Maximum Willis Coupling in Acoustic Scatterers

Willis coupling in acoustic materials defines the cross-coupling between strain and velocity, analogous to bianisotropic phenomena in electromagnetics. While these phenomena have been garnering significant attention in recent years, to date their effects have been considered mostly perturbative. Here, we derive general bounds on the Willis response of acoustic scatterers, show that these bounds can be reached in suitably designed scatterers, and outline a systematic venue for the realistic implementation of maximally bianisotropic acoustic inclusions. We then employ these inclusions to realize acoustic metasurfaces for bending and steering of sound with unitary efficiency.

Figure 1

Physical interpretation of bianisotropic coupling for an acoustic bianisotropic particle. (a) A general bianisotropic particle located in a standing acoustic wave where the applied velocity has its maximum and the pressure is almost zero. (b) A general acoustic particle: Here, 2A indicates the total volume velocity coming out from the upper outlet and 2B indicates the total volume velocity getting into the lower outlet at a specific instant. (c) Symmetric acoustic scatterer: Due to the constant width of the channel in the inclusion, the volume velocity for the upper and lower outlets should be identical. In this case, the inclusion is nonbianisotropic because the background velocity field does not excite the monopole component. (d) Example of an asymmetric acoustic inclusion: Since the lower outlet is closed, the total volume velocity for the lower outlet is zero. By decomposing the zero-volume velocity into -A and A, we see that the background volume velocity excites both dipole and monopole components, respectively.

Figure 2

Comparison of the cross-coupling terms for different biansiotropic inclusions providing different levels of bianisotropic coupling. The radius of all inclusions is 5 cm. (a) Inclusion with constant channel width providing no bianisotropic coupling. (b) Inclusion with different channel width realizing moderate bianisotropic coupling. (c) Inclusion with only one outlet realizing maximum bianisotropy. (d) Inclusion with resonance in two directions.

Figure 3

(a) Schematic of the designed acoustic metagrating, and simulated distribution of incident and reflected pressure fields. Here the period of the grating is chosen to be 15 cm to ensure ($-1$), (0) and (1) modes are the only modes which can carry out energy, and all other modes are evanescent waves. The biansiotropy inclusion is 4 cm in radius. For a normally incident plane wave, by choosing proper inclusion, unitary reflections can be achieved as shown in the left panel. (b) Normalized reflection spectrum for different Floquet channels. When the operation frequency is near 2404 Hz, all incident energy is reflected into $-1$ Floquet channel and unitary reflection is achieved.