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Inverse design method of microscatterer array for realizing scattering field intensity shaping

Wang Zhi-Peng Wang Bing-Zhong Liu Jin-Pin Wang Ren

Inverse design method of microscatterer array for realizing scattering field intensity shaping

Wang Zhi-Peng, Wang Bing-Zhong, Liu Jin-Pin, Wang Ren
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  • It is a novel and interesting idea to inversely design the scattering structure with the desired scattering field intensity distribution in a given target area as the known information. The inverse design method proposed in this paper does not need to be optimized, and the spatial distribution and dielectric constant distribution of the micro-scatterer array can be quickly analytically calculated according to the desired scattering field intensity in the target area. First, based on the spatial Fourier transform and angular spectrum transformation, the plane wave sources required in all directions are inversely obtained from the electric field intensity distribution required in the target area. Then, based on the theory of induced source, a method of irradiating the array of all-dielectric micro-scatterers with incident electromagnetic field to generate the required plane wave source is proposed. The scattering fields generated by these micro-scatterers will be superimposed on the target area to achieve the desired scattering field strength intensity. Finally, according to the proposed inverse design theory model, a specific three-dimensional (3D) design is carried out. In the 3D example, we study the scattering field intensity distribution of the point-focused shape of the square surface target area, and show an all-dielectric micro-sphere distribution design. Its spatial distribution and permittivity distribution are both obtained through the rapid analytical calculation of the desired scattered field intensity shape in the target area. Finally, based on the principle of linear superposition, we quickly and easily generate the complex shapes of “I”, “T”, and “X” in the target area. The satisfactory results of full-wave simulation show that the proposed inverse design method is effective and feasible.
      Corresponding author: Wang Bing-Zhong, bzwang@uestc.edu.cn ; Wang Ren, rwang@uestc.edu.cn
    [1]

    Fink M, Prada C, Wu F, Cassereau D 1989 Proceedings, IEEE Ultrasonics Symposium Montreal, Canada, October 3−6, 1989 p681

    [2]

    Azar L, Shi Y, Wooh S C 2000 NDT&E Int. 33 189

    [3]

    Zhao X Y, Gang T 2008 Ultrasonics 49 126

    [4]

    张碧星, 王文龙 2008 物理学报 57 3613

    Zhang B X, Wang W L 2008 Acta Phys. Sin. 57 3613

    [5]

    郑莉, 郭建中 2016 物理学报 65 044305

    Zheng L, Guo J Z 2016 Acta Phys. Sin. 65 044305

    [6]

    Shan L, Wen G Y 2014 IEEE Trans. Antennas Propag. 62 5565

    [7]

    Wang X Y, Yang G M, Wen G Y 2014 Microwave. Opt. Technol. Lett. 56 2464

    [8]

    Nepa P, Buffi A 2017 IEEE Antennas Propag. Mag. 59 42

    [9]

    Elmer M, Jeffs B D, Warnick K F, Fisher J R, Norrod R D 2012 IEEE Trans. Antennas Propag. 60 903

    [10]

    Guo S, Zhao D, Wang B Z 2019 International Conference on Microwave and Millimeter Wave Technology (ICMMT) Guangzhou, China, May 19−22, 2019 p1

    [11]

    Zhao D, Zhu M 2016 IEEE Antennas Wirel. Propag. Lett. 1 5

    [12]

    Zhao D, Guo F, Guo S, Wang B Z 2018 International Conference on Microwave and Millimeter Wave Technology (ICMMT) Chengdu, China, May 7−11, 2018 p1

    [13]

    Bellizzi G G, Crocco L, Iero D A M, Isernia T 2017 International Workshop on Antenna Technology: Small Antennas, Innovative Structures, and Applications (iWAT) Athens, March 1−3, 2017 p162

    [14]

    Bellizzi G G, Bevacqua M T, Crocco L, Isernia T 2018 IEEE Trans. Antennas Propag. 66 4380

    [15]

    Alu A 2009 Phys. Rev. B 80 245115

    [16]

    Yu N, Genevet P, Kats M A, Aieta F, Tetienne J P, Capasso F, Gaburro Z 2011 Science 334 333

    [17]

    Pfeiffer C, Grbic A 2013 Phys. Rev. Lett. 110 197401

    [18]

