## 2 Derivation of Coexistence Conditions for CPA and Laser Modes in a Single Acoustic PIM Slab  
Figure 1 illustrates the structure under study. A PIM slab of thickness \(d\) is sandwiched between two semi-infinite background layers. Without loss of generality, the background material is set to be air with a mass density of \(\rho_{0}=1.21\,\mathrm{kg/m^{3}}\) and a velocity of \(c_{0}~{}=~{}343\) m/s. In terms of PIM material parameters, both bulk compressibility \(\beta_{s}=i\rho_{r}\beta_{0}\) and mass density \(\rho=-i\rho_{r}\rho_{0}\) are purely imaginary numbers with conjugate phase factors, where \(\beta_{0}\) and \(\rho_{0}\) are the corresponding parameters of air, and \(\beta_{r}\) and \(\rho_{r}\) are relative parameters which are real numbers. It can be found that the speed of sound \(c=\pm\sqrt{\left(\rho\beta_{s}\right)^{-1}}\) and the refractive index \(n=\pm\sqrt{\rho\beta_{s}/\rho_{0}\beta_{0}}\) are real numbers. The blue rays indicate the incident waves, while the black and red rays indicate the scattered waves in the PIM slab and the background medium, respectively. Due to the opposite sign of the imaginary part of the bulk compressibility and mass density, the material of PIM contains both gain and loss elements. If the gain element dominates, the incident wave may excite the transmitted and reflected waves so strongly that the injected waves can be ignored. This corresponds to the acoustic laser mode, as shown in Figure 1a. In contrast, if the loss element dominates, the coherent waves incident from both sides can be completely absorbed by the PIM slab without any scattered waves. This is equivalent to the CPA mode, as shown in Figure 1b. By analyzing the scattering characteristics of the PIM slab with thickness of \(d\) in air, combined with the boundary conditions, the dispersion relation of the acoustic laser mode can be ascribed as  
\[\left(\frac{1-\eta}{1+\eta}\right)^{2}=e^{i2k_{x}{{}^{\prime}}d}. \tag{1}\]  
Correspondingly, the dispersion relation for the acoustic CPA mode is  
\[\left(\frac{1-\eta}{1+\eta}\right)^{2}=e^{-\eta 2k_{x}{{}^{\prime}}d}. \tag{2}\]  
Here, \(k_{0}=2\pi/\lambda\) is the wave vector in air, \(\eta=\rho k_{x}/\rho_{0}k_{x}{{}^{\prime}}\) is the relative impedance of the PIM-air interface, and \(k_{x}=\sqrt{k_{0}^{2}-k_{y}^{2}}\), \(k_{x}{{}^{\prime}}=\sqrt{n^{2}k_{0}^{2}-k_{y}{{}^{2}}}\) are the \(x\) directional wave vector in the air and PIM media, respectively. (For detailed derivation, see Appendix A). By comparing Equations (1) and (2), it can be found that in general, the laser mode and the CPA mode appear in different situations. That is, for given material parameters, the laser mode and the CPA mode have different wave vectors. However, when \(\left(\frac{1-\eta}{1+\eta}\right)^{2}=e^{i2k_{x}{{}^{\prime}}d}=e^{-i2k_{x}{{} ^{\prime}}d}\), the CPA mode and the laser mode can be satisfied simultaneously. After further solving, we can get two sets of analytical solutions, which are  
\[\left\{ \begin{array}{l}\rho_{r}=\sqrt{\frac{\frac{\pi^{2}}{k_{0}^{2}d^{2}} \left(2m+\frac{1}{2}\right)^{2}}{(1-\sin^{2}\theta)}}=\frac{\pi}{k_{0}d}\times (2m+\frac{1}{2})/\cos\theta\\ \beta_{r}=[\frac{\pi^{2}}{k_{0}^{2}d^{2}}(2m+\frac{1}{2})^{2}+\sin^{2}\theta]/ [\frac{\pi}{k_{0}d}\times(2m+\frac{1}{2})/\cos\theta]\end{array} \right. \tag{3}\]  
and  
\[\left\{ \begin{array}{l}\rho_{r}=\sqrt{\frac{\pi^{2}}{k_{0}^{2}d^{2}}\left(2m -\frac{1}{2}\right)^{2}}{(1-\sin^{2}\theta)}=\frac{\pi}{k_{0}d}\times(2m- \frac{1}{2})/\cos\theta\\ \beta_{r}=[\frac{\pi^{2}}{k_{0}^{2}d^{2}}(2m-\frac{1}{2})^{2}+\sin^{2}\theta] /[\frac{\pi}{k_{0}d}\times(2m-\frac{1}{2})/\cos\theta]\end{array} \right.. \tag{4}\]  
Here, \(m\) is an arbitrary integer related to the mode order. (For detailed derivation, see Appendix B). Based on the above analytical results, further numerical simulations of the coexistence of the laser mode and the CPA mode were carried out using COMSOL Multiphysics software (version 4.2).