Scattering Properties of an Acoustic Anti-Parity-Time-Symmetric System and Related Fabry–Perot Resonance Mode

Abstract

The pursuit of artificial structures exhibiting unusual acoustic properties is a major scientific endeavor, in which anti-parity-time ($\mathcal{PT}$) symmetry has been coming into view recent years. At the same time, with the emergence of new acoustic metamaterials, the classical Fabry–Perot resonance mode also exhibits fascinating scattering features similar to those of the anti-$\mathcal{PT}$-symmetric system. We derive the generalized relation for the scattering parameters of an acoustic anti-$\mathcal{PT}$-symmetric system with a transfer matrix, including conjugate bidirectional reflection coefficients and pure real feature of transmission coefficient. In the absence of the real or the imaginary (representing gain/loss) part of the refractive index, the anti-$\mathcal{PT}$-symmetric system degrades into a pair of complementary media, resulting in the bidirectional total transmission. Moreover, a Fabry–Perot resonance mode exhibiting conjugate bidirectional reflection coefficients and a pure imaginary transmission coefficient has been demonstrated. Our results are meaningful for guiding the experimental test of an acoustic anti-$\mathcal{PT}$-symmetric system and the design of associated bidirectional response prototype devices.

1. Introduction

Since the discovery of non-Hermitian Hamiltonian $\mathcal{H}$ that commute with the $\mathcal{PT}$ operator, a property that can lead to real-energy eigenvalues [1,2,3], intriguing developments in quantum mechanics have been made in recent years. Owing to the analogy between the Schrödinger equation in quantum mechanics and the classical wave equation, the concept of $\mathcal{PT}$ symmetry was further introduced into photonics [4,5,6,7,8,9] and acoustics [10,11,12,13,14,15,16,17]. By utilizing balanced gain and loss satisfying $n(-x)=n^{*}\left(x\right)$, many intriguing acoustic phenomena and applications have been predicted and observed, such as acoustic cloaks [10], invisible sensors [11,12], flexible control of exceptional points [13], unidirectional transparency [14], tunability of piezoelectric semiconductors [15,16] and periodic $\mathcal{PT}$-symmetric structures [17]. Additionally, while the stringent loss–gain balance can be circumvented by implementing unbalanced loss-only structures [18,19], the absence of gain gives rise to the intrinsic loss of overall energy.

Recently, another closely related striking symmetry, known as anti-$\mathcal{PT}$ symmetry, broadened the family of non-Hermitian symmetries [20,21,22,23,24,25,26,27,28,29]. Since it requires either gain or loss, the anti-$\mathcal{PT}$ symmetry had been demonstrated based on positive and negative index materials [20], nonlinear structure [21], flying atom beams [22], electrical circuit resonators [23], multiwaveguide systems [26], hybrid $\mathcal{PT}$ and anti-$\mathcal{PT}$ symmetry [27], and optical fibre [28]. Such systems lead to various intriguing effects, such as continuous lasing spectrum [20], energy difference conserving dynamics [23], coherent switch [24], and highly tunable Fano resonances [27]. However, due to the lack of efficient experimental methodology, the acoustic anti-$\mathcal{PT}$-symmetric system has not been implemented yet, either theoretically or experimentally. On the other hand, metamaterials have been widely demonstrated to control acoustic waves, but previous efforts were mostly based on modulating the real parts of the constitutive parameters [30,31,32,33,34]. The acoustic anti-$\mathcal{PT}$-symmetric structure, if it is realized based on metamaterials, may extend the study of metamaterials into the realm of complex constitutive parameters.