    Grbic A, Jiang L, Merlin R 2008 Science 320 511

    [19]

    Imani M F, Grbic A 2013 IEEE Trans. Antennas Propag. 61 5425

    [20]

    Grbic A, Merlin R, Thomas E M, Imani M F 2011 Proceedings of the IEEE 99 1806

    [21]

    Khorasaninejad M, Chen W T, Devlin R C, Oh J, Zhu A.Y, Capasso F 2016 Science 352 1190

    [22]

    Li L, Liu H, Zhang H, Xue W 2018 IEEE Trans. Ind. Electron. 65 3230

    [23]

    Yu S, Liu H, Li L 2019 IEEE Trans. Ind. Electron. 66 3993

    [24]

    Chen X D 2018 Computational Methods for Electromagnetic Inverse Scattering (Hoboken: Wiley-IEEE Press) p24

    [25]

    Kong J A 1990 Electromagnetic Wave Theory (New York: Wiley-Interscience) pp482−483

  • 图 1  逆向设计示意图

    Figure 1.  Schematic diagram of inverse design.

    图 2  微球阵列设计示意图 (a) 三维视角图; (b)俯视图; (c)正视图; (d)侧视图

    Figure 2.  Schematic of micro-sphere array design: (a) 3-D view; (b) top view; (c) front view; (d) side view.

    图 3  微球相对介电常数分布

    Figure 3.  Spheres relative permittivity of array distribution.

    图 4  目标区域归一化点聚焦形状散射场分布图 (a) 三维视角图; (b) 俯视图

    Figure 4.  Normalized scattering field distribution of focused shape in target area: (a) 3-D view; (b) top view.

    图 5  目标区域三条线上的归一化散射场分布图 (a) y = 0处场分布; (b) x = 0处分布; (c) y = x处场分布

    Figure 5.  Normalized scattering field distribution on three special lines in target area: (a) A cut view in y = 0; (b) a cut view in x = 0; (c) a cut view in y = x.

    图 6  目标区域归一化复杂形状散射场分布图 (a) 相对原点沿向x方向右平移沿z方向上平移2${\lambda _0}$的点聚焦形状散射场; (b) “I”形状; (c) “T”形状; (d) “X”-形状

    Figure 6.  Normalized scattering field intensity distribution of complex shape in target area: (a) focused shaped field moving 2${\lambda _0}$ to the right and top relative to the origin; (b) “I”-shaped; (c) “T”-shaped; (d) “X”-shaped.

    图 7  “I”形散射场分布的微散射体阵列示意图 (a) 三维视角图; (b)俯视图; (c)正视图; (d)侧视图

    Figure 7.  Schematic of micro-sphere array design with “I”-shaped: (a) 3-D view; (b) top view; (c) front view; (d) side view.

  • [1]

    Fink M, Prada C, Wu F, Cassereau D 1989 Proceedings, IEEE Ultrasonics Symposium Montreal, Canada, October 3−6, 1989 p681

    [2]

    Azar L, Shi Y, Wooh S C 2000 NDT&E Int. 33 189

    [3]

    Zhao X Y, Gang T 2008 Ultrasonics 49 126

    [4]

    张碧星, 王文龙 2008 物理学报 57 3613

    Zhang B X, Wang W L 2008 Acta Phys. Sin. 57 3613

    [5]

    郑莉, 郭建中 2016 物理学报 65 044305

    Zheng L, Guo J Z 2016 Acta Phys. Sin. 65 044305

    [6]

    Shan L, Wen G Y 2014 IEEE Trans. Antennas Propag. 62 5565

    [7]

    Wang X Y, Yang G M, Wen G Y 2014 Microwave. Opt. Technol. Lett. 56 2464

    [8]

    Nepa P, Buffi A 2017 IEEE Antennas Propag. Mag. 59 42

    [9]

    Elmer M, Jeffs B D, Warnick K F, Fisher J R, Norrod R D 2012 IEEE Trans. Antennas Propag. 60 903

    [10]

    Guo S, Zhao D, Wang B Z 2019 International Conference on Microwave and Millimeter Wave Technology (ICMMT) Guangzhou, China, May 19−22, 2019 p1

    [11]

    Zhao D, Zhu M 2016 IEEE Antennas Wirel. Propag. Lett. 1 5

    [12]

