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A method of rapidly designing graphene-based terahertz diffusion surface

Wang Jian Zhang Chao-Yue Yao Zhao-Yu Zhang Chi Xu Feng Yang Yuan

Su Xin, Huang Tian-Ye, Wang Jun-Zhuan, Liu Yuan, Zheng You-Liao, Shi Yi, Wang Xiao-Mu. Circular photogalvanic effect. Acta Phys. Sin., 2021, 70(13): 138501. doi: 10.7498/aps.70.20210498
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A method of rapidly designing graphene-based terahertz diffusion surface

Wang Jian, Zhang Chao-Yue, Yao Zhao-Yu, Zhang Chi, Xu Feng, Yang Yuan
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  • Electromagnetic diffusion surface can reduce the radar cross section, thus profiting stealth of targets. Terahertz diffusion surface has a wide prospect in the field of next-generation radar and communication, promising to act as a kind of intelligent smart skin. In this paper, utilizing the excellent tunable properties of graphene in the terahertz band, a hybrid structure of graphene and metal which has inverse phase response of reflecting waves is proposed. The reflection phase switches in the mechanism of resonant modes and can be controlled efficiently by the bias voltage. Meanwhile, unlike metal materials, graphene has a non-negligible loss characteristic, which leads the response amplitudes corresponding to the two different switching states to be inconsistent with each other. According to the interference and superposition principle of electromagnetic field, it is not conducive to eliminating the coherent far-field, leading to an unsatisfactory diffusion result. In this paper, we present a “molecular” structure by secondary combination of the above-mentioned reverse phase element states, and take it as the basic element of the diffusion surface. Finally, we use particle swarm optimization to optimize the arrangement of “molecular” structures. The final diffusion surface consists of a combinatorial design of “molecules” rather than randomly distributed reflection units. In addition, molecules designed artificially have similar amplitude responses but different phase responses, which improves the convergence speed and reduces the computation quantity during algorithm evolution. The method of designing molecular structure, described in this paper, is simple, rapid and widely applicable, which effectively improves the amplitude-to-phase modulation ability of graphene metasurface against electromagnetic waves. When diffuse reflection optimization is applied to most of graphene metasurfaces, the method described in this paper can achieve the results that are the same as or even better than the results after a large number of iterations of traditional particle swarm optimization in the most computation-efficient manner. The results show that the dynamic diffusion surface designed by this method has the advantages of fast convergence speed and small far-field peak.
      Corresponding author: Xu Feng, feng.xu@njupt.edu.cn
    • Funds: Project suported by the Open Fund of the National and Local Joint Engineering Laboratory of RF Integration and Micro-Assembly Technology, China (Grant No. KFJJ20180205), the Science and Research Start-up Fund for Introducing Talents of Nanjing University of Posts and Telecommunication, China (Grant No. NY218113), and the Nanjing University of Posts and Telecommunication Science Foundation, China (Grant No. NY219077).

    具有低散射特性的太赫兹电磁表面在成像、雷达、新型光源等领域具有广阔的应用前景[1-6], 而吸波与漫反射是低散射电磁表面的两种主要机理[7-9]. 其中, 漫反射是利用电磁波的相干原理将散射场能量分散到自由空间各个方向当中, 从而将其峰值最小化[10]. 在微波频段, 利用成熟的印刷电路板(PCB)贴片结构即可便捷地获得反射场具有$ \mathrm{\pi }/2 $$ \mathrm{\pi } $及其他相位差的单元, 进一步还可借助微波二极管对单元反射相位进行动态灵活控制[11]. 由于损耗很小, 贴片结构可以近似认为全反射, 不同相位响应的单元反射幅度可视为均匀一致. 另一方面, 漫反射电磁表面需要对单元结构进行排列设计, 一种途径是遍历寻找最优值[12], 另一种是借助优化算法, 如模拟退火、粒子群优化[13]、遗传算法[14,15]等. 为取得更加理想的低散射效果, 可以在单元结构内引入损耗材料, 将吸波与漫反射结合[16], 不同编码单元依然能够实现理想的相位差, 但反射场幅值不一致, 不利于远场相干相消, 阵列设计优化算法面临很大挑战[17].

