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Recent development of metasurfaces indicates that achieving high efficiency requires nonlocal designs where the coupling between constituent units is fully considered. However, most metasurfaces for elastic waves are still designed as local structures based on the Generalized Snell’s Law (GSL), which ignores the coupling between sub-units, thus often resulting in low efficiency. In order to design nonlocal structures for flexural wave in thin elastic plate, a previously proposed method based on the multi-port structural model (MPSM) for acoustic metasurfaces is extended in this work. Using this method, anomalous reflector and anomalous refractor, each with a large diffraction angle and planar focuser with large numerical aperture for flexural waves in thin elastic plates, are designed. As the first example, an anomalous reflector or anomalous refractor for flexural wave on an infinite free thin elastic plate with elastic cylinder pairs assembled symmetrically on both surfaces is considered. The design target is to optimize the heights of the cylinder pairs, by which anomalous reflection or refraction for flexural wave in plate can be realized. It is shown that by modelling the structure as an MPSM, configurations with the desired functionalities can be efficiently determined. The three dimensional finite element simulations show that even for structures with a deflection angle as large as 80°, the proposed anomalous reflectors and refractors can achieve near-unity efficiency. By the same method, a planar focuser is further designed. It is shown that by optimizing the heights of each cylinder pair, the normally incident flexural wave can be focused on the incident side or the transmitting side of the metasurface with arbitrary focal length. It is found that the focusing efficiency of our nonlocal designs is significantly higher than that of their GSL-based counterparts, particularly for the structures with numerical apertures approaching unity. This work not only presents an effective design method for nonlocal metasurfaces of flexural waves in thin elastic plates, but also provides two efficient nonlocal structures with broad application prospects in sensing, energy harvesting, and other fields. -
Keywords:
- flexural wave in thin plate /
- nonlocal metasurface /
- multi-port structural model /
- anomalous reflection /
- planar focusing
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图 1 (a) 超表面结构示意图, 其中, 左侧子图示意出弹性柱体(绿色)对称地安装于弹性薄板正、反两个表面; 空心盒子标出了一个单元, 图中带箭头的线段示意出沿$ {\theta _{\text{i}}} $方向入射的平面波被反射/透射至沿$ {\theta _{\text{r}}} $/$ {\theta _{\text{t}}} $方向的情况, 右侧图展示了单元的结构细节, 其中元胞的横向尺寸、薄板的厚度和两排圆柱之间的间距分别用符号$ {a_0} $, $ {h_0} $和$ {d_{\mathrm{r}}} $表示, 柱体的半径用$ {r_{cy}} $表示, $ {h_n} $表示位于薄板正、反表面上第$ n $对圆柱的高度; (b) 单元结构俯视图, 图中深色区域示意出多端口模型, 带箭头的线段表示薄板中的输入/输出端口, $ A_1^ \pm , A_2^ \pm , \cdots , A_6^ \pm $为端口中的波模式振幅, 绿色圆圈为弹性柱体, $ H_{kn}^ \pm $为第$ n $个柱体中的第$ k $支波模式(端口)的振幅; (c) 弹性圆柱中可与薄板弯曲振动相耦合的3支模式的形变示意图, 从左到右分别为两支剪切模(标记为$ {S_x} $和$ {S_y} $)及一支伸缩模式(标记为$ {L_z} $)
Figure 1. (a) Schematic view of the structure for the metasurface, in the left panel, elastic rods (in green) are assembled symmetrically on both side of the plate, arrows in color show the effect of deflecting a plane wave from $ {\theta _{\text{i}}} $ to $ {\theta _{\text{r}}} $ or $ {\theta _{\text{t}}} $ direction, the unit cell of the structure is shown by a hollow box, the right panel shows the detailed structure of a single unit cell, the symbols $ {a_0} $, $ {h_0} $, and $ {d_{\text{r}}} $ represent respectively the size of the unit cell in $ x $ direction, the thickness of the plate, and the distance between two rows of cylinders. The radius of the cylinder is denoted by $ {r_{cy}} $, the height of the $ n $-th pair of cylinders is denoted by $ {h_n} $; (b) top view of the unit cell. The multi-port structural model is marked in deep color. The red lines indicate the ports in the plate. The amplitudes of the modes in these ports are denoted by $ A_1^ \pm , A_2^ \pm , \cdots , A_6^ \pm $, respectively, circles in green are elastic cylinders, the amplitude of the $ k $-th mode in the $ n $-th cylinder is denoted by $ H_{kn}^ \pm $; (c) the deformation of three vibration modes in cylinders that can couple with the flexural wave in thin plate, from the left to the right are the shear modes ($ {S_x} $ and $ {S_y} $) and longitudinal mode ($ {L_z} $), respectively.