    Zhao D, Guo F, Guo S, Wang B Z 2018 International Conference on Microwave and Millimeter Wave Technology (ICMMT) Chengdu, China, May 7−11, 2018 p1

    [13]

    Bellizzi G G, Crocco L, Iero D A M, Isernia T 2017 International Workshop on Antenna Technology: Small Antennas, Innovative Structures, and Applications (iWAT) Athens, March 1−3, 2017 p162

    [14]

    Bellizzi G G, Bevacqua M T, Crocco L, Isernia T 2018 IEEE Trans. Antennas Propag. 66 4380

    [15]

    Alu A 2009 Phys. Rev. B 80 245115

    [16]

    Yu N, Genevet P, Kats M A, Aieta F, Tetienne J P, Capasso F, Gaburro Z 2011 Science 334 333

    [17]

    Pfeiffer C, Grbic A 2013 Phys. Rev. Lett. 110 197401

    [18]

    Grbic A, Jiang L, Merlin R 2008 Science 320 511

    [19]

    Imani M F, Grbic A 2013 IEEE Trans. Antennas Propag. 61 5425

    [20]

    Grbic A, Merlin R, Thomas E M, Imani M F 2011 Proceedings of the IEEE 99 1806

    [21]

    Khorasaninejad M, Chen W T, Devlin R C, Oh J, Zhu A.Y, Capasso F 2016 Science 352 1190

    [22]

    Li L, Liu H, Zhang H, Xue W 2018 IEEE Trans. Ind. Electron. 65 3230

    [23]

    Yu S, Liu H, Li L 2019 IEEE Trans. Ind. Electron. 66 3993

    [24]

    Chen X D 2018 Computational Methods for Electromagnetic Inverse Scattering (Hoboken: Wiley-IEEE Press) p24

    [25]

    Kong J A 1990 Electromagnetic Wave Theory (New York: Wiley-Interscience) pp482−483

  • Citation:
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  • Received Date:  01 June 2020
  • Accepted Date:  12 August 2020
  • Available Online:  12 December 2020
  • Published Online:  05 January 2021

Inverse design method of microscatterer array for realizing scattering field intensity shaping

    Corresponding author: Wang Bing-Zhong, bzwang@uestc.edu.cn
    Corresponding author: Wang Ren, rwang@uestc.edu.cn
  • Institute of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, China

Abstract: It is a novel and interesting idea to inversely design the scattering structure with the desired scattering field intensity distribution in a given target area as the known information. The inverse design method proposed in this paper does not need to be optimized, and the spatial distribution and dielectric constant distribution of the micro-scatterer array can be quickly analytically calculated according to the desired scattering field intensity in the target area. First, based on the spatial Fourier transform and angular spectrum transformation, the plane wave sources required in all directions are inversely obtained from the electric field intensity distribution required in the target area. Then, based on the theory of induced source, a method of irradiating the array of all-dielectric micro-scatterers with incident electromagnetic field to generate the required plane wave source is proposed. The scattering fields generated by these micro-scatterers will be superimposed on the target area to achieve the desired scattering field strength intensity. Finally, according to the proposed inverse design theory model, a specific three-dimensional (3D) design is carried out. In the 3D example, we study the scattering field intensity distribution of the point-focused shape of the square surface target area, and show an all-dielectric micro-sphere distribution design. Its spatial distribution and permittivity distribution are both obtained through the rapid analytical calculation of the desired scattered field intensity shape in the target area. Finally, based on the principle of linear superposition, we quickly and easily generate the complex shapes of “I”, “T”, and “X” in the target area. The satisfactory results of full-wave simulation show that the proposed inverse design method is effective and feasible.

    • 灵活控制物理场, 使其在空间中聚焦或呈现某种形状在诸多领域中具有重大意义以及潜在应用价值, 如无损检测、通信、医疗、无线功率传输等. 最早在声学领域中, 利用时间反演(time-reversal, TR)技术[1]、声学相控阵技术[2-5]等方法对空间中声场进行聚焦及整形. 近年来对电磁场在空间中的控制、整形引起了学者们的广泛研究, 并取得一系列的重要成果.