In this paper, we propose a synthetic system bridging positive-index material (PIM), negative-index material (NIM), and $\mathcal{PT}$-symmetric acoustics. Their refractive index is anti-symmetric under combined $\mathcal{PT}$ operations, i.e., $n(-x)=-n^{*}\left(x\right)$, with balanced PIM and NIM. In addition, we require that the mass density satisfies $\rho (-x)=-\rho \left(x\right)$ [20]. The imaginary part of $n\left(x\right)$ is symmetric, and can be positive (loss), negative (gain), zero, or any complicated spatial function. We found that they can display intriguing features, such as conjugate bidirectional reflection coefficients and pure real transmission coefficient, spontaneous phase transition of the scattering matrix, and a flat total transmission band. Subsequently, we further expand the scattering properties of the anti-$\mathcal{PT}$-symmetric system and design a Fabry–Perot resonance mode with conjugate bidirectional reflection coefficients but pure imaginary transmission coefficient. Note that this Fabry–Perot resonance mode and classical anti-$\mathcal{PT}$-symmetric system are comparatively different, no matter the geometric structure or the implementation method, and they are related to each other only through scattering properties. Our results, comprising conjugate bidirectional reflection coefficients and pure real/imaginary transmission coefficient are expected to have a huge application advantage in the virtual imaging and stealth of acoustic waves.

2. Generalized Relation

We generally begin with the scattering matrix S, which can be defined by [14]

$$\left[\begin{array}{c}{p}_1^{-}\\ p_2^{+}\end{array}\right]=S\left[\begin{array}{c}{p}_1^{+}\\ p_2^{-}\end{array}\right]\equiv \left[\begin{array}{cc}{r}_L& t\\ t& r_R\end{array}\right]\left[\begin{array}{c}{p}_1^{+}\\ p_2^{-}\end{array}\right],$$

(1)

where $p_{1,2}^{+}$ and $p_{1,2}^{-}$ represent the sound pressure spreading along the forward and backward direction, respectively, with the subscript 1 (2) denoting the PIM (NIM) side, as illustrated in Figure 1a. $r_L$ ($r_R$) is the reflection coefficient from the PIM (NIM) side of the system, and t is the transmission coefficient from both sides of the system.

In accordance with ref. [35], we obtain the overall transfer matrix $\Gamma ={\Gamma }_1{\Gamma }_2$, where ${\Gamma }_{1,2}$ is the transfer matrixes of the PIM (NIM). Each of the transfer matrixes can be expressed by

$${\Gamma }_m=\left[\begin{array}{cc}cos\left(k_{m}L\right)& i\sin \left(k_{m}L\right)·Z_m\\ i\sin \left(k_{m}L\right)/Z_m& \cos \left(k_{m}L\right)\end{array}\right],$$

(2)

where $m=1,2$, and L is the length of the PIM (NIM) region. In this anti-$\mathcal{PT}$-symmetric system, we have $k_{1,2}=k_0{n}_{1,2}$, where $k_0$ is the wave vector in the surrounding medium (air) and $k_{1,2}$ represents the wave vector in the PIM (NIM). Thus, $n_{1,2}=\pm \alpha +i\beta$ is the refractive index of the PIM (NIM), and $\alpha$ and $\beta$ are the real and imaginary parts of the refractive index, which has been shown in Figure 1b. Another prerequisite for anti-$\mathcal{PT}$-symmetric system is ${\rho }_1=-{\rho }_2={\rho }_0$; thus $Z_1={\rho }_0{c}_0/n_1$, $Z_2=-{\rho }_0{c}_0/n_2$, and $Z_{1,2}$ represents the impedance of the PIM (NIM).