    在太赫兹频段, 金属及衬底的损耗依然很小, 漫反射电磁表面设计方法与微波频段类似[18], 但是微波二极管等相位控制部件在太赫兹频段难以正常发挥功能, 不利于动态灵活调控. 而石墨烯材料在太赫兹频段具有损耗较小、感性可调特点, 在太赫兹波束调控方面已有较成熟的研究[19,20]. 进一步地, 将石墨烯与金属结构协同设计, 有望实现幅度与相位高效调控、实现太赫兹散射可调的电磁功能表面[21]. 本文提出了一种太赫兹低散射电磁表面的设计方法, 首先设计了一种石墨烯与金属协同的太赫兹反射场调控单元结构, 该结构能够实现反射场反相控制功能. 接着, 针对反射场幅度不一致的问题, 研究了一种改进的粒子群优化(PSO)算法实现阵列结构排布方式的快速计算, 得到电磁表面的图案, 并且与文献[18]金属结构不同的是, 石墨烯自身具备一定的损耗特性, 结合漫反射, 该电磁表面有望获得更低的低散射特性.

    本文设计了一种基于谐振模式开关机理的石墨烯/金属协同结构单元, 其由石墨烯贴片、金属电极、金属过孔、介质隔层、馈电网络、金属背板构成. 通过馈电网络可对各个不同单元上的石墨烯加载偏置电压, 进而改变其化学势参数, 由其组成的阵列如图1(a)所示. 而图1(b)表示了太赫兹波照射下所设计结构处于非谐振/谐振状态条件时的电流分布示意. 其原理是: 在未加载偏置电压下, 结构处于非谐振状态, 石墨烯近似于一种“透明”材料, 太赫兹反射场的相位变化由介质及背板贡献而得, 感应电流分布在金属背板上; 而当结构加载偏置电压后, 石墨烯化学势的随之升高、导电特性增强, 结构上的感应电流可以沿着过孔过渡到石墨烯表层, 形成环形电流, 结构产生磁响应谐振, 并引发反射场相位的骤变.

    Figure 1.  (a) Unit cell of graphene diffuse reflecting surface; (b) the current distribution under resonance/non-resonance condition.

    考虑金属材质为金(电导率${\sigma }_{\mathrm{A}\mathrm{u}}=4.561\times {10}^{7} \mathrm{S}/\mathrm{m}$), 工作频率$\lambda =1\;\mathrm{T}\mathrm{H}\mathrm{z}$, 经过设计, 单元周期 D = 60 μm, 石墨烯贴片的缝隙$ g $$ D $关系为$g= D/10$, 介质层厚度${t}_{\mathrm{d}}=30$ μm, 等效介电常数${\epsilon }_{\mathrm{r}}= 2.349$, 金属电极为边长$a=10$ μm 的正方形. 在太赫兹频段下石墨烯的电导率$ {\sigma }_{\mathrm{g}\mathrm{r}} $根据Drude模型计算而得, 考虑弛豫时间$ \tau =0.5\mathrm{p}\mathrm{s} $, 化学势分别为0与0.8 eV, 石墨烯的表面阻抗$ {Z}_{\mathrm{g}\mathrm{r}}=1/{\sigma }_{\mathrm{g}\mathrm{r}} $的色散曲线如图2所示.

    Figure 2.  Dispersion curves of graphene surface impedance: (a) Real part; (b) imaginary part.

    在此基础上, 使用CST Studio电磁全波仿真工具可得该模型对应反射场的幅度与相位响应, 如图3所示.

    Figure 3.  Reflecting response of amplitude (a) and phase (b) of graphene diffuse reflecting surface unit cell.

    图3可知, 在化学势为0时, 石墨烯类似一种损耗轻微的“透明”材料, 与没有石墨烯(w/o graphene曲线)时相比, 石墨烯/金属协同结构的幅度、相位特性接近一致. 而当化学势上升到0.8 eV时, 石墨烯/金属协同结构开始出现明显的谐振, 如图蓝色箭头标注所示. 可知, 在太赫兹频段下, 对于图1(a)所示的结构, 通过改变化学势的方法能够激励石墨烯/金属协同结构特定的谐振模式, 最终获得理想的幅度调制和相位调制效果.

