Abnormal topological refraction into free medium at subwavelength scale in valley phononic crystal plates

In this work we propose a topological valley phononic crystal plate and we extensively investigate the refraction of valley modes into the surrounding homogeneous medium. This phononic crystal includes two sublattices of resonators (A and B) modeled by mass-spring systems. We show that two edge states confined at the AB/BA and BA/AB type domain walls exhibit different symmetries in physical space and energy peaks in the Fourier space. As a result, distinct refraction behaviors, especially through an armchair cut edge, are observed. On the other hand, the decay depth of these localized topological modes, which is found to be solely determined by the relative resonant strength between the scatterers, significantly affects the refraction patterns. More interestingly, the outgoing traveling wave through a zigzag interface becomes evanescent when operating at deep subwavelength scale. This is realized by tuning the average resonant strength. We show that the evanescent modes only exist along a particular type of outlet edge and that they can couple with both topological interface states. We also present a near-ideal monopole and a dipole emitter based on our phononic structure.

Figure 1

(a) Elastic PnC plate with honeycomb lattice of mass-spring resonators, primitive unit cell, and first irreducible Brillouin zone (IBZ). (b) Dirac frequency (solid line) and bulk band gap range (shaded area) at a perturbation level of $\pm 0.1\gamma$ as a function of the average mass $\gamma =({\gamma }_A+{\gamma }_B)/2$. Band structures of the (c) unperturbed $\mathrm{\Delta }\gamma =0$ and (d) perturbed $\mathrm{\Delta }\gamma =0.2$ PnCs when $\gamma =2$. The right panel of (d) displays the phase distributions and the time-averaged energy flux (black arrows) of the lower and upper gapped states at valley $K$.

Figure 2

(a) Band structure of a supercell consisting of 20 layers with $\gamma =2$ and $\mathrm{\Delta }\gamma =-0.1\gamma$ type of PnCs sandwiched in between two domains made of 10 layers with $\gamma =2$ and $\mathrm{\Delta }\gamma =+0.1\gamma$. (b) The eigenmodes and (c) the corresponding Fourier spectra of the symmetric (S) and antisymmetric (A) valley topological states. (d) Displacement magnitudes of mode S along the perpendicular direction of the topological interface for various average masses $\gamma$ and perturbation levels $\mathrm{\Delta }\gamma$.

Figure 3

Topologically protected refraction of valley states into the surrounding HP through different outlet edges and the corresponding Fourier spectra. The refraction of (a) symmetric and (b) antisymmetric valley states through a zigzag interface. (c),(d) the same as (a) and (b) but with an armchair interface. The Fourier transformations are performed in a square region of $[0,20a]\times [-10a,10a]$, and the structural parameters are $\gamma =2,\mathrm{\Delta }\gamma =\pm 0.1\gamma$.

Figure 4

Refracted patterns of topological S and A valley modes with different decay depths ($\gamma =2,\left|\mathrm{\Delta }\gamma \right|/\gamma =0.2,0.5$).

Figure 5

(a) Band structure of a supercell with ST type edges. (b) The out-of-plane displacement field of eigenmodes E1 and E2. (c) The same as (a) but with LT type edges. (d) Reference band structure of the pure topological PnC. (e) Coupling between topological valley edge states and evanescent modes in the deep subwavelength regime ($\gamma =8,\mathrm{\Delta }\gamma =\pm 0.1\gamma$).

Figure 6

Nearly ideal topological (a),(b) monopole and (c),(d) dipole emitters at the subwavelength scale ($\gamma =8,\mathrm{\Delta }\gamma =\pm 0.1\gamma$).