      对电磁辐射场的控制可以分为近场和远场两类: 近场聚焦天线利用相位共轭原理将天线辐射的电磁场聚焦在天线近场区域中的某个点[6-8]; 采用相控阵及天线阵列技术可以对远场进行调控整形[9,10]. 在最近的文献报道中, TR场整形技术[11,12]、多目标最优约束功率方法[13]和优化的多目标TR技术[14]可以使天线阵列在远场产生任意期望的电场强度形状或图案. 然而, TR方法依赖于信道探测和优化, 需要不断重复“探测-聚焦”过程, 且依赖于大量的信号源或复杂多径环境, 优化方法也需要长时间的优化计算, 不利于场整形的快速实现.

      电磁散射场的控制在电磁学领域中同样具有重要意义, 并且在许多功能上有着广泛的应用价值, 如电磁隐身[15]、波前整形和波束形成等[16,17]. 其中, 使散射场在目标区域呈现某种形态, 即散射场强整形的研究也有了一系列的进展. 在近年的文献报道中多采用近场聚焦板[18-20]和超表面[21-23]实现散射场的近场点聚焦, 然而报道中的设计方法依赖于复杂的超表面阵列, 难以通过理论计算快速设计整形结构.

      本文提出了一种根据目标区域期望散射电场强度分布快速逆向设计电磁微散射体阵列的方法. 首先, 基于空间傅里叶变换与角谱变换, 从目标区域期望的电场强度分布出发逆向求取各方向上所需的平面波. 接下来, 介绍了感应源的概念并给出非磁性介电微散射体的感应源模型, 提出利用入射电磁场照射全介电微散射体阵列来产生所需平面波的方法. 这些微散射体所产生的散射场将在目标区域叠加合成期望的场强分布. 最后, 根据提出的逆向设计理论模型进行了具体的案例设计, 展示了三维(three-dimensions, 3-D)情况下的设计案例. 研究了方形面目标区域期望的点聚焦形状的散射场强分布, 展示了一种全介电微球分布设计, 它的空间分布及介电常数分布都是通过目标区域处期望的散射场强形状进行快速解析计算求得. 最后基于线性叠加原理在目标区域产生了“I”, “T”, “X”形的复杂图形. 全波仿真结果与使用本文提出的逆向设计方法解析计算的结果符合良好, 这表明该方法是有效且可行的.

    2.   逆向设计原理
    • 以面目标区域中心为原点, 在空间中建立直角坐标系, 面目标区域位于z = 0平面上. 考虑正入射情况, 即波长为${\lambda _0}$y极化单色平面波沿+z方向垂直目标区域入射, 设计者期望在目标区域实现y极化同相分布的$E(x, y)$散射电场强(由于只关注$E(x, y)$的幅度分布, 因此其为标量), 如图1所示.

      Figure 1.  Schematic diagram of inverse design.

      为了构建期望的散射场强分布, 在目标区域周围放置介电微散射体阵列作为散射结构. 阵列中微散射体在数学模型中可看作点散射体, 其同原点的距离设为${r_{i, j}}$, 在${r_{i, j}}$距离上微散射体同原点的连线与+z轴夹角编号为${\theta _i}$, 其连线在xoy面上投影与+x轴夹角编号为${\varphi _j}$, 相对介电常数记作$\varepsilon _{\rm{r}}^{i, j}$, 其中i = 1, ···, N; j = 1, ···, M. 逆向设计任务的核心在于将目标区域期望散射场强分布$E(x, y)$作为已知信息, 逆向求出介电微散射体的空间分布及其相对介电常数${\varepsilon _{\rm{r}}}^{i, j}$分布, 从而快速完成微散射体阵列的设计.

    • 采用${{\rm{e}}^{ - {\rm{j}}\omega t}}$时谐变化惯例. 在目标区域处, 设计者期望的散射场强度分布$E(x, y)$作为已知量. 利用空间傅里叶变换, 其空谱为

      其中${k_x}$${k_y}$分别是自由空间中波矢量${{{k}}_0}$xy分量的模. 波数${k_0}^2 < {k_x}^2 + {k_y}^2$时, 凋落波无法传播至远区, 因此, 在本研究中仅考虑利用${k_x}^2 + {k_y}^2 \leqslant {k_0}^2$空间谱的情况.