    根据场的干涉与叠加原理, 基于石墨烯/金属协同结构单元设计的反射表面阵列结构的远场方向图表达式为

    $ f\left(\theta , \phi \right)={f}_{\mathrm{e}}\left(\theta , \phi \right)AF\left(\theta , \phi \right)\text{, } $

    (1)

    $\begin{split} & AF(\theta , \phi)\\ =& \sum _{m=1}^{M}\sum _{n=1}^{N}\mathrm{exp}\Big\{\!-\mathrm{j}\Big[\!\Big(m-\frac{1}{2}\Big){k}_{0}d\sin\theta \cos\phi \\ &+\Big(n-\frac{1}{2}\Big){k}_{0}d\sin\theta \sin\phi +\varphi (m, n)\Big]\Big\},\\[-15pt] \end{split}$

    (2)

    其中$ {f}_{\mathrm{e}}\left(\theta , \phi \right) $表示单元的方向图函数; $ \theta $$ \phi $分别表示极角与方位角; $ M $$ N $分别代表阵列的行数与列数, 本文实验中使用正方形阵列, 故$ M=N $; $ d $表示xy方向上元素的周期, $ {k}_{0} $是自由空间波数. 图4(a)图4(b)分别表示石墨烯化学势分别为0和0.8 eV时, 石墨烯/金属协同单元结构的方向图函数.

    Figure 4.  Pattern function of unit cell under different chemical potentials of graphene: (a) 0 eV; (b) 0.8 eV.

    本文提出的动态漫反射表面的设计方法如图5所示. 其中, 根据图1结构与图3结果, 定义两种反相单元表示为“原子”A与B, 在工作频率为1 THz时其反射幅度归一化后比值${R}_{\mathrm{A}}:{R}_{\mathrm{B}}= 1:0.81$, 如图5(a)所示. 同时, 将构成单个“分子”的A, B两种“原子”数量比例$ {N}_{\mathrm{A}}:{N}_{\mathrm{B}} $选取为$ \mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\left({R}_{\mathrm{B}}:{R}_{\mathrm{A}}\right)=4:5 $, 即设计“分子”中共包含9个“原子”. 在排列组合方面, 尽可能减少同相“原子”的相邻情形, 本文共设计五种“分子”结构, 如图5(b)所示.

    Figure 5.  Design method of dynamic diffusion reflecting surface of graphene: (a) “Atom” property; (b) “lattice” structure; (c) algorithm flowchart.

    使用粒子群算法对“分子”结构进行阵列优化流程图如图5(c)所述: 初始输入具有一定粒子个数的种群, 而粒子在本文指的是漫反射阵列的图案, 其由“分子”以随机方式初始化分布组成. 在已知阵列结构条件下根据(1)式计算远场数据, 并记录远场的峰值, 将其作为判断粒子优化质量的参数. 算法根据本次计算峰值与历史最优值比较, 计算函数根据当前最优解以及种群的全局最优解来更新粒子的速度和方向; 经过多次迭代后, 当种群全局最优解的位置满足收敛最小界限时结束, 以此时最优解对应的“分子”排布作为漫反射阵列图案. 优化结束后, 先对输出的种群内的全部粒子求取远场峰值, 而后将这组远场峰值作为粒子群输出参数求其方差与均值, 以验证优化方案的质量.

    粒子群算法的粒子分布方差和均值衡量了粒子距离数据集合中心的大小关系和每次迭代的收敛情况, 其方差与均值分别代表着粒子的离散程度与粒子精度, 方差越小说明粒子离散程度小、算法收敛能力越强; 均值越小, 代表着反射阵列对入射电磁波的漫反射能力越强, 优化结果的性能更好. 定义粒子方差为

    $ {s}^{2}\left(x\right)=\frac1{n-1} {\sum\limits _{i=1}^{n}{({x}_{i}-{\bar x})}^{2}}\text{, } $

    (3)

    粒子均值为

    $ \bar{x}= \frac 1n {\sum\limits _{i=1}^{n}{x}_{i}}, $

    (4)

    其中$ n $为本文实验中设置的粒子总数, 即有$ n $个随机的阵列排布方式参与优化; $ x $为粒子在优化过程结束后的远场峰值, 即根据优化结束后得到的最优“分子”排布方式, 对阵列仿真后得到的远场峰值.

    每个种群个体数量对应“分子”阵列的规模$ M\times M $, 而每个“分子”的构成方式为图5(b)中随机抽取产生.