From the $ABCD$ matrix of the system, we calculate the scattering matrix S as

$$r_L=a/\Delta ,r_R=b/\Delta ,t=2/\Delta ,$$

(3)

where $\Delta ={\delta }_1+{\delta }_2$, $a={\delta }_4-i{\delta }_3$, $b={\delta }_4+i{\delta }_3$. After straightforward calculations, and assuming $\xi =k_{0}L$, we obtain

$$\begin{array}{cc}\hfill & {\delta }_1=\frac{2}{N}\left[{\beta }^2\cos \left(2\alpha \xi \right)+{\alpha }^2\cosh \left(2\beta \xi \right)\right],\hfill \\ \hfill & {\delta }_2=\frac{1}{N}\left[\left(1-N\right)\beta sin\left(2\alpha \xi \right)-\left(1+N\right)\alpha \sinh \left(2\beta \xi \right)\right],\hfill \\ \hfill & {\delta }_3=-\frac{P}{N}\left[\cos \left(2\alpha \xi \right)-\cosh \left(2\beta \xi \right)\right],\hfill \\ \hfill & {\delta }_4=\frac{1}{N}\left[\left(1+N\right)\beta \sin \left(2\alpha \xi \right)-\left(1-N\right)\alpha \sinh \left(2\beta \xi \right)\right].\hfill \end{array}$$

(4)

Here $N={\alpha }^2+{\beta }^2$; $P=2\alpha \beta$. Since ${\delta }_1$, ${\delta }_2$, ${\delta }_3$, and ${\delta }_4$ are all real quantities, so is $\Delta$. Further, we get

$$r_L^{*}=r_R,\mathrm{Im}\left(t\right)=0.$$

(5)

Another necessary standpoint for exploring the scattering properties of the system is the eigenvalues of the scattering matrix, which could be expressed as

$${\lambda }_{1,2}=\frac{1}{2}\left[(r_L+r_R)\pm \sqrt{{(r_L-r_R)}^2+4t^2}\right]=\frac{{\delta }_4}{\Delta }\pm \frac{\sqrt{4-{\delta }_3^2}}{|\Delta |}.$$

(6)

Each eigenvalue quantitatively describes how the incident waves are scattered, since the ratio between the amplitudes of the scattered and incident waves is $|{\lambda }_{1,2}|$. Equation (3) is the fundamental consequence of this work. It is valid for all 1D acoustic systems with anti-$\mathcal{PT}$ symmetry, and results in spontaneous phase transition and total transmission band.

3. Spontaneous Phase Transition

Figure 2a illustrates the logarithmic amplitudes of the reflection and transmission coefficients of the anti-$\mathcal{PT}$-symmetric system with index $n_{1,2}=\pm 0.8+i0.05$, where a sharp peak located at the pole (namely, $\Delta =0$ and $\xi =21.55$) can be observed for $|r_L|$, $|r_R|$ and $\left|t\right|$. According to Equation (3), $\Delta =0$ causes the reflected and transmitted waves to be trapped for a long time and get amplified extremely by the local gain, finally resulting in the acoustic equivalent lasing, where the transmittance and reflectance both tend to infinity. Not surprisingly, we can find the abrupt change of $\pi$ in phases at the pole in Figure 2b, thus the delay times for all reflection and transmission coefficients show a delta function behavior as the variable $\xi$ is varied. Besides, the phases of two reflections are opposite to each other, and the phase of transmission is either 0 or $\pi$, which was predicted in Equation (5). The peak is also located at $\xi =21.55$ for the curve of eigenvalues, as shown in Figure 2c, and the interchange of the two eigenvalues results from a change in the sign of $\Delta$ from positive to negative (see Equation (6)). The other point of interest is at $\xi =35.33$ (namely, ${\delta }_3=2$), which is referred to as an exceptional point, a spontaneous phase transition occurs for the eigenvalues of the scattering matrix. Generally, when $\xi <35.33$ (${\delta }_3<2$), $|{\lambda }_{1,2}|$ are different, implying that the scattering matrix is in the symmetric phase (white background region). However, when $\xi >35.33$ (${\delta }_3>2$), ${\lambda }_1^{*}={\lambda }_2$, indicating that the scattering matrix is under the symmetry-broken phase (light yellow background region).