      将空间谱域变换到角谱域, 可以得到角谱为

      事实上, 角谱的物理意义可理解为目标区域期望的散射场分布是由从不同方向入射而来的多束感应平面波叠加而成的. 设从$(\theta, \varphi )$方向入射而来的感应平面波的幅度和相位分别为$\tilde A(\theta, \varphi )$$\tilde P(\theta, \varphi )$, 则角谱可以表示为

      在设计理论模型中, 本文利用目标区域远区的感应源来近似产生感应平面波. 由于感应源在远区产生的场会随着距离衰减, 定义幅度衰减系数$\varsigma (r)$, 这样感应源激励幅值和初始相位可以表示为:

      接下来引入感应源的概念. 对于满足瑞利散射条件的非磁性且介电常数各向同性的微散射体, 入射波产生的振荡电场作用于散射体内部的电荷, 可以使其感应出一个小的辐射电偶极子[24]:

      其中${ \xi}$代表电极化张量, 它取决于微散射体的相对介电常数${\varepsilon _{\rm{r}}}$、大小、形状. ${{{E}}_{{\rm{in}}}}$代表入射电场.

      因此, 位于(${r_{i, j}},{\theta _i},{\varphi _j}$)处的微散射体的散射场可以写成电偶极子辐射场的形式, 其在原点(原点位于微散射体的远区)的电场表达式为

      这里不妨设照射微散射体的入射平面波幅度为1, 因此微散射体处的入射场为

      根据(4)式—(8)式, 可得微散射体与原点距离${r_{i, j}}$以及${ { \xi} _{i, j}}$:

      其中${ { I}_3}$对应的矩阵形式是3 × 3的单位对角矩阵, ${u_0}$为真空磁导率, $\omega $为角频率.

      根据不同角度入射的感应平面波幅度$\tilde A({\theta _i}, {\varphi _j})$及相位$\tilde P({\theta _i}, {\varphi _j})$就可以计算出微散射体与原点距离${r_{i, j}}$以及需要的${ { \xi} _{i, j}}$. 该理论模型从目标区域期望散射场出发, 进而推导出了所需散射结构的相关参数分布, 是一种“由场到结构”的设计方法, 因此称之为“逆向设计方法”. 该方法可以指导设计者快速求取阵列中微散射体位置以及相对介电常数分布, 在下一节设计案例中将具体详细阐述.

    3.   设计案例及全波仿真结果
    • 设计者期望在目标区域处获得散射场为均匀激发产生的准贝塞尔光束的横截面分布, 该分布在目标区域表现为在中心原点处点聚焦的形式, 其表达式为

      将(11)式代入(2)式中, 可得到其角谱为

      因此可得各个角谱分量大小均为常数${\lambda _0}{\rm{/2}}$.

      设计中选用半径为a ($a \ll \lambda $)相对介电常数为${\varepsilon _{\rm{r}}}$的各向同性介电球, 则(6)式中电极化张量${ \xi}$的解析表达为[25]

      联合(10)式和(12)式, 即可求得微球的相对介电常数分布:

      根据提出的理论模型, 本文设计了一个如图2(a)所示的121个介电微球组成的阵列来对目标区域进行散射场强整形. 阵列中的介电微球分别以等$\Delta \sin \theta $与等$\Delta \varphi $间隔放置, 以保证在空间谱域中进行均匀采样, 从而使微散射体的散射场在目标区域合成均匀. 由于阵列中介电微球产生的散射场须同相到达目标区域, 即$\tilde P(\theta )$为常数, 因此得${r_{i, j}}(1 + \cos {\theta _i})$为常数, 记为p($p \gg \lambda $). 令${\theta _1}=\sigma $, ${\varphi _1} = 0$且有${\theta _i} > {\theta _{i + 1}}$, ${\varphi _j} < {\varphi _{j + 1}}$, 则介电微球的排布满足以下规则:

      Figure 2.  Schematic of micro-sphere array design: (a) 3-D view; (b) top view; (c) front view; (d) side view.

      算例中取N = 16, M = 8. 以+z方向为观察视角, 图2 (b)图2(d)分别展示了俯视、正视、侧视图. 需说明的是, 当i = 16时, 在坐标(0, 0, p/2)处仅有一个介电微球. 此阵列中的介电微球半径均为$a = 1/20{\lambda _0}$, 取$p=30{\lambda _0}$, $\sigma = 3{{\rm{0}}^ \circ }$, 入射平面波的工作频率为2 GHz.