图 2 (a)优化得到的$ {\theta _{{\text{r, t}}}} = $60°, 70°, 80°结构的$ {h_n} $和$ h_n' $值(近入射侧一排柱体的高度标记为$ {h_n} $, 远入射侧一排的柱体高度用$ h_n' $标记); (b) $ {\theta _{{\text{r, t}}}} = $60°结构板中面散射波$ z $分量位移实部的场分布图, 左侧子图对应于异常透射, 右侧子图对应于异常反射, 图中白色箭头示意平面波的传播方向; (c)用于对比的基于GSL的结构构型(左)以及${\theta _{\text{r}}} = {60^ \circ }$结构的$ z $分量反射位移场分布(右)
Figure 2. (a) The optimized $ {h_n} $/$ h_n' $ value for structures with $ {\theta _{{\text{r, t}}}} = $60°, 70° and 80°, the height of the cylinders in the row closed to the incident side is denoted by $ {h_n} $, while those in the other row is denoted by $ h_n' $; (b) the field distribution of the real part of the $ z $-component displacement on the middle plane of the plate for the structure with $ {\theta _{{\text{r, t}}}} = $60°, the left and right panels correspond respectively to the anomalous refraction and anomalous reflection, the white arrows indicate the direction of the plane wave; (c) configuration of the GSL structure for comparison (left) and the displacement field of the scattering wave for the structure with ${\theta _{\text{r}}} = {60^ \circ }$ (right).
图 3 (a)平面聚焦超表面的俯视图, 结构由对称在薄板的正、反两表面上分别安装两排间距为$ {d_{\text{r}}} $的弹性柱体(标注为绿色圆圈)构成, 两排弹性柱体均等间距地分布于$ - \dfrac{{{a_0}}}{2} < x < \dfrac{{{a_0}}}{2} $间, $ \left( {0, f} \right) $, $ \left( {0, - f} \right) $为反射、透射式聚焦的焦点坐标(薄板中面上), 图中带箭头的线段示意了波传播的轨迹; (b)优化得到的构成超表面圆柱体的高度, 其中$ {h_n} $为近入射侧一排柱体的高度, $ h_n' $为远入射侧一排柱体的高度. 左、右两列子图分别对应于$ {a_0} = 10{\lambda _0} $和$ {a_0} = 20{\lambda _0} $的结构, 上、下两行子图分别对应于反射及透射式聚焦结构
Figure 3. (a) Top view of the planar focuser. The structure is constructed by symmetrically assembling two rows of cylinders with a space of $ {d_{\text{r}}} $ on each surface of the thin plate, on each surface, the elastic cylinders are arranged equidistantly in $ - \dfrac{{{a_0}}}{2} < x < \dfrac{{{a_0}}}{2} $, $ \left( {0, f} \right) $ and $ \left( {0, - f} \right) $ are the focal point of the reflection-type and refraction-type structure, arrows indicate the wave propagation trajectories; (b) the optimized height of the elastic cylinders for the structure, the height of the cylinders in the row closed to the incident side is denoted by $ {h_n} $, the ones in the other row is denoted by $ h_n' $, panels on the left (right) -hand side are for structures with $ {a_0} = 10{\lambda _0} $ ($ 20{\lambda _0} $), panels on top (bottom) are for reflection (refraction) -type structures.
图 4 (a)平面聚焦超表面在垂直入射平面波下的散射波强度场分布, 该场值由$ {r_{\text{f}}} = {\left| {{{{A_{\text{s}}}}}/{{{A_{\text{i}}}}}} \right|^2} $定义, 其中$ {A_{\text{s}}} $和$ {A_{\text{i}}} $分别为散射波和入射波的振幅; (b)焦平面上的$ {r_{\text{f}}} $值分布, 图中仅显示了$ \left| x \right| \leqslant 1.5{\lambda _0} $区域中的值, 左上子图中的$ {x_1} $和$ {x_2} $示意出了主峰的边缘
Figure 4. (a) Field distribution of the scattered wave from the planar focuser under the normally incident plane wave. The fields value is defined by $ {r_{\text{f}}} = {\left| {{{{A_{\text{s}}}}}/{{{A_{\text{i}}}}}} \right|^2} $, where $ {A_{\text{s}}} $ and $ {A_{\text{i}}} $ are the amplitudes of the scattered and incident waves, respectively; (b) the $ {r_{\text{f}}} $ values along the focal plane, only the value in region $ \left| x \right| \leqslant 1.5{\lambda _0} $ is shown. In the top-left panel, edges of the main peak are marked by $ {x_1} $ and $ {x_2} $.
