Topological Acoustics

The manipulation of acoustic wave propagation in fluids has numerous applications, including some in everyday life. Acoustic technologies frequently develop in tandem with optics, using shared concepts such as waveguiding and metamedia. It is thus noteworthy that an entirely novel class of electromagnetic waves, known as “topological edge states,” has recently been demonstrated. These are inspired by the electronic edge states occurring in topological insulators, and possess a striking and technologically promising property: the ability to travel in a single direction along a surface without backscattering, regardless of the existence of defects or disorder. Here, we develop an analogous theory of topological fluid acoustics, and propose a scheme for realizing topological edge states in an acoustic structure containing circulating fluids. The phenomenon of disorder-free one-way sound propagation, which does not occur in ordinary acoustic devices, may have novel applications for acoustic isolators, modulators, and transducers.

Figure 1

A two-dimensional acoustic topological insulator and its band structure. (a) Triangular acoustic lattice with lattice constant $a$. $a=0.2\mathrm{m}$ in the following calculation. Inset: unit cell containing a central metal rod of radius $r_1=0.2a$, surrounded by an anticlockwise circulating fluid flow (flow direction indicated by red arrows) in a cylinder region of radius $r_2=0.4a$. (b) Band structures of the acoustic lattice without the circulating fluid flow (red curves, $\mathrm{\Omega }=2\pi \times 0\mathrm{rad}/\mathrm{s}$) and with fluid flow (blue curves, $\mathrm{\Omega }=2\pi \times 400\mathrm{rad}/\mathrm{s}$). In the gapped band structure, the bands have Chern number $\pm 1$ (blue labels). Left inset: enlarged view of Dirac cone. Right lower inset: the first Brillouin zone. (c) Frequency splitting as a function of the angular velocity of the cylinder in each unit cell. The degeneracy at the Dirac point with frequency ${\omega }_0=0.577\times 2\pi c_a/a$ (992 Hz) is removed for $\mathrm{\Omega }\ne 0$.

Figure 2

Acoustic one-way edge states. (a) Dispersion of the one-way acoustic edge states (red curves) occurring in a finite strip of the acoustic lattice, for $\mathrm{\Omega }=2\pi \times 400\mathrm{rad}/\mathrm{s}$. The left and right red curves correspond to edge states localized at the bottom and top of the strip. (b),(c) The normalized acoustic pressure $p$ for a left-propagating acoustic edge state at frequency ${\omega }_0=0.577\times 2\pi c_a/a$ (992 Hz) for $\mathrm{\Omega }=2\pi \times 400\mathrm{rad}/\mathrm{s}$ (b) and $\mathrm{\Omega }=2\pi \times 200\mathrm{rad}/\mathrm{s}$ (c). Lattice parameters are the same as in Fig. 1.

Figure 3

Demonstration of the robustness of acoustic one-way edge states against disorder. Topological protection requires the waves to be fully transmitted through an acoustic cavity (a), a $Z$-shape bend along the interface (b), and a 180-deg bend (c). The operating frequency is ${\omega }_0=0.577\times 2\pi c_a/a$ (992 Hz) and $\mathrm{\Omega }=2\pi \times 400\mathrm{rad}/\mathrm{s}$. Lattice parameters are the same as in Fig. 1.