Controlling Sound in Non-Hermitian Acoustic Systems

The concept of non-Hermitian physics has inspired numerous intriguing phenomena in classical wave systems. By introducing judicious arrangements of gain and loss media, non-Hermitian modulation in acoustic metamaterials can give rise to many extraordinary wave-matter interactions that cannot be achieved naturally, which have been verified theoretically and experimentally by a series of previous works. Here, we aim to review recent developments in this topic. First of all, we explain the basic concepts and mathematical tools to deal with non-Hermitian acoustics by studying some pedagogical examples. Then, we present some interesting works that demonstrate the superior abilities of non-Hermitian modulation in wave manipulation. Also, we pay attention to the study of topological systems with non-Hermitian modulation, with a special emphasis on nontrivial topological states induced by non-Hermiticity alone. Finally, we present an outlook on the potential directions and applications for future works. We hope this review can provide a better understanding to stimulate research on acoustics beyond the Hermitian regime.

Figure 1

Non-Hermitian acoustics in the closed-cavity system. Two identical acoustic resonators with eigenfrequency ${\omega }_0$ are connected by a narrow tube. Existence of the tube can introduce coupling to split the eigenmode, while the amplitude of coupling can be adjusted by changing the size of the opening of the tube, d. (a) Upper panel shows, in the Hermitian case, two passive resonators (depicted in blue) are connected by a narrow tube. Dipole mode with eigenfrequency ${\omega }_0$ will be split into odd mode and even mode with frequencies of ${\omega }_{+}$ and ${\omega }_{-}$. Lower panel shows the change of splitting eigenmodes as a function of d. (b) Upper panel shows, in the non-Hermitian case, a passive resonator (depicted in blue) and a lossy resonator (depicted in green) connected by a narrow tube. Fields localized in one of the resonators in the PT-broken phase. Lower panel shows the real and imaginary parts of eigenfrequency that change as a function of non-Hermitian modulation, $\gamma$. (c) Upper panel shows, in the non-Hermitian case, an amplified resonator (depicted in yellow) and a lossy resonator (depicted in green) connected by a narrow tube. Lower panel shows real and imaginary parts of the eigenfrequency that change as a function of non-Hermitian modulation, $\gamma$.

Figure 2

Non-Hermitian acoustics in an open scattering system. (a) Two areas with the same length, $L\mathrm{ =\; 5}\mathrm{cm}$, are located in the middle of a straight pipe symmetrically. One (depicted in yellow) has the gain modulation [${\rho }_g={\rho }_0$ and $c_g=c_0(1-i\gamma )$] and the other one (depicted in green) has the loss modulation [${\rho }_l={\rho }_0$ and $c_l=c_0(1+i\gamma )$]. Acoustic waves can propagate from both sides. (b) Transmission and reflection coefficients are retrieved from the scattering properties. (c) Eigenvalues of the scattering matrix in two different formulas versus frequency.

Figure 3

Non-Hermitian acoustics in multiple-cavity systems. (a) Four-state system that consists of four coupled acoustic resonators with asymmetric loss modulation used to exhibit multiple EPs. (b) Formation pattern of multiple EPs is mapped in the parameter space. (c) Three-state system that consists of three coupled acoustic resonators with non-Hermitian modulation used to realize the hybrid topological invariant. (d),(e) Eigenvalue Riemann surfaces show the encircling of the exceptional nexus in different parameter spaces. (a),(b) Reprinted with permission. © 2016 APS. (c),(d) Reprinted with permission. © 2020 AAAS.

Figure 4

PT acoustics with gain and loss modulation. (a) Schematic diagram of acoustic PT-symmetric structure. (b) Invisible acoustic sensor realized by active elements with PT symmetry. (c) Acoustic cloak enabled by the PT-symmetric metasurface. Blue dashed line represents the lossy part, while the red dashed line represents the amplified parts. (d) Acoustic pressure field of the rigid rod cloaked by the PT-symmetric metasurface. (e) Experimental approach to accessing the EP via active acoustic elements. (f) Conceptual diagrams of the incidences from the loss side and gain side. (g) Transmission and reflection coefficients for incidences from the loss side and gain side versus varying spacing between them. (a) Reprinted with permission. © 2014 APS. (b) Reprinted with permission. © 2015 Springer Nature. (c),(d) Reprinted with permission. © 2019 AAAS. (e)–(g) Reprinted with permission. © 2016 Springer Nature.