    为验证本文提出的方法的有效性, 定义漫反射表面优化效率为

    $ \rho =-20\mathrm{log}({{R}_{\mathrm{o}\mathrm{p}}}/{{R}_{\mathrm{m}\mathrm{a}\mathrm{x}}})\text{, } $

    (5)

    其中$ {R}_{\mathrm{m}\mathrm{a}\mathrm{x}} $表示所有单元反射同相位情况下的阵列结构的反射场峰值, $ {R}_{\mathrm{o}\mathrm{p}} $表示经过基于“分子”结构的粒子群算法优化后的漫反射表面散射场的峰值.

    首先考虑“分子”阵列的规模M = 30, 使用本文提出的算法进行优化, 粒子总数设置为11, 结果如图6(a)所示, 远场峰值可以在30次迭代内缩收敛至–25 dB, 最终收敛为可达到 –25.8 dB, 漫反射效果好, 远场分布图与收敛后对应的的阵列结构如图6(b)图6(c)所示, 其中远场峰值为42.9 V/m, 而阵列由图5(b)所述“分子”为基本元素组成.

    Figure 6.  (a) Iterative convergence process of diffusion calculation; (b) far field distribution of diffusion surface after optimization; (c) array structure of reflecting surface.

    为了证明本文提出的方法具有普适性, 另选取9 × 9, 18 × 18, 27 × 27三种规模的“分子”阵列进行漫反射优化. 并且, 为了便于对比, 同时使用传统的PSO算法进行计算, 对比结果如表1所列, 优化过程如图7所示.

    Table 1.  Effect comparison between the traditional PSO and the proposed method in this paper.
    阵列
    规模
    传统PSO 本文方法
    远场峰
    值/(V·m–1)
    优化效
    率/dB
    远场峰
    值/(V·m–1)
    优化效
    率/dB
    9 × 9 10.08 –16.80 6.72 –21.00
    18 × 18 27.60 –20.74 19.60 –23.71
    27 × 27 45.00 –23.54 34.00 –25.97
     | Show Table
    DownLoad: CSV
    Figure 7.  Comparisons between the traditional PSO and the proposed method in this paper for three sizes of “molecular” arrays: (a) 9 × 9; (b) 18 × 18; (c) 27 × 27.

    可以直观地看出, 利用“分子”算法设计不同规模的石墨烯太赫兹动态漫反射表面, 均体现较好的鲁棒性, 计算而得的远场峰值缩减程度与优化效率明显优于传统的PSO算法. 在计算速度方面, 本文方法也体现出比传统PSO更大的优势, 以9 × 9规模优化实验为例, 在两种算法使远场峰值收敛至 –17 dB左右时, 本文方法在5次迭代以内即可完成, 而传统PSO需要迭代大约20次. 且由于传统PSO较强的随机性, 迭代结果难以达到最优化.

    此外, 从粒子群优化的质量这一角度分析结果, 横向比较各种规模阵列优化后的粒子群分布情况. 以算法第200次迭代所产生的粒子均值和方差做分析, 对比结果如表2所列, 与之对应的粒子分布如图8所示.

    Table 2.  Particle distribution comparisons between the traditional PSO and the proposed method in this paper.
    阵列规模 粒子方差 粒子均值
    传统PSO 本文方法 传统PSO 本文方法
    9 × 9 1.12 0.19 11.34 7.09
    18 × 18 2.04 1.65 28.45 21.67
    27 × 27 28.69 10.25 49.57 38.04
     | Show Table
    DownLoad: CSV
    Figure 8.  Particle distribution comparisons between the traditional PSO algorithm and the proposed method for three sizes of “molecular” arrays: (a) 9 × 9; (b) 18 × 18; (c) 27 × 27.

    通过比较优化结果粒子群的方差与均值, 能够比较优化算法的离散程度与粒子精度. 本文方法方差均不同程度地小于传统PSO优化结果, 即本文方法粒子离散程度较小, 粒子分布更集中在最优解附近, 使得粒子权重计算更有效, 算法预估优化结果更优; 并且, 本文方法优化结果粒子均值也更小, 即对入射太赫兹电磁波的反射幅度更小, 优化精度更高, 所产生的编码阵列漫反射性能更优, 符合较低方差对优化结果的预估. 故本文方法相比传统PSO算法更具优势.

