Acoustic Purcell effect induced by quasibound state in the continuum

Graphical abstract

Abstract

We study theoretically and experimentally the acoustic Purcell effect induced by quasi-bound states in the continuum (quasiBICs). A theoretical framework describing the acoustic Purcell effect of a resonant system is developed based on the system’s radiative and dissipative factors, which reveals the critical emission condition for achieving optimum Purcell factors. We show that the quasiBICs contribute to highly confined acoustic field and bring about greatly enhanced acoustic emission, leading to strong Purcell effect. Our concept is demonstrated via two coupled resonators supporting a Friedrich–Wintgen quasiBIC, and the theoretical results are validated by the experiments observing emission enhancement of the sound source by nearly two orders of magnitude. Our work bridges the gap between the acoustic Purcell effect and acoustic BICs essential for enhanced wave-matter interaction and acoustic emission, which may contribute to the research of high-intensity sound sources, high-quality-factor acoustic devices and nonlinear acoustics.

1. Introduction

The Purcell effect predicts that spontaneous radiation is not an intrinsic property of matter, but is determined by the interaction of matter with its surrounding environment [1], [2], [3], [4], [5], [6]. Although this notion was first explored in quantum systems [7], [8], [9], it fundamentally reveals that the emission of sources can be modulated actively by engineering the surrounding environments, which has aroused broad interests in electromagnetic [10], [11], [12], [13], [14], [15], elastic [16], [17] and acoustic [18], [19] wave systems. The acoustical counterpart of the Purcell effect in quantum emission has been proposed via Mie-like resonances [18] and Helmholtz resonances [19], and shown to be important for improving the low-frequency emission of loudspeakers and generating strong acoustic collimated waves [18], [19], [20], [21]. A major challenge in achieving strong acoustic Purcell effect is that realistic acoustic systems usually suffer from non-trivial dissipative (thermal-viscous) losses and exhibit considerably low quality factors ($Q$), which severely restricts the highest Purcell factor one may access. More importantly, despite the importance of thermal-viscous losses, a comprehensive theoretical model that incorporates the effect of dissipative losses for describing the acoustic Purcell effect has not been developed so far.

In this study, we establish a theoretical framework with the effects of radiative and dissipative losses being considered and systematically investigate the acoustic Purcell effect. As we will show in the following, the acoustic Purcell factor is governed by the system’s radiative and dissipative factors ($Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$), and there exists a critical condition that signifies the combinations of $Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$ for the optimum Purcell factor values. To improve the upper bound of the Purcell effect, we should simultaneously decrease the values of $Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$. However, it is rather difficult and even conflicting in realistic acoustic resonant structures (shown later). We subsequently manifest that this conflict can be well solved if we introduce the concept of quasiBIC-induced Purcell effect. Bound states in the continuum (BICs) are peculiar states perfectly confined with no radiation yet lie inside a continuous spectrum of radiating waves [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33]. Acoustic BICs, early explored in the 1960s [34], [35], [36], exhibit infinite $Q$ factors and feature complete isolation, which can provide high $Q$ factors [37], [38], [39], [40], [41] and meanwhile allow access to external radiation when turning into quasiBICs [22], [42], [43], [44], therefore providing an efficient pathway for the construction of strong Purcell effect. To demonstrate our concept, a two-state system supporting a Friedrich–Wintgen quasiBIC together with its cavity-based practical implementation is presented, offering tunable and low $Q_{\text{leak}}^{-1}$ to approach the critical emission condition and consequently promoting the acoustic Purcell effect. The presented concept of quasiBIC-induced Purcell effect and the developed framework would open up an avenue to explore high-intensity, high-$Q$ and nonlinear-acoustic devices.

2. Methods

2.1. Numerical simulations

All simulations in this paper are performed with the commercial finite element software COMSOL Multiphysics with the preset “Pressure Acoustics, Frequency Domain” module. In the simulations, the domain material is air (the static air density, $\rho =1.21$ $\text{kg}/{\mathrm{m}}^3$, the sound speed, $c=343$ $\mathrm{m}/\mathrm{s}$, the dynamic viscosity of air, $\mu =1.81\times {10}^{-5}$ $\mathrm{N}\mathrm{S}/{\mathrm{m}}^2$, and the preset environmental temperature, $T=293.15\mathrm{K}$ (20 ${}^{\circ }\mathrm{C}$). The detailed information of the simulations are presented in Supplementary Note 1 and Note 2.