图 5 $ {a_0} = 10{\lambda _0} $反射式聚焦结构的(a) $ {r_{{\text{ff}}}} $及(b) $ \eta $值随归一化入射波长$ {\lambda / {{\lambda _0}}} $的变化关系
Figure 5. (a) $ {r_{{\text{ff}}}} $ and (b) $ \eta $ value as a function of the normalized working wavelength $ {\lambda/ {{\lambda _0}}} $ for the reflection-type focuser with $ {a_0} = 10{\lambda _0} $.
表 1 4种不同结构的数值孔径 (NA) 、焦点处强度比($ {r_{{\text{ff}}}} $)、焦平面上主峰的半高宽(HWMP), 以及聚焦效率($ \eta $)
Table 1. Numerical aperture (NA), intensity ratio at the focal point ($ {r_{{\text{ff}}}} $), half height width of the main peak on the focal plane (HWMP), and focusing efficiency ($ \eta $) for four different structures.
聚焦类型 横向长度 NA $ {r_{{\text{ff}}}} $ HWMP $ \eta $/% 反射式 $ 10{\lambda _0} $ 0.928 29.92 $ 0.372{\lambda _0} $ 64.36 反射式 $ 20{\lambda _0} $ 0.980 48.01 $ 0.375{\lambda _0} $ 48.97 透射式 $ 10{\lambda _0} $ 0.928 23.66 $ 0.396{\lambda _0} $ 67.82 透射式 $ 20{\lambda _0} $ 0.980 38.59 $ 0.414{\lambda _0} $ 67.71 
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[1] Chi Z J, Du Y C, Huang W H, Tang C X 2018 J. Appl. Phys. 124 124901
Google Scholar
[2] Glybovski S B, Tretyakov S A, Belov P A, Kivshar Y S, Simovski C R 2016 Phys. Rep. -Rev. Sec. Phys. Lett. 634 1
[3] Leonhardt U 2006 Science 312 1777
Google Scholar
[4] Schurig D, Mock J J, Justice B J, Cummer S A, Pendry J B, Starr A F, Smith D R 2006 Science 314 977
Google Scholar
[5] Smolyaninov I I, Narimanov E E 2010 Phys. Rev. Lett. 105 067402
Google Scholar
[6] Yu N F, Genevet P, Kats M A, Aieta F, Tetienne J P, Capasso F, Gaburro Z 2011 Science 334 333
Google Scholar
[7] Zhang L, Chen X Q, Liu S, Zhang Q, Zhao J, Dai J Y, Bai G D, Wan X, Cheng Q, Castaldi G, Galdi V, Cui T J 2018 Nat. Commun. 9 4334
Google Scholar
[8] Ma G C, Yang M, Xiao S W, Yang Z Y, Sheng P 2014 Nat. Mater. 13 873-878
Google Scholar
[9] Tang K, Qiu C Y, Ke M Z, Lu J Y, Ye Y T, Liu Z Y 2014 Sci. Rep. 4 6517
Google Scholar
[10] Xie Y B, Wang W Q, Chen H Y, Konneker A, Popa B I, Cummer S A 2014 Nat. Commun. 5 5553
Google Scholar
[11] Li Y, Jiang X, Liang B, Cheng J C, Zhang L K 2015 Phys. Rev. Appl. 4 024003
Google Scholar
[12] Zhu Y F, Zou X Y, Li R Q, Jiang X, Tu J, Liang B, Cheng J C 2015 Sci. Rep. 5 10966
Google Scholar
[13] Jiang X, Li Y, Liang B, Cheng J C, Zhang L K 2016 Phys. Rev. Lett. 117 034301
Google Scholar
[14] Xie Y B, Shen C, Wang W Q, Li J F, Suo D J, Popa B I, Jing Y, Cummer S A 2016 Sci. Rep. 6 35437
Google Scholar
[15] Melde K, Mark A G, Qiu T, Fischer P 2016 Nature 537 518
Google Scholar
[16] Díaz-Rubio A, Li J F, Shen C, Cummer S A, Tretyakov S A 2019 Sci. Adv. 5 eaau7288
Google Scholar
[17] Epstein A, Eleftheriades G V 2016 Phys. Rev. Lett. 117 256103
Google Scholar
[18] Ra'di Y, Sounas D L, Alù A 2017 Phys. Rev. Lett. 119 067404
Google Scholar
[19] Li J F, Song A L, Cummer S A 2020 Phys. Rev. Appl. 14 044012
Google Scholar
[20] Peng X Y, Li J F, Shen C, Cummer S A 2021 Appl. Phys. Lett. 118 061902
Google Scholar
[21] Chiang Y K, Quan L, Peng Y G, Sepehrirahnama S, Oberst S, Alù A, Powell D A 2021 Phys. Rev. Appl. 16 064014
Google Scholar
[22] Craig S R, Su X S, Norris A, Shi C Z 2019 Phys. Rev. Appl. 11 061002
Google Scholar
[23] Mei J, Fan L J, Hong X B 2023 Appl. Phys. Express 16 077002