    虽然利用传统PSO对石墨烯电磁表面可实现漫反射优化, 但是运用所述分子算法后, 可高效进行原PSO难以完成的大规模优化, 且效率高于传统PSO方法. 本文所述的“分子”结构设计方法简单, 设计速度快, 适用范围广, 有效地增强了石墨烯超表面对电磁波的幅度相位调制能力. 在大部分石墨烯超表面单元进行漫反射优化时, 只需应用此方法, 就能够以最节省计算资源的方式, 获取与传统PSO大量迭代后相同甚至更优的效果. 其原因在于在优化前向阵列加入了固定的"原子"数量这一条件, 且人工设计的每个分子都具有相似的幅度响应, 而仅相位响应不同, 这样提高了算法进化过程中的收敛速度、减少了计算量. 利用分子算法结合PSO对石墨烯阵列结构进行漫反射优化, 优化效率高, 且在对不同规模阵列优化时, 优化效率十分稳定, 随机性小于传统PSO. 因此本文方法能够最大程度地排除传统PSO算法中过高的随机性对优化结果的干扰, 利于对所设计的石墨烯超表面单元进行快速验证.

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  • 图 1  (a) 石墨烯漫反射表面单元结构; (b) 谐振/非谐振状态下电流分布

    Figure 1.  (a) Unit cell of graphene diffuse reflecting surface; (b) the current distribution under resonance/non-resonance condition.

    图 2  石墨烯表面阻抗色散曲线 (a) 实部; (b) 虚部

    Figure 2.  Dispersion curves of graphene surface impedance: (a) Real part; (b) imaginary part.

    图 3  石墨烯漫反射表面单元结构的反射幅度(a)与相位(b)响应

    Figure 3.  Reflecting response of amplitude (a) and phase (b) of graphene diffuse reflecting surface unit cell.

    图 4  石墨烯化学势不同情况下单元结构的方向图函数 (a) 0 eV; (b) 0.8 eV

    Figure 4.  Pattern function of unit cell under different chemical potentials of graphene: (a) 0 eV; (b) 0.8 eV.

    图 5  石墨烯动态漫反射表面的设计方法 (a) “原子”性质; (b) “分子”图案; (c) 算法流程

    Figure 5.  Design method of dynamic diffusion reflecting surface of graphene: (a) “Atom” property; (b) “lattice” structure; (c) algorithm flowchart.

    图 6  (a) 基于“分子”方法的漫反射计算迭代收敛过程; (b) 优化后产生漫反射表面远场分布; (c) 漫反射表面的阵列结构

    Figure 6.  (a) Iterative convergence process of diffusion calculation; (b) far field distribution of diffusion surface after optimization; (c) array structure of reflecting surface.

    图 7  三种规模的“分子”阵列情形下使用传统PSO算法与本文提出的方法优化过程的对比 (a) 9 × 9; (b) 18 × 18; (c) 27 × 27

    Figure 7.  Comparisons between the traditional PSO and the proposed method in this paper for three sizes of “molecular” arrays: (a) 9 × 9; (b) 18 × 18; (c) 27 × 27.

    图 8  三种规模的“分子”阵列情形下使用传统PSO算法与本文提出的方法的粒子分布对比 (a) 9 × 9; (b) 18 × 18; (c) 27 × 27

    Figure 8.  Particle distribution comparisons between the traditional PSO algorithm and the proposed method for three sizes of “molecular” arrays: (a) 9 × 9; (b) 18 × 18; (c) 27 × 27.

    表 1  传统PSO算法与本文提出的方法的效果对比

    Table 1.  Effect comparison between the traditional PSO and the proposed method in this paper.

    阵列
    规模
    传统PSO 本文方法
    远场峰
    值/(V·m–1)
    优化效
    率/dB
    远场峰
    值/(V·m–1)
    优化效
    率/dB
    9 × 9 10.08 –16.80 6.72 –21.00
    18 × 18 27.60 –20.74 19.60 –23.71
    27 × 27 45.00 –23.54 34.00 –25.97
    DownLoad: CSV

    表 2  传统PSO算法与本文提出的方法的粒子分布比较

    Table 2.  Particle distribution comparisons between the traditional PSO and the proposed method in this paper.

    阵列规模 粒子方差 粒子均值
    传统PSO 本文方法 传统PSO 本文方法
    9 × 9 1.12 0.19 11.34 7.09
    18 × 18 2.04 1.65 28.45 21.67
    27 × 27 28.69 10.25 49.57 38.04
    DownLoad: CSV
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Metrics
  • Abstract views:  6448
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Publishing process
  • Received Date:  01 July 2020
  • Accepted Date:  10 August 2020
  • Available Online:  19 January 2021
  • Published Online:  05 February 2021

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