2.2. Sample fabrications and measurements

The experimental sample is fabricated by 3D-printing technology using laser stereolithography (SLA, 140 µm) with photosensitive resin (UV curable resin) being the base material (manufacturing precision 0.1 mm). In the measurements, a loudspeaker is mounted at the bottom center of one of the cavities, and the sample is connected to a waveguide with the same cross-section. Porous sound-absorbing materials are placed at the waveguide’s end to create a nonreflecting boundary. A 1/4-inch condenser microphone (Brüel & Kjær Type 4187) is situated at the designated position to test the emitted sound. The detailed experimental setup is presented in Supplementary Note 3.

3. Results and discussion

3.1. Concept of the quasiBIC-induced acoustic Purcell effect

We start by introducing the concept of the quasiBIC-induced acoustic Purcell effect (Fig. 1). For quasiBIC-supporting systems, when acoustic sources are placed inside, the emitted waves in the continuum frequency spectrum will only receive trivial or negligible influence due to the systems’ normal radiation rate, while the emitted waves at the quasiBIC will be greatly enhanced thanks to the strongly confined fields within the systems [45], [46]. Generally, a resonant mode of an open system can be characterized by the system’s radiative and dissipative factors ($Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$) that describe the relations of the energy stored to the energy leaked and dissipated, respectively. Thus, a theoretical framework interpreting the mode-induced acoustic Purcell effect based on $Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$ would be generic and comprehensive.

Fig. 1.

Concept of the quasiBIC-induced acoustic Purcell effect. In a quasiBIC-supporting system (light-blue domain) with a sound source inside, the emitted waves at the quasiBIC (green) will be greatly confined and lead to strong enhancement. The emitted waves in the continuum frequency spectrum (grey-shaded, blue, red) will receive insignificant influence. $x_0$ marks the front surface of the system.

3.2. Purcell factor of an acoustic resonant system

As depicted in Fig. 2a, we consider an acoustic resonant system constructed with an effective medium. A source is placed at the bottom radiating through the interface located at $x_0$ to the outside. From the decomposed sound field given in the caption of Fig. 2a, $Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$ can be deduced as [47]: (see Supplementary Note 4 for details)

$$\begin{array}{c}\hfill Q_{\text{leak}}^{-1}=\frac{c{\left|P_{\mathrm{T}}\right|}^2}{{\omega }_0{L}_{\mathrm{e}}({\left|P_{\mathrm{A}2}\right|}^2+{\left|P_{\mathrm{B}2}\right|}^2)}\end{array}$$ (1)

$$\begin{array}{c}\hfill Q_{\text{loss}}^{-1}=\frac{c{\epsilon }^2{\left|P_{\mathrm{A}2}\right|}^2}{{\omega }_0{L}_{\mathrm{e}}({\left|P_{\mathrm{A}2}\right|}^2+{\left|P_{\mathrm{B}2}\right|}^2)}\end{array}$$ (2)

Fig. 2.

Theoretical framework of the acoustic Purcell effect. (a) Wave analysis of a resonant system whose structure is theoretically treated as an effective medium and characterized by $Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$ of the dominating resonance. The sound source is at the bottom of the system. $P_{\mathrm{I}}$ represents the acoustic pressure amplitudes of waves generated by the source with a constant intensity. $P_{\mathrm{T}}$ represents the acoustic pressure amplitudes of waves emitted to the outside of the resonant system. $P_{\mathrm{A}1}$, $P_{\mathrm{A}2}$, $P_{\mathrm{B}1}$ and $P_{\mathrm{B}2}$ are the pressure amplitudes of decomposed waves at the bottom and front boundaries of the system. (b) Calculated $F_{\text{peak}}$ with the variation of $Q_{\text{leak}}^{-1}$ in the absence of $Q_{\text{loss}}^{-1}$. (c) Calculated $F_{\text{peak}}$ as functions of $Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$. The white dashed line indicates the critical emission condition.

where $L_{\mathrm{e}}$ is the effective propagation length for waves from sound source to the system’s front surface, $c$ the sound speed, and $\epsilon$ the intrinsic-dissipation factor [48] (see Supplementary Note 4 for details).

