag. 62(12), 6536–6541 (2014). [CrossRef]
6. T. J. Cui, M. Q. Qi, X. Wan, J. Zhao, and Q. Cheng, “Coding metamaterials, digital metamaterials and programmable metamaterials,” Light: Sci. Appl. 3(10), e218 (2014). [CrossRef]
7. S. Liu, T. J. Cui, Q. Xu, D. Bao, L. Du, X. Wan, W. X. Tang, C. Ouyang, X. Y. Zhou, and H. Yuan, “Anisotropic coding metamaterials and their powerful manipulation of differently polarized terahertz waves,” Light: Sci. Appl. 5(5), e16076 (2016). [CrossRef]
8. S. Liu, T. J. Cui, L. Zhang, Q. Xu, Q. Wang, X. Wan, J. Q. Gu, W. X. Tang, M. Qing Qi, and J. G. Han, “Convolution operations on coding metasurface to reach flexible and continuous controls of terahertz beams,” Adv. Sci. 3(10), 1600156 (2016). [CrossRef]
9. S. Liu, L. Zhang, Q. L. Yang, Q. Xu, Y. Yang, A. Noor, Q. Zhang, S. Iqbal, X. Wan, and Z. Tian, “Frequency-dependent dual-functional coding metasurfaces at terahertz frequencies,” Adv. Opt. Mater. 4(12), 1965–1973 (2016). [CrossRef]
10. S. Iqbal, S. Liu, G. D. Bai, M. Furqan, H. A. Madni, and T. J. Cui, “Dual-band 2-bit coding metasurface for multifunctional control of both spatial waves and surface waves,” J. Opt. Soc. Am. B 36(2), 293–299 (2019). [CrossRef]
11. S. Iqbal, S. Liu, J. Luo, L. Zhang, H. A. Madni, and T. J. Cui, “Controls of transmitted electromagnetic waves for diverse functionalities using polarization-selective dual-band 2-bit coding metasurface,” J. Opt. 22(1), 015104 (2020). [CrossRef]
12. S. Iqbal, M. R. Akram, M. Furqan, H. A. Madni, M. I. Khan, and G. X. Shu, “Broadband and High-Efficiency Manipulation of Transmitted Vortex Beams via Ultra-Thin Multi-Bit Transmission Type Coding Metasurfaces,” IEEE Access 8, 197982–197991 (2020). [CrossRef]
13. S. Liu and T. J. Cui, “Concepts, working principles, and applications of coding and programmable metamaterials,” Adv. Opt. Mater. 5(22), 1700624 (2017). [CrossRef]
14. Q. Ma and T. J. Cui, “Information metamaterials: bridging the physical world and digital world,” PhotoniX 1(1), 1 (2020). [CrossRef]
15. Q. Ma, Q. Xiao, Q. R. Hong, X. Gao, V. Galdi, and T. J. Cui, “Digital coding metasurfaces: from theory to applications,” IEEE Antennas Propag. Mag. 64(4), 96–109 (2022). [CrossRef]
16. Y. B. Li, L. L. Li, B. B. Xu, W. Wu, R. Y. Wu, X. Wan, Q. Cheng, and T. J. Cui, “Transmission-type 2-bit programmable metasurface for single-sensor and single-frequency microwave imaging,” Sci. Rep. 6(1), 23731 (2016). [CrossRef]
17. W. Xiang, Q. Q. Mei, Y. C. Tian, and T. J. Cui, “Field-programmable beam reconfiguring based on digitally-controlled coding metasurface,” Sci. Rep. 6(1), 20663 (2016). [CrossRef]
18. L. Li, T. J. Cui, J. Wei, S. Liu, J. Ding, W. Xiang, B. L. Yun, M. Jiang, C. W. Qiu, and Z. Shuang, “Electromagnetic reprogrammable coding-metasurface holograms,” Nat. Commun. 8(1), 197 (2017). [CrossRef]
19. Q. Ma, G. D. Bai, H. B. Jing, C. Yang, L. Li, and T. J. Cui, “Smart metasurface with self-adaptively reprogrammable functions,” Light: Sci. Appl. 8(1), 98 (2019). [CrossRef]
20. H. Wang, Z. Qin, L. L. Huang, Y. F. Li, R. Z. Zhao, H. Q. Zhou, H. Y. He, J. Q. Zhang, and S. B. Qu, “Metasurface with dynamic chiral meta-atoms for spin multiplexing hologram and low observable reflection,” PhotoniX 3(1), 10 (2022). [CrossRef]