If we flip the sign of $\beta$, i.e., changing local gain into loss, the system merely undergoes a time reversal, and the phase transition of the eigenvalues happens at exactly the same location, but not for the scattering amplitudes as well as phases. Figure 2d plots the curves of logarithmic amplitudes with refractive index $n_{1,2}=\pm 0.8-i0.05$. Distinguished from from the case of $\beta >0$, the curves have no longer peaks; that is, there is no pole of the scattering matrix, and correspondingly the abrupt change of scattering phase disappears (see Figure 2e). We can observe a sharp valley in Figure 2f, which is at the same position as the peak of $\xi =21.55$ in Figure 2c. According to Equation (6), this valley corresponds to the condition of ${\lambda }_2=0$; in other words, ${\delta }_3^2+{\delta }_4^2=4$ at this point. In fact, elements in Equation (4) have an intrinsic equivalence relation of

$$({\delta }_1^2+{\delta }_3^2)-({\delta }_2^2-{\delta }_4^2)=4,$$

(7)

and the symmetry of the system before and after $\beta$ changes its sign also comes from this. This means, in the case of $\beta =0.05$, the amplitude of the eigenvalues is the reciprocal of that in the case of $\beta =-0.05$, namely, ${\left|\lambda \right||}_{\beta =0.05}{\left|\lambda \right||}_{\beta =-0.05}=1$. Similarly, the system also undergoes a symmetric phase ($\xi <35.33$, ${\delta }_3>-2$, white background region) and a symmetry-broken phase ($\xi >35.33$, ${\delta }_3<-2$, light yellow background ground). Moreover, the position of the exceptional point is consistent with that of $\beta >0$ (see Figure 2f).

4. Total Transmission Band

The phase symmetry discussed above is a general property of all 1D anti-$\mathcal{PT}$-symmetric systems, independent of whether the system has net gain or loss. Besides, there is one exception, which occurs when the local gain or loss is zero, i.e., $\beta =0$. In this case, an anti-$\mathcal{PT}$-symmetric system is always in the symmetric phase. Another exception not mentioned in previous reports is the case of $\alpha =0$, which can also lead to the symmetric phase of the anti-$\mathcal{PT}$-symmetric system.

More strikingly, relations (4) now become

$${\delta }_1=2,{\delta }_2=0,{\delta }_3=0,{\delta }_4=0,$$

(8)

in these two cases, and we have

$$r_L=r_R=0,t=1,$$

(9)

which is independent of the complexity and size of the system or the frequency of the incident wave. Further explorations show that this phenomenon is robust upon a slight breakdown of the anti-$\mathcal{PT}$ symmetry or in the presence of a small $\beta \ne 0$. A total transmission band then forms if the anti-$\mathcal{PT}$ symmetry can be maintained over a finite-frequency range, and a pulse transmitted within this frequency window will be exactly the same as the initial pulse, with no pulse distortion or shrinkage or expansion. This phenomenon is independent of the propagation direction, in contrast to the one-way invisibility found in the $\mathcal{PT}$-symmetric system [11,13].

As an example, we plot the variation of the amplitudes of the reflection and transmission coefficients for $\alpha =0.8$, $\beta =0$ (Figure 3a) and $\alpha =0$, $\beta =0.05$ (Figure 3d), respectively. A unity transmittance and a zero reflectance can be found in the entire frequency range in these two figures. Furthermore, we demonstrate the bidirectional total transmission with finite element simulations, as shown in Figure 3b,c,e,f. Here, $L=0.4$ m, and $f=1715$ Hz. In both cases of $\beta =0$ and $\alpha =0$, it is found that the pressure amplitudes in the left and right background media are both 1 Pa, which are equal to the incident amplitude. The amplitudes and phases of the transmitted waves are exactly the same as those of the incident wave regardless of incident direction, which verifies the effect of bidirectional total transmission.