The acoustic Purcell factor ($F$) can be defined as $|P_{\mathrm{T}}/P_{\mathrm{I}}|{}^2$, which indicates the enhancement of the source’ emitted sound power to the radiation field contributed from the presented structures [16], [17], [18], [19], [45], [46]. With the derivations in Supplementary Note 4, the link between the quality factors of a resonant mode and the peak acoustic Purcell factor can be obtained as:

$$\begin{array}{c}\hfill F_{\text{peak}}=1+\frac{2}{\pi Q_{\text{leak}}^{-1}}+2\sqrt{\frac{1}{\pi Q_{\text{leak}}^{-1}}+\frac{1}{{\left(\pi Q_{\text{leak}}^{-1}\right)}^2}}\end{array}$$ (3)

$$\begin{array}{ccc}\hfill F_{\text{peak}}& & =\frac{Q_{\text{leak}}^{-1}}{Q_{\text{leak}}^{-1}+Q_{\text{loss}}^{-1}}+\frac{2Q_{\text{leak}}^{-1}}{\pi {(Q_{\text{leak}}^{-1}+Q_{\text{loss}}^{-1})}^2}\hfill \\ & & +2Q_{\text{leak}}^{-1}\sqrt{\frac{1}{\pi {(Q_{\text{leak}}^{-1}+Q_{\text{loss}}^{-1})}^3}+\frac{1}{{\pi }^2{(Q_{\text{leak}}^{-1}+Q_{\text{loss}}^{-1})}^4}}\hfill \end{array}$$ (4)

where $F_{\text{peak}}$ is the Purcell factor at the dominating mode of the system, which indicates the highest value of $F$ in the frequency spectrum. Eqs. 3 and 4 correspond to the conditions in the absence of and in the presence of dissipative losses, respectively. Compared with the theoretical model based on the acoustic density of states (DOS) [18], the theoretical model established allows the consideration of the effect of dissipative (thermal-viscous) losses. Eq. 4 manifests that $F_{\text{peak}}$ is determined by the mutual effect of $Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$ and reaches its optimum under the critical emission condition that reads:

$$\begin{array}{c}\hfill Q_{\text{leak}}^{-1}=Q_{\text{loss}}^{-1}(1+\pi Q_{\text{loss}}^{-1})\end{array}$$ (5)

It is noteworthy that the Purcell factors described in Eqs. 3 and 4 don’t contain a term of “acoustic mode volume”, which is different from the commonly-used theoretical model in describing Purcell factors in quantum systems. The reason for this may be that the acoustic Purcell factor here is derived based on a simple one-dimensional effective-medium system.

The developed framework reveals important insights of the Purcell effect for an acoustic system. On the one hand, in a hypothetical lossless scenario ($Q_{\text{loss}}^{-1}=0$), $F_{\text{peak}}$ increases with the decrease of $Q_{\text{leak}}^{-1}$, where an infinitely small $Q_{\text{leak}}^{-1}$ leads to an infinite emission enhancement (Fig. 2b). This suggests the great potential of quasiBIC systems, featured with extremely low $Q_{\text{leak}}^{-1}$, in stimulating the strong acoustic Purcell effect. Note that in photonic systems where the dissipative losses could often be tuned to a relatively low level (compared to acoustic systems) by carefully selecting the base material, the dissipative losses can be neglected when using a quasiBIC to realize strong emission enhancement [33]. However, in practical acoustic systems with non-trivial thermal-viscous losses, the effect of $Q_{\text{loss}}^{-1}$ becomes crucial (see Supplementary Note 5), and optimum emission enhancement takes place under the critical emission condition (white dashed line in Fig. 2c). Evidently, to achieve a larger Purcell factor, it will be ideal that $Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$ of the system can be simultaneously modulated to lower values.

Nevertheless, the above requirements for strong Purcell effect are rather challenging in acoustic resonant systems due to the contradiction between low $Q_{\text{leak}}^{-1}$ and low $Q_{\text{loss}}^{-1}$. For commonly-seen acoustic resonators such as Helmholtz resonators (HRs), a lower $Q_{\text{leak}}^{-1}$ means less energy leaking from the openings, which subsequently demands narrower openings (necks) of the HRs. A narrower opening, in turn, leads to stronger thermal-viscous boundary layer effect [48], [49] and brings about higher dissipative losses (larger $Q_{\text{loss}}^{-1}$) [19], [46], thereby preventing us from realizing a large acoustic Purcell factor (exemplified by the red arrow in Fig. 2c). This contradiction between $Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$ of HRs is specified in detail in Supplementary Note 6.