Figure 5

Non-Hermitian modulation enables asymmetric transmissions and reflections through the metasurface structure. (a) Illustration of tunable asymmetric transmission induced by loss modulations. (b) Calculated acoustic pressure fields of positive and negative transmissions for three cases: gradient-index metasurface without loss modulation, gradient-index metasurface with a loss of 0.14, and $\pm 1$ orders of the diffraction mode with a loss of 0.14. (c) Schematic diagram of asymmetric reflection at the EP. (d) Scattering fields for high-efficiency asymmetric reflection with which the wave can be retroreflected from one side while being absorbed perfectly from the opposite side. (a),(b) Reprinted with permission. © 2017 APS. (c),(d) Reprinted with permission. © 2019 APS.

Figure 6

Non-Hermitian modulation enables unidirectional wave-vector manipulation. (a) Unidirectional acoustic focusing by using the passive PT-symmetric metamaterial structure. Left and right parts are simulated and measured fields, respectively. Acoustic wave incident from the bottom is reflected and generates a focused field. Acoustic wave incident from the top side is absorbed due to dissipation in the PT-symmetric potentials. (b) Single-sided acoustic beam splitter under balanced gain and loss modulation. Acoustic wave incident from the left (black solid line) will be split into two beams (red and green solid lines). For the wave incident from the top side, there will be total transmission without wave-front distortion. (a) Reprinted with permission. © 2018 APS. (b) Reprinted with permission. © 2020 APS.

Figure 7

Asymmetric absorption around the EP. (a) Two Helmholtz resonators are loaded on the waveguide. (b) Asymmetric absorption via two small tubes embedded in the square cavity. Transmission and reflection coefficients of the sample shown in (b) for left incidence (c) and right incidence (d). (e) Experimental setup of coupled resonators. (f) Asymmetric absorption of the sample shown in (e) versus frequency for both incidence directions. (a) Reprinted with permission. © 2017 APS. (b)–(d) Reprinted with permission. © 2021 Springer. (e),(f) Reprinted with permission. © 2018 APS.

Figure 8

(a) Continuous-pressure fields of the disordered system with and without non-Hermitian modulation. (b) Discretized model of the disordered system with and without non-Hermitian modulation. (c) Picture of the passive PT-symmetric ring cavity. (d) Amplitude of pressure in the cavity with and without non-Hermitian modulation. (e) Phase of pressure in the cavity with and without non-Hermitian modulation. (a),(b) Reprinted with permission. © 2018 Springer Nature. (c)–(e) Reprinted with permission. © 2020 Springer Nature.

Figure 9

Acoustic topological insulator with non-Hermitian modulation. (a) Acoustic second-order topological insulator composed of the cylinders embedded in the background medium. Each unit contains four cylinders, and by expanding or shrinking these four cylinders, the topological property of the sample can be adjusted. Green and yellow denote the cylinders with gain and loss modulation respectively. (b) Simulated eigenmodes of edge states (the upper panel) and corner states (the lower panel). (c) Acoustic valley topological insulator composed of a hexagonal lattice. Gray and white circles represent the cylinders with gain or loss modulations respectively. (d) Amplitude and phase fields for the valley states. (e) Intensity fields of valley states excited by a chiral source under the non-Hermitian modulation. (a),(b) Reprinted with permission. © 2019 APS. (c)–(e) Reprinted with permission. © 2018 APS.

Figure 10

Non-Hermiticity-induced topological insulators. (a) 1D non-Hermitian topological insulator coupled with acoustic resonators. Simulated and measured spectra in one of the (b) bulk and (c) edge resonators. (d) 2D non-Hermitian topological insulator coupled with acoustic resonators. Measured intensity profile for (e) corner mode and (f) edge mode. (a) –(c) Reprinted with permission. © 2020 APS. (d)–(f) Reprinted with permission. © 2021 Springer Nature.

Figure 11

Non-Hermitian skin effect in acoustics. (a) Sketch of the basic unit with nonreciprocal coupling. (b) Intensity distribution of the eigenstates of the lattice structure with 20 sites.