When we observe the sound field (both amplitudes and phases) inside the anti-$\mathcal{PT}$-symmetric system under these two cases, something interesting happens. First, in each case, the pressure amplitudes inside the system are symmetric to $x=0$, while the phases are opposite numbers to each other. This means that at the same position, the sound pressures are conjugate to each other, which is also caused by the antisymmetry of the system. Second, in Figure 3b,c, where $\beta =0$, the sound field behaves as a regular wave with a wavelength of $\lambda =c_0/\left(\alpha f\right)=1.25{\lambda }_0$. However, in Figure 3e,f, where $\alpha =0$, the sound field is represented by a region of uniform intensity with infinite wavelength, corresponding to the nearly constant phase distribution inside the anti-$\mathcal{PT}$-symmetric system.

5. Fabry–Perot Resonance Mode

The emergence of active acoustic metamaterials [36] provides a new form of acoustic media; that is, in addition to the distributed parameter materials described above, we can also use active metamaterials to obtain media morphology similar to internal admittance, as achieved in previous reports [11,12]. Of course, through the design of the circuit, the admittance medium can be gain, loss, or even lossless.

We further constructed a Fabry–Perot resonance mode, as shown in Figure 4. According to ref. [35], the transfer matrix of an internal admittance can be expressed by

$$T_a=\left[\begin{array}{cc}1& 0\\ Y_a& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ u/Z_0& 1\end{array}\right],$$

(10)

where $u=Y_a/Y_0$ represents the admittance ratio between the internal medium and air. Thus, we obtain the overall transfer matrix T of the Fabry–Perot resonance mode as

$$T=T_1{T}_0{T}_a=\left[\begin{array}{cc}cos\eta & isin\eta ·Z\\ isin\eta /Z& cos\eta \end{array}\right]\left[\begin{array}{cc}cos\mu & isin\mu ·Z_0\\ isin\mu /Z_0& cos\mu \end{array}\right]\left[\begin{array}{cc}1& 0\\ u/Z_0& 1\end{array}\right].$$

(11)

Here, $T_1$ represents the transfer matrix of the lossless medium (LLM). $\mu =k_{0}d$, and $k_0$ is the wave vector in the air. Meanwhile, $\eta =kL$, and $k=k_0\alpha$ is the wave vector in the lossless medium. Z and $Z_0$ are the impedance of LLM and air, respectively.

From the $ABCD$ matrix of the system, we calculate the scattering matrix $S^{\prime }$, in which

$$r_L^{\prime }=a^{\prime }/{\Delta }^{\prime },r_R^{\prime }=b^{\prime }/{\Delta }^{\prime },t=2/{\Delta }^{\prime }.$$

(12)

Here, ${\Delta }^{\prime }=u{\Theta }_1+(2cos\eta +i{\Lambda }_1)·e^{i\mu }$, $a^{\prime }=-u{\Theta }_2+i{\Lambda }_2·e^{i\mu }$, and $b^{\prime }=-u{\Theta }_1+i{\Lambda }_2·e^{-i\mu }$. We assume $z=Z/Z_0$, and

$$\begin{array}{cc}\hfill & {\Theta }_1=cos\eta ·e^{i\mu }+isin\eta \left(cos\mu ·z+isin\mu /z\right),\hfill \\ \hfill & {\Theta }_2=cos\eta ·e^{-i\mu }-isin\eta \left(cos\mu ·z-isin\mu /z\right),\hfill \\ \hfill & {\Lambda }_1=sin\eta ·\left(z+1/z\right),\hfill \\ \hfill & {\Lambda }_2=sin\eta ·\left(z-1/z\right).\hfill \end{array}$$

(13)

Further calculation shows that when

$$u=i\frac{2cos\eta cos\mu -(z+1/z)sin\eta sin\mu }{cos\eta sin\mu +zsin\eta cos\mu },$$

(14)

we obtain

$${r^{\prime }}_L^{*}=r_R^{\prime },\mathrm{Re}\left(t^{\prime }\right)=0.$$

(15)

