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} $)

Fig. 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)

Fig. 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

Fig. 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

Fig. 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

Fig. 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} $.