21. Z. Gu, Q. Ma, C. Liu, Q. Xiao, X. Gao, T. Yan, L. Miao, L. Li, and T. J. Cui, “High-resolution programmable metasurface imager based on multilayer perceptron network,” Adv. Opt. Mater. 10(18), 2200619 (2022). [CrossRef]
22. L. Chen, Q. Ma, S. S. Luo, F. J. Ye, H. Y. Cui, and T. J. Cui, “Touch-programmable metasurface for various electromagnetic manipulations and encryptions,” Small 18(45), 2203871 (2022). [CrossRef]
23. Q. Xiao, W. Gao, Q. Ma, Z. Gu, Y. Zhou, C. Liu, Z. A. Huang, X. Wan, L. Li, Y. Li, and T. J. Cui, “Electromagnetic brain-computer-metasurface holography,” ACS Photonics (to be published). [CrossRef]
24. T. J. Cui, S. Liu, G. D. Bai, and Q. Ma, “Direct transmission of digital message via programmable coding metasurface,” Research 2019, 2584509 (2019). [CrossRef]
25. X. Wan, C. K. Xiao, H. Huang, Q. Xiao, W. Xu, Y. H. Li, J. Eisenbeis, J. W. Wang, Z. A. Huang, Q. Cheng, S. Jin, T. Zwick, and T. J. Cui, “Joint modulations of electromagnetic waves and digital signals on a single metasurface platform to reach programmable wireless communications,” Engineering 8, 86–95 (2022). [CrossRef]
26. Q. Ma, W. Gao, Q. Xiao, L. S. Ding, T. Y. Gao, Y. J. Zhou, X. X. Gao, T. Yan, C. Liu, Z. Gu, X. Kong, Q. H. Abbasi, L. Li, C.-W. Qiu, Y. Li, and T. J. Cui, “Directly wireless communication of human minds via non-invasive brain-computer-metasurface platform,” eLight 2(1), 11 (2022). [CrossRef]
27. L. Zhang, M. Z. Chen, W. Tang, J. Y. Dai, L. Miao, X. Y. Zhou, S. Jin, Q. Cheng, and T. J. Cui, “A wireless communication scheme based on space- and frequency-division multiplexing using digital metasurfaces,” Nat. Electron. 4(3), 218–227 (2021). [CrossRef]
28. C. Liu, Q. Ma, Z. J. Luo, Q. R. Hong, Q. Xiao, H. C. Zhang, L. Miao, W. M. Yu, Q. Cheng, L. L. Li, and T. J. Cui, “A programmable diffractive deep neural network based on a digital-coding metasurface array,” Nat. Electron. 5(2), 113–122 (2022). [CrossRef]
29. Z. H. Zhao, Y. Wang, C. S. Guan, K. Zhang, Q. Wu, H. Y. Li, J. Liu, S. N. Burokur, and X. M. Ding, “Deep learning-enabled compact optical trigonometric operator with metasurface,” PhotoniX 3(1), 15 (2022). [CrossRef]
30. X. Gao, Q. Ma, Z. Gu, W. Y. Cui, C. Liu, J. Zhang, and T. J. Cui, “Programmable surface plasmonic neural networks for microwave detection and processing,” Nat. Electron. 6(4), 319–328 (2023). [CrossRef]
31. L. L. Wang, D. G. Fang, and W. X. Sheng, “Combination of genetic algorithm and fast Fourier transform for synthesis of arrays,” Microw. Opt. Technol. Lett. 37(1), 56–59 (2003). [CrossRef]
32. Y. K. Chen, S. Yang, and Z. Nie, “Synthesis of satellite footprint patterns from time-modulated planar arrays with very low dynamic range ratios,” Int. J. Numer. Model. 21(6), 493–506 (2008). [CrossRef]
33. J. X. Li and R. Chen, “Fast computation of planar phased array pattern based on FFT,” J. Microw. 25(6), 12–17 (2009).
34. G. K. Mahanti and A. Chatterjee, “Combination of fast Fourier transform and self-adaptive differential evolution algorithm for synthesis of phase-only reconfigurable rectangular array antenna,” Ann. Telecommun. 69(9-10), 515–527 (2014). [CrossRef]
35. L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp Z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969). [CrossRef]
36. D. Reddy and V. V. Rao, “Two-dimensional chirp Z-transform,” Signal Process. 4(1), 17–24 (1982). [CrossRef]
