Broadband sound barriers with bianisotropic metasurfaces

Noise is a long standing societal problem that has recently been linked to serious health consequences. Despite decades of research on noise mitigation techniques, existing methods have significant limitations including inability to silence broadband noise and shield large volumes. Here we show theoretically and experimentally that acoustic bianisotropic materials with non-zero strain to momentum coupling are remarkably effective sound barriers. They surpass state-of-the-art sound isolators in terms of attenuation, bandwidth, and shielded volume. We implement our barriers with very compact active meta-atoms that owe their small size to their local response to external sound. Moreover, our active approach is not constrained by feedback stabilization requirements, in stark contrast with all traditional active sound control systems. Consequently, bianisotropic sound barriers have the potential to revolutionize noise control technologies and provide much needed solutions to an increasingly important and difficult challenge.

Consider a sound source placed in front of the conventional wall shown in Supplementary   Fig. 1a. A window is cut in the wall and replaced with a bianisotropic metasurface composed of meta-atoms whose functional behavior is summarized in Supplementary Fig. 1b. The metaatom response to the incident wave p i is described in terms of polarizabilities following a standard procedure [1][2][3][4] . Specifically, the meta-atom is modeled as a set of sources represented as spheres placed in a background fluid of characteristic impedance Z 0 . Each source responds to either the monopole moment associated to the local pressure p loc or the local particle velocity v loc associated to the dipole moment produced by the incident wave. In response to p loc and v loc the sources generate either a monopole moment (non-zero local pressure field) or a dipole moment (non-zero local particle velocity).
The bianisotropic meta-atom presented here is composed of three sources that account for the conventional monopole-to-monopole, dipole-to-dipole, and bianisotropic monopole-to-dipole couplings. We assume a zero dipole-to-monopole coupling, thus breaking the reciprocity conditions in unbiased passive media [4][5][6][7] . We further assume that the conventional polarizabilities α m , α d = α d I are isotropic and the bianisotropic polarizability is α md = α mdx , where I is the second order unit tensor andx is the unit vector pointing in the direction perpendiculat on the metasurface, as illustrated in the Supplementary Fig. 1a. Our goal is to find the pressure field transmitted through the metasurface p t in terms of α m , α d , α md , and the incident wave p i .
The local fields produced by the three types of polarizabilities are given by The dipole-to-dipole and monopole-to-dipole inclusions produce pure dipole moments (net particle velocity but zero local pressure), and the monopole-to-monopole inclusion produces pure monopole moments (net pressure but zero local particle velocity), therefore the local pressure and particle velocity are written Equations (1) and (2) lead to the following local field expressions where v i is the particle velocity associated with the incident wave.
The total pressure travelling through the metasurface is the superposition of the incident field p i and the fields scattered by the three inclusions where the transmitted pressure fields propagating in the positivex direction produced by the sources are obtained from equations (1) and (3) and are given by, respectively In the above equations we took into account that the ratio pressure-to-particle-velocity is the characetristic impedance of the background fluid Z 0 . Plugging the above expressions in equation (4) leads to the following expression of the transmitted pressure in the positivex direction We can decompose the transmitted pressure into a bianisotropic pressure p b due to the bianisotropic polarizability α md and a conventional pressure p c due to the purely monopole-tomonopole α m and dipole-to-dipole α d metasurface responses. The bianisotropic and conventional pressures are obtained from equation (6) and are given by Active systems based on sensor-driver pairs have potential instability issues due to the feedback loops caused by 1) the close proximity of the sensor and driver (intra-meta-atom coupling), and 2) reflections of the sound produced by the dipole driver from the environment coupling back into the sensor (meta-atom-environment coupling). We show in this section that the active bianisotropic meta-atom is inherently stable even when embedded in complex environments and does not require any electronic feedback stabilization techniques that reduce the effectiveness of traditional active sound control techniques.
This remarkable stability comes from the requirement to realize the non-zero monopole-to-dipole Willis coupling term. Namely, the meta-atom should sense the local pressure and generate a local velocity field that, in turn, has no contribution to the local pressure field. This design constraint translates into an active meta-atom in which the sensing and driven transducers are mechanically decoupled, i.e. the field produced by the driver is not sensed by the sensor. As shown in Supplementary Note 2, we realize this requirement by placing the pressure sensing transducer (i.e. monopole sensor) into the plane of anti-symmetry of the driven transducer (i.e. a dipole source).
This geometry makes the dipole invisible to the sensing monopole, thus cancelling the intra-metaatom feedback. Therefore, no additional circuitry is needed to stabilize the local feedback loop.
This is in clear contrast with related approaches described in the literature in which active sound and vibration control performance and bandwidth are limited by stability constraints 11,12 .
To show that the meta-atom is stable in the presence of external scatterers, we consider the test case of a meta-atom placed inside a one-dimensional waveguide. This scenario simulates an infinite metasurface (see Fig. 3a). We analyze the metasurface dynamic response to an incident wave of pressure p i using the flow graph presented in Fig. 3b. Various components of the meta-atom and waveguide are represented through their impulse responses. Specifically, s, g, and r represent the impulse responses of the sensing, active, and driven elements respectively. The driver-sensor feedback path is modeled by the impulse response f , and the environment is modeled through the impedances Z r and Z t looking towards the wave source (reflection side) and away from the source (transmission side), or, equivalently, through the reflection coefficients Γ r and Γ t defined as shown in the figure in terms of Z r , Z t , and the background characteristic impedance Z 0 . The passive reflection and transmission coefficients are defined as R = p r /p i and T = p t /p i , respectively. From the block diagram we can express the transfer function p act /p i and look at its stability requirements.
For maximum sound isolation we choose the active impulse response g = −T/(sr) which leads to no transmission through the metasurface in the ideal case in which f = 0. With this choice of g, the Nyquist criterion becomes |T || f −Γ r +Γ t | < 1. Therefore, a sufficient condition for stability Ideally, f = 0 but meta-atom fabrication imperfections makes f non-zero. To measure | f |, we performed two experiments. In both experiments we interrupt the direct path between g and r in Fig. 3b. The two on-board speakers are driven by a broadband chirp, e, having a Gaussian envelope and a 3 dB band of 2 kHz -4kHz. The signal fed back to the front microphone, s 1 = esr f g, is recorded and compared against the signal received by the sensing monopole assembly when a third speaker, identical to the two on-board speakers, is placed 1 cm in front of the meta-atom and is driven by the same electrical signal e. The signal measured after the gain element g in the second experiment is s 2 = esrg. It follows that f = s 1 /s 2 . Figure 3c shows the measured | f |. The amplitude | f | is highly sub-unitary in the octave 2 kHz -4 kHz, and assures a stable meta-atom behavior for any passive environment.
Even in high gain media the meta-atom remains stable. For example, if the meta-atom is designed to have a passive transmission coefficient of magnitude |T | ≈ 0.1 which is the scenario analyzed in this paper, it follows that the condition for stability becomes |Γ r | + |Γ t | < 9.8. In the most unfavorable scenario |Γ r | = |Γ t | = 1 and the stability condition is fulfilled. Remarkably, even in high gain media in which |Γ r | > 1 and/or |Γ t | > 1 the meta-atom remains stable as long as