3.3. Construction of a Friedrich–Wintgen quasiBIC-supporting system

In this study, we find that the contradiction is not inherent and can be well solved by constructing a Friedrich–Wintgen quasiBIC-supporting system (Fig. 3a), which enables the Purcell factors to theoretically reach infinite without essential contradictory restrictions. The unit cell of the periodic quasiBIC-supporting system consists of two slightly detuned resonators (cavities) with the same cross-section but different lengths (Fig. 3b). Based on the presented system, we can modulate $Q_{\text{leak}}^{-1}$ from zero (a pure BIC) to a target value via tuning the length difference between the two cavities ($l_{\mathrm{A}}$ and $l_{\mathrm{B}}$). In the previous study [42], a similar system was presented for extreme sound confinement of incoming sound waves from outside, while in this work we investigate the emission property of a sound source inside the quasiBIC-supporting system. The modulation of $Q_{\text{leak}}^{-1}$ can be described by the temporal coupled-mode theory (CMT) [22], [42], [50], [51], [52] (see Supplementary Note 7), from which we can obtain the Hamiltonian matrix of the presented system that $-i\partial A/\partial t=HA$, $A=({\bar{a}}_{\mathrm{A}},{\bar{a}}_{\mathrm{B}}){}^{\mathrm{T}}$, with:

$$\begin{array}{c}\hfill H=\left(\begin{array}{cc}{\omega }_{\mathrm{A}}& 0\\ 0& {\omega }_{\mathrm{B}}\end{array}\right)+i\left(\begin{array}{cc}{\gamma }_{\mathrm{A}}& \sqrt{{\gamma }_{\mathrm{A}}{\gamma }_{\mathrm{B}}}\\ \sqrt{{\gamma }_{\mathrm{A}}{\gamma }_{\mathrm{B}}}& {\gamma }_{\mathrm{B}}\end{array}\right)\end{array}$$ (6)

Fig. 3.

Friedrich–Wintgen quasiBIC-supporting system. (a) Schematic of the system consisting of periodic unit cells. (b) The modulation of the unit cells with acoustic sources. The length of Cavity A ($l_{\mathrm{A}}$) is fixed. The length of Cavity B ($l_{\mathrm{B}}$) is tunable. (c) Calculated (CMT) and simulated $Q_{\text{leak}}$ with the variation of $\Delta l$. The infinite $Q_{\text{leak}}$ at $\Delta l=0$ demonstrates the existence of a BIC. (d) $F_{\text{peak}}$ with the modulation of $\Delta l$ and the hypothetical free variation of $Q_{\text{loss}}^{-1}$. The analyzed system’s geometric parameters are set to $w=46$ mm, $h=100$ mm, $b=8$ mm and $l_{\mathrm{A}}=120$ mm. The white dashed line indicates the critical emission condition.

where ${\tilde{a}}_{\mathrm{A}\left(\mathrm{B}\right)}$ ($=a_{\mathrm{A}\left(\mathrm{B}\right)}{e}^{i\omega t}$) represents the resonance mode of Cavity A(B), ${\omega }_{\mathrm{A}\left(\mathrm{B}\right)}$ the resonant angular frequency of Cavity A(B), ${\gamma }_{\mathrm{A}\left(\mathrm{B}\right)}$ ($={\omega }_{\mathrm{A}\left(\mathrm{B}\right)}{Q}_{\text{leakA}\left(\mathrm{B}\right)}^{-1}/2$) the corresponding radiative decay rate. Then, the two eigenvalues of $H$ can be deduced as [53]:

$$\begin{array}{c}\hfill {\sigma }_{\pm }={\omega }_{\text{ave}}+i{\gamma }_{\text{ave}}\pm \sqrt{{\mu }^2+{({\omega }_{\text{diff}}+i{\gamma }_{\text{diff}})}^2}\end{array}$$ (7)