This is a quite different but similar and important conclusion that means we have achieved $\pm \pi /2$ phase transition of transmission from the classical anti-$\mathcal{PT}$-symmetric case while ensuring the conjugacy of reflection coefficients from both sides. Figure 5 illustrates the evolution process of the corresponding scattering matrixes of the Fabry–Perot resonance mode as $\eta$ changes from 0 to $\pi$ by fixing $z=2$. It can be seen that the scattering properties of the system are independent of the single refractive index $\alpha$ or length L of the LLM, and depend on the parameter $\eta$, which is jointly determined by $\alpha$ and L. In other words, even if $\alpha$ or L of the LLM changes, the scattering properties will remain unchanged as long as $\eta$ is constant. For fixed $\eta$, the phase of the transmission coefficient can only be $-\pi /2$ or $\pi /2$, and there are phase transitions with the size of $\pi$ at several specific positions. As $\mu$ increases, the phase transition point of the transmission coefficient repeats with a period of $2\pi$, which is completely different from the result in the classical anti-$\mathcal{PT}$-symmetric system, where the phase transition of the transmission coefficient will occur only once ($\beta >0$) or not ($\beta <0$) (see Figure 2).

When $\eta$ comes to 0 (Figure 5a), it means the LLM does not exist. At the point of $\mu =\pi /2$, we achieve the transmission resonance with $r_L^{\prime }=r_R^{\prime }=0$ and $t^{\prime }=-i$. Meanwhile, at the point of $\mu =3\pi /2$, another type of transmission resonance with $r_L^{\prime }=r_R^{\prime }=0$ and $t^{\prime }=i$ appears. On the other hand, a scattering phenomenon similar to the soft boundary occurs at $\mu =\pi$, which means $r_L^{\prime }=r_R^{\prime }=-1$ and $t^{\prime }=0$. At this time, the reflection phase and $\mu$ are strictly linear, which may have important applications in acoustic sensing. With the increase of $\eta$ (Figure 5b,c), the transmission phase transition point is red-shifted, and the phase changes from linear to nonlinear with respect to $\mu$. For $0<\eta <\pi$, the scattering phenomenon of soft boundary still exists, but the abnormal transmission resonance point completely disappears. We also figure out that when $\eta =\pi /2$ (Figure 5c), the soft boundary points fall on $\mu =\pi /2$ and $3\pi /2$, which is equivalent to the sound wave experiencing half or one wavelengths, but the scattering features of a soft boundary are obtained due to the addition of internal admittance. When $\eta$ comes to $\pi$ (Figure 5d), the reflection coefficients and the amplitude of the transmission coefficient remain the same as that of $\eta =0$; however, the phase of the transmission coefficient is reversed. That means we get $\mathrm{Arg}\left(t^{\prime }\right)=\pi /2$ when $0<\mu <\pi$ and $\mathrm{Arg}\left(t^{\prime }\right)=-\pi /2$ when $\pi <\mu <2\pi$. Similar to the situation of $\eta =0$, the reflection phase and $\mu$ are strictly linear.

In order to further capture the distinctive phenomena of the Fabry–Perot resonance mode, we plot their pressure field distributions under several particular parameter combinations. Firstly, we set the structural parameters as $f=1715$ Hz, $L=0.45$ m, $d=0.5$ m, $Z=2Z_0$, and $c=2c_0$, so that $\eta =kL=9\pi /4\sim \pi /4$ and $\mu =k_{0}d=5\pi \sim \pi$ in Figure 5. According to Equations (12)–(14), after straightforward calculations, we get $u=i$, $r_L^{\prime *}=r_R^{\prime }=(1-i4)/7$, and $t^{\prime }=i4\sqrt{2}/7\approx i0.808$. The corresponding pressure fields (upper panels) and the pressure amplitude and phase distributions (lower panels) are shown in Figure 6a,b. A plane wave with amplitude of 1 Pa is incident on the Fabry–Perot resonance mode from the left and right sides, respectively. In both cases, the transmission coefficient can be found as $i0.808$ without real part, which is consistent with the result of theoretical calculation. On the other hand, the conjugate reflection coefficients can also be observed in the figures.

