图 1  (a) 二维正方晶格声子晶体的能带结构, 插图是原胞示意图; (b) M点附近的三维能带结构, 对应于图(a)中的虚线区域. 橡胶与水的质量密度和声速分别为: $ \rho =1.3\times {10}^{3}\;\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3} $, $ v=500\;\mathrm{m}/\mathrm{s} $; $ {\rho }_{0}=1.0\times {10}^{3}\;\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3} $, $ {v}_{0}=1500\;\mathrm{m}/\mathrm{s} $

Fig. 1.  (a) Bulk band structure of a two-dimensional phononic crystal with a square lattice, consisting of a rubber cylinder in water. Inset: the unit cell. (b) 3 D bulk band structure around the M point, corresponding to the dashed region in (a). Here, the lattice constant and the radius of the cylinder are $ a=1\;\mathrm{m} $, and $ R=0.15 a $, respectively. The mass densities and sound velocity of the rubber and water are: $ \rho =1.3\times {10}^{3}\;\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3} $, $ v=500\;\mathrm{m}/\mathrm{s} $; and $ {\rho }_{0}=1.0\times {10}^{3}\;\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3} $, $ {v}_{0}=1500\;\mathrm{m}/\mathrm{s} $, respectively.

图 2  (a) 倾斜角$ \alpha ={70}^ {\circ} $的斜方晶格体系的能带结构, 左插图是二维斜方晶格声子晶体的原胞, 右插图表示虚线区域的放大能带结构; (b)斜方晶格的第一布里渊区; (c) 线性狄拉克点附近的三维能带结构, 对应图(a)中的虚线区域

Fig. 2.  (a) Bulk band structure of an oblique lattice with the tilted angle $ \alpha ={70}^ {\circ} $, Inset: the unit cell (left); the enlarged band structure around the Dirac point near ${ {K}}_{1}$ point (right); (b) first Brillouin zone of the oblique lattice; (c) 3D bulk band structure around the Dirac point, corresponding to the dashed region in (a).

图 3  (a) 正方晶格声子晶体沿x方向的投影能带; (b) 倾斜角$ \alpha ={70}^ {\circ} $的斜方晶格声子晶体沿x方向的投影能带$, \mathrm{I}\mathrm{m}\left(Z\right) $表示表面阻抗的虚部; (c) 由上述两个声子晶体构成的沿x方向界面的界面态色散关系, 粉色线表示界面态色散; (d)正方晶格和斜方晶格声子晶体构成的沿x方向的界面(左图), 频率为$ 937.4\;\mathrm{H}\mathrm{z} $的界面态本征声压场分布(右图)

Fig. 3.  (a)Projected band structures along the $ {k}_{x} $ direction of the phononic crystals with a square lattice; (b) projected band structures along the $ {k}_{x} $ direction of phononic crystals with an oblique lattice of $ \alpha ={70}^ {\circ} $, Im(Z) represents the imaginary part of surface impedance; (c)interface state dispersion along the $ {k}_{x} $ direction of the interface constructed by two phononic crystals with the square and oblique lattices, the pink lines denote the interface states; (d) the interface constructed by two phononic crystals with the square and oblique lattices(left), the eigen pressure field distribution of the interface state at $ 937.4\;\mathrm{H}\mathrm{z}\left(\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\right) $.

图 4  倾斜角$ {\rm{\alpha }}={70}^ {\circ} $的斜方晶格声子晶体在$ {k}_{x}=0.6\;{\text{π}}/a $ (a) 和$ {k}_{x}=0.85\;{\text{π}}/a $(b)时沿$ {k}_{y} $方向的体能带; (c) 在$ {k}_{x}=0.85\;{\text{π}}/a $时, 正方晶格声子晶体沿$ {k}_{y} $方向的体能带, 其中红色区域和蓝色区域分别表示$ \mathrm{I}\mathrm{m}\left(Z\right) < 0 $和$ \mathrm{I}\mathrm{m}\left(Z\right) > 0 $

Fig. 4.  Bulk band structures along the $ {k}_{y} $ direction of the phononic crystal with an oblique lattice with $ {\rm{\alpha }}={70}^ {\circ} $ for $ {k}_{x}=0.6\;{\text{π}}/a $(a) and $ {k}_{x}=0.85\;{\text{π}}/a $ (b); (c) bulk band structures along the $ {k}_{x} $ direction of the phononic crystal with a square lattice for $ {k}_{x}=0.85\;{\text{π}}/a $. The red and blue regions represent $ \mathrm{I}\mathrm{m}\left(Z\right) < 0 $ and $ \mathrm{I}\mathrm{m}\left(Z\right) > 0 $, respectively.

图 5  由$ \alpha ={50}^ {\circ} $斜方晶格与正方晶格声子晶体构成的沿x方向界面的界面态色散, 红色线和绿色线分别表示两个共同带隙中的界面态色散

Fig. 5.  Interface state dispersion along the $ {k}_{x} $ direction of the interface constructed by two phononic crystals with the square and oblique lattices with $ \alpha ={50}^ {\circ} $, the red line and the green line represent the interface state dispersion in the two common band gaps, respectively.

图 6  铁柱子在环氧树脂中周期性排列构成二维声子晶体 (a) 二维正方晶格声子晶体的能带结构; (b) 倾斜角$ \alpha ={70}^ {\circ} $的斜方晶格体系的能带结构; (c)由上述两个声子晶体构成的沿x方向界面的界面态色散关系(左), 粉色线表示界面态色散, 频率为$ 529.6\;\mathrm{H}\mathrm{z} $的界面态本征位移场分布(右)

Fig. 6.  Two-dimensional phononic crystals are constructed by steel cylinders in epoxy: (a) Bulk band structure of a square lattice; (b) bulk band structure of an oblique lattice with the tilted angle $ \alpha ={70}^ {\circ} $; (c) the interface state dispersion along the $ {k}_{x} $ direction of the interface constructed by these two phononic crystals(Left), the pink line denotes the interface states, the eigen displacement field distribution of the interface state at $ 529.6\;\mathrm{H}\mathrm{z}\left(\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\right) $.