where ${\omega }_{\text{ave}}=({\omega }_{\mathrm{A}}+{\omega }_{\mathrm{B}})/2$, ${\gamma }_{\text{ave}}=({\gamma }_{\mathrm{A}}+{\gamma }_{\mathrm{B}})/2$, ${\omega }_{\text{diff}}=({\omega }_{\mathrm{A}}-{\omega }_{\mathrm{B}})/2$ and ${\gamma }_{\text{diff}}=({\gamma }_{\mathrm{A}}-{\gamma }_{\mathrm{B}})/2$. $\mu =i\sqrt{{\gamma }_{\mathrm{A}}{\gamma }_{\mathrm{B}}}$ is the coupling coefficient of the presented system. The radiative decay rate of the system (${\gamma }_{\mathrm{s}}$) at the quasiBIC can be introduced as the imaginary part of ${\sigma }_{-}$. As a proof-of-concept demonstration, we consider a structure with $w=46$ mm, $h=100$ mm, $b=8$ mm and $l_{\mathrm{A}}=120$ mm. And with the tuning of $l_{\mathrm{B}}$, it is observed from Eq. 7 that the system is in the vicinity of a BIC when the length difference of the two cavities ($\Delta l=l_{\mathrm{B}}-l_{\mathrm{A}}$) approaches zero (Fig. 3c).

Apart from the free tuning of $Q_{\text{leak}}^{-1}$ shown in Fig. 3c, the system’s $Q_{\text{loss}}^{-1}$ can be modulated by changing the cross-sectional area of the cavities, where a wider cross-sectional area leads to a smaller $Q_{\text{loss}}^{-1}$. The system’s $Q_{\text{loss}}^{-1}$ can be calculated from the reflection property of the system [49], [54], [55], [56], [57], which is given by [58], [59]:

$$\begin{array}{c}\hfill Q_{\text{loss}}^{-1}=Q_{\text{leak}}^{-1}\left(\frac{1+r_0}{1-r_0}\right)\end{array}$$ (8)

where $r_0$ is the reflection coefficient of the system at resonance under normal incidence. To examine intuitively how the Purcell effect of the system is governed by the dissipative losses and the quasi-BIC, we analyze the structure in Fig. 3c and assume an arbitrarily tunable $Q_{\text{loss}}^{-1}$. From the diagram of $F_{\text{peak}}$ shown in Fig. 3d, it can be observed that the presented quasiBIC-supporting system has the potential to achieve optimum $F_{\text{peak}}$ under arbitrary $Q_{\text{loss}}^{-1}$ via changing $\Delta l$. In addition, an acoustically rigid waveguide can create a series of mirror images of the cavity unit, which is equivalent to an infinitely repeated arrangement of the building blocks. Consequently, the presented system can also achieve an infinite $Q_{\text{leak}}$ (BIC) with one or more units in a rigid waveguide. In Supplementary Note 8, we analyze a unit cell with hard boundaries of a radiation channel, and the results only exhibit trivial differences from the results in Fig. 3c and d.

3.4. Experimental demonstration of the quasiBIC-induced acoustic Purcell effect

To verify our theoretical analysis, experiments are conducted in a steel waveguide (tube) with a length of 1460 mm and a wall thickness of 8 mm. The cross-section of the waveguide is a square with a side length of 100 mm. A unit cell of the presented structure (Fig. 3b) is fabricated by 3D-printing technology. The detailed experimental setup is presented in Supplementary Note 3. When the two cavities have a length difference of 9.5 mm ($\Delta l=9.5$ mm, $Q_{\text{loss}}^{-1}$ = 0.0037, $Q_{\text{leak}}^{-1}$ = 0.0043), large emission enhancement with a Purcell factor of 82 is experimentally achieved at 564 Hz (Fig. 4a). This result proves the existence of the strong acoustic Purcell effect induced by the quasiBIC. In addition, in the case that $\Delta l=0$ mm, the inferior coupling condition of $Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$ suppresses the acoustic Purcell effect and the emitted waves are unable to be enhanced, which is in accordance with the results demonstrated in Fig. 3d. To further validate our results and better illustrate the acoustic Purcell effect, simulations are conducted with the commercial finite element software COMSOL Multiphysics. The simulation results agree well with the experimental results, where $F_{\text{peak}}$ reaches 84.7 at 568.6 Hz. The frequency shift of $F_{\text{peak}}$ between the experiment and the simulation may result from the manufacturing error of the experimental sample and other environmental factors such as temperature. Besides, the simulated acoustic pressure fields shown in Fig. 4b confirm that the quasi-BIC contributes to the confined and intensive wave energy inside the system and finally leads to the largely enhanced emitted waves [45], [46]. Furthermore, the acoustic DOS of the sound field with and without the presented quasiBIC-supporting system is calculated in Supplementary Note 9, which shows a significant increase of acoustic DOS contributed by the presented system.