Google Scholar
[24] Hou Z L, Fang X S, Li Y, Assouar B 2019 Phys. Rev. Appl. 12 034021
Google Scholar
[25] Ni H Q, Fang X S, Hou Z L, Li Y, Assouar B 2019 Phys. Rev. B 100 104104
Google Scholar
[26] Ren J, Hou Z L 2023 Phys. Rev. Appl. 20 044004
Google Scholar
[27] Ren J, Hou Z L 2024 Phys. Rev. Appl. 22 014040
Google Scholar
[28] Nakamura K, Kobayashi Y, Oda K, Shigemura S 2023 Sustainability 15 4846
Google Scholar
[29] Ahamad S, Soman R, Malinowski P, Wandowski T 2023 Conference on Health Monitoring of Structural and Biological Systems XVII SPIE Long Beach, CA, 124880E
[30] Lee T G, Jo S H, Seung H M, Kim S W, Kim E J, Youn B D, Nahm S, Kim M 2020 Nano Energy 78 105226
Google Scholar
[31] Kim S Y, Bin Oh Y, Lee J S, Kim Y Y 2023 Mech. Syst. Sig. Process. 186 109867
Google Scholar
[32] Mei J, Fan L J, Hong X B 2022 Crystals 12 901
Google Scholar
[33] Lee S W, Shin Y J, Park H W, Seung H M, Oh J H 2021 Phys. Rev. Appl. 16 064013
Google Scholar
[34] Li L X, Su K, Liu H X, Yang Q, Li L, Xie M X 2023 J. Appl. Phys. 133 105103
Google Scholar
[35] Ruan Y D, Liang X 2021 Inter. J. Mech. Sci. 212 106859
Google Scholar
[36] Zhang X B, Li L, Li K L, Liu T, Zhang J, Hu N 2023 Appl. Acoust. 202 109170
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[37] Yang H G, Feng K, Li R, Yan J 2022 Front. Phys. 10 909318
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[38] Kim S Y, Lee W, Lee J S, Kim Y Y 2021 Mech. Syst. Sig. Process. 156 107688
Google Scholar
[39] Yuan S M, Gao T, Chen A L, Wang Y S 2025 Phys. Lett. A 529 130081
Google Scholar
[40] Oh Y B, Kim S Y, Cho S H, Lee J S, Kim Y Y 2024 Inter. J. Mech. Sci. 262 108750
Google Scholar
[41] Su G Y, Du Z L, Jiang P, Liu Y Q 2022 Mech. Syst. Sig. Process. 179 109391
Google Scholar
[42] Packo P, Norris A N, Torrent D 2019 Phys. Rev. Appl. 11 014023
Google Scholar
[43] Jang S V, Lee S W, Oh J H 2023 Phys. Rev. Appl. 19 024036
Google Scholar
[44] Jiang M, Wang Y F, Assouar B, Wang Y S 2023 Phys. Rev. Appl. 20 054020
Google Scholar
[45] Jin Y B, Wang W, Khelif A, Djafari-Rouhani B 2021 Phys. Rev. Appl. 15 024005
Google Scholar
[46] Wang W, Iglesias J, Jin Y B, Djafari-Rouhani B, Khelif A 2021 Apl Mater. 9 051125
Google Scholar
[47] Lee G, Choi W, Ji B, Kim M, Rho J 2024 Adv. Sci. 11 2198
[48] Li M Z, Hu Y B, Cheng J L, Chen J L, Li Z, Li B 2024 Inter. J. Mech. Sci. 268 109048
Google Scholar
[49] Lin B Z, Li J R, Lin W, Ma Q F 2024 Appl. Sci. -Basel 14 2717
Google Scholar
[50] Peng H C, Fan L J, Mei J 2024 J. Appl. Phys. 135 033102
Google Scholar
[51] Peng H C, Mei J 2024 Phys. Rev. Appl. 21 034007
Google Scholar
[52] Torrent D, Mayou D, Sánchez-Dehesa J 2013 Phys. Rev. B 87 115143
Google Scholar
[53] Zhu H F, Patnaik S, Walsh T F, Jared B H, Semperlotti F 2020 Proc. Natl. Acad. Sci. U. S. A. 117 26099
Google Scholar
[54] Jin Y B, El Boudouti E, Pennec Y, Djafari-Rouhani B 2017 J. Phys. D-Appl. Phys. 50 425304
Google Scholar
[55] Moriyama H, Masuda N, Osaka Y 2006 Proc. Sch. Eng. Tokai Univ. (Engl. Ed. ) (Japan) 46 111
[56] Taghavipour S, Kharkovsky S, Kang W H, Samali B, Mirza O 2017 Smart Mater. Struct. 26 104009
Google Scholar
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