      根据(14)式以及期望场强分布, 可以解析计算出阵列中介电微球的${\varepsilon _{\rm{r}}}$, 其分布如图3所示. 可以看到实现目标区域点聚焦散射场的介电微球${\rm{\varepsilon}} _{\rm{r}}^{i, j}$j无关, 具有旋转对称性, 这是因为同一${\varphi _j}$上的介电微球与原点的距离${r_{i, j}}$相等.

      Figure 3.  Spheres relative permittivity of array distribution.

      下面使用全波仿真软件FDTD.Solution来验证上述求解结果的可行性. 图4展示了面目标区域中期望形成的点聚焦散射场强分布, 从3-D视角以及俯视视角可以看到, 在设定的方向面目标区域中得到了需要的点聚焦散射场分布.

      Figure 4.  Normalized scattering field distribution of focused shape in target area: (a) 3-D view; (b) top view.

      y = 0, x = 0, y = x三条特殊线上比较期望的散射场分布与全波仿真的结果, 结果如图5所示. 可以看到, 目标区域中的点聚焦散射场强分布仿真结果与期望结果有些微失真, 这是由于微散射体产生的散射波不是理想平面波导致的, 并且散射体之间的微弱的多次散射效应也会造成不良的影响.

      Figure 5.  Normalized scattering field distribution on three special lines in target area: (a) A cut view in y = 0; (b) a cut view in x = 0; (c) a cut view in y = x.

      假设期望的散射场强分布分别沿着xy轴正方向平移两个波长的点聚焦形状, 根据傅里叶变换的空间平移特性, 即将散射体阵列分别沿着xy轴正方向方向平移$2{\lambda _0}$长度即可. 图6 (a)展示了全波仿真结果, 其散射场强的分布与预期完全符合. 在此基础上结合线性叠加原理, 可以在目标区域处便捷地实现复杂的期望散射场形状. 如图6 (b)图6(d)所示, 由基本的点聚焦形状散射场叠加获取了3种不同形状的复杂散射场形状. 特别说明的是, 在目标区域处实现复杂的期望散射场形状需要将微球阵列平行于xoy面进行对应的平移叠加. 图7以“I”形散射场形状为例展示了其微散射体的空间分布情况.

      Figure 6.  Normalized scattering field intensity distribution of complex shape in target area: (a) focused shaped field moving 2${\lambda _0}$ to the right and top relative to the origin; (b) “I”-shaped; (c) “T”-shaped; (d) “X”-shaped.

      Figure 7.  Schematic of micro-sphere array design with “I”-shaped: (a) 3-D view; (b) top view; (c) front view; (d) side view.

      图6中良好的仿真结果可以看出, 点聚焦形状的散射场在目标区域组合形成复杂的期望散射场形状是有效且可行的.

    4.   讨 论
    • 从各个方向上入射的感应平面波合成了目标区域期望实现的散射场强形状, 而入射波照射到微散射体阵列后感应的等效源产生的场只有传播到远区后才可近似看作平面波, 这表明了目标区域必须处于微散射体阵列的远区. 包含高谱域信息的凋落波无法传播至远区, 这也解释了本文提出的逆向设计理论模型仅考虑利用${k_x^2} + {k_y^2} \leqslant {k_0^2}$空间谱的情况. 由于高谱域信息的损失, 这也使得本研究无法在目标区域实现超空间分辨率的散射场整形, 如设计案例中所展示的点聚焦场, 即为提出的逆向设计方法可以获得的面目标区域处最大空间分辨率的场, 其中心线上的分布为sinc函数形状.

    5.   结 论
    • 研究中提出的逆向设计方法可以根据目标区域期望的散射场强快速解析计算微散射体阵列的空间分布及介电常数分布. 根据该方法在3-D情况下进行了具体的逆向设计案例. 在3-D算例中研究了方形面目标区域期望的点聚焦形状的散射场强分布案例, 展示了一种全介电微球阵列的设计过程, 并且基于线性叠加原理快速简便地在目标区域产生了“I”, “T”, “X”形的复杂图形. 良好的全波仿真结果表明, 本文提出的逆向设计方法是有效且可行的. 后续的研究中可以考虑利用优化算法对微散射体空间位置与介电常数分布进行优化, 从而使目标区域散射场强分布更加精确, 本文提出的逆向设计方法可以为优化提供高质量的初值.

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