Secondly, we change the structure parameters to $L=0.4$ m and $d=0.45$ m; thus, $\eta =2\pi \sim 0$ and $\mu =9\pi /2\sim \pi /2$. In this case, we get $u=0$, $r_L^{\prime }=r_R^{\prime }=0$, and $t^{\prime }=-i$, as shown in Figure 6c,d. Analogously, if $d=0.55$ m, $\mu$ changes to be $11\pi /2\sim 3\pi /2$, we renewedly obtain $u=0$, $r_L^{\prime }=r_R^{\prime }=0$, and $t^{\prime }=i$ in Figure 6e,f. A noticeable feature is that at these two transmission resonances, the internal admittance becomes 0, which means the internal admittance does not exist (red dotted lines in Figure 6c–f), and the transmission resonance of the mode degenerates into the transmission resonance of a single LLM. However, this does not mean that the internal admittance is useless. In the actual manufacture of corresponding acoustic devices, the presence of internal admittance is necessary to modify the scattering characteristics of the system to achieve the desired target.

6. Conclusions and Discussion

In conclusion, we investigated the generalized relation for the scattering parameters of an anti-$\mathcal{PT}$-symmetric system using the transfer matrix, which is directly related to the experimental parameters of the proposed system. The results show that a 1D anti-$\mathcal{PT}$-symmetric system has intrinsic scattering properties of $r_L^{*}=r_R$, and $\mathrm{Im}\left(t\right)=0$. In addition, the spontaneous phase transition of the scattering matrix has been figured out under the condition of ${\delta }_3=2$ ($\beta >0$) or ${\delta }_3=-2$ ($\beta <0$). Further exploration indicates that once the real or imaginary part of refractive index of the medium degenerated to 0, the system would enter the total transmission band ($r_L=r_R=0$, and $t=1$); however, their intrinsic physical mechanisms are different. Finally, we design a Fabry–Perot resonance mode, having scattering properties (${r^{\prime }}_L^{*}=r_R^{\prime }$, and $\mathrm{Re}\left(t^{\prime }\right)=0$) similar to but more peculiar than the classical anti-$\mathcal{PT}$-symmetric system, and its unitary transmission features change to $r_L^{\prime }=r_R^{\prime }=0$ and $t^{\prime }=\pm i$.

Numerically, this method can also handle the $\mathcal{PT}$-symmetric system (corresponding to the anti-$\mathcal{PT}$-symmetric system) or non-Hermitian system without $\mathcal{PT}$ symmetry (corresponding to the Fabry–Perot resonance mode). Thanks to the active acoustic metamaterials, PIM and NIM in the anti-$\mathcal{PT}$-symmetric system can be realized by designing suitable circuits [11,12], and the internal admittance in the Fabry–Perot resonance mode can be demonstrated through the thermoacoustic effect [37]. Our method can be applied to different fields such as electronics, optics, microwaves and elastic metamaterials. Especially in recent years, preliminary studies have been made on $\mathcal{PT}$ symmetry in the elastic wave system [38,39,40]. As a counterpoint, anti-$\mathcal{PT}$ symmetry may be a new research hotspot in this field. In view of the correlation between elastic and sound waves, elastic metamaterials may also provide potential ideas for experimental realization of acoustic anti-$\mathcal{PT}$ symmetry, and vice versa. It is also noted that recent progress in time-varient metamaterials [41] offers a new view of the Fabry–Perot resonance mode, in which the internal admittance varies according to time. This design ability over the anti-$\mathcal{PT}$-symmetric system and related Fabry–Perot resonance mode opens a new avenue for developing asymmetric wave transport devices, important for directional imaging, sensing, logic devices and other applications.