Fig. 4.

Experimental demonstration of the quasiBIC-induced acoustic Purcell effect. (a) Measured and simulated results of acoustic Purcell factors ($F_{}$) for $\Delta l=9.5$ mm (red) and $\Delta l=0$ mm (blue). (b) Simulated distributions of acoustic pressure (colors) and acoustic intensity (denoted by green arrows) of a normal tube and the presented two-state system. $S_{\text{in}}$ denotes the sound source. (c) Calculated, Measured and simulated $F_{\text{peak}}$ as a function of $\Delta l$. The variations of $Q_{\text{leak}}^{-1}$ for the systems with $w=46$ mm and $w=96$ mm are demonstrated in Supplementary Note 10.

It is worth mentioning that the Purcell factors of the presented quasiBIC-supporting system can be consistently enhanced by further increasing the cross-sectional area of the two cavities (smaller $Q_{\text{loss}}^{-1}$). For instance, when the structure size is changed to $w=96$ mm and $h=200$ mm, the simultaneously decreased $Q_{\text{loss}}^{-1}$ (0.002) and $Q_{\text{leak}}^{-1}$ (0.0022) give rise to an increased $F_{\text{peak}}$ reaching above 150 at $\Delta l=9.5$ mm (Fig. 4c). Compared to another efficient pathway to induce the strong Purcell effect by a Mie-like resonance whose potential Purcell factor is restricted by a finite $Q_{\text{leak}}$ [18], the quasiBIC-induced Purcell effect theoretically removes such a restriction.

4. Conclusion

We have presented the concept of the quasiBIC-induced acoustic Purcell effect, which provides a feasible and efficient pathway to achieve extreme emission enhancement. The acoustic Friedrich–Wintgen quasiBIC-supporting system overcomes the contradiction that exists in conventional acoustic structures and lifts the major upper limit of the acoustic Purcell effect. We have also developed a theoretical framework based on two fundamental factors ($Q_{\text{leak}}^{-1}$ and $Q_{\text{loss}}^{-1}$), which uncovers the comprehensive physical picture of the acoustic Purcell effect and reveals the critical emission condition. The good agreement among the experimental, simulation, and theoretical results proves the reliability of our framework. Our work opens up avenues for the study of the quasiBIC-induced acoustic Purcell effect for extreme emission enhancement, which would inspire the studies in the acoustic Purcell effect and BICs, and may find applications in high-intensity and nonlinear acoustics.

Declaration of competing interest

The authors declare that they have no conflicts of interest in this work.

Biographies

Sibo Huang received the B.S. degree in physics from Tongji University in 2016. He received the Ph.D. degree in acoustics at Tongji University and the Ph.D. degree in mechanical engineering at the Hong Kong Polytechnic University. His current research interests include acoustic metamaterials, acoustic metasurfaces and noise control.

Tuo Liu(BRID: 08103.00.89658) received his Ph.D. degree in mechanical engineering from the Hong Kong Polytechnic University in 2018. He is currently a full professor at the Institute of Acoustics, Chinese Academy of Sciences. His research interests include physical acoustics, acoustic artificial structures, and non-Hermitian acoustics.

Yong Li(BRID: 05925.00.17551) received his Ph.D. degree in acoustics from Nanjing University in 2013. He is currently a full professor at Tongji University. His current research interests focus on acoustic functional materials, non-Hermitian acoustics, and advanced broadband absorbers/liners.

Jie Zhu(BRID: 05219.00.59330) received his Ph.D. degree in materials engineering from the Pennsylvania State University in 2008. He is currently a full professor at Tongji University. His research focuses on structured acoustic materials and metamaterials, and acoustic imaging technology and system.

Appendix A. Supplementary materials

Supplementary Data S1

Supplementary Raw Research Data. This is open data under the CC BY license http://creativecommons.org/licenses/by/4.0/