## 2 Derivation of Coexistence Conditions for CPA and Laser Modes in a Single Acoustic PIM Slab
Without loss of generality, the width of the PIM slab is set to \(2\lambda\) and the black vertical lines indicate the outer boundary of the PIM slab. As a periodic boundary condition is set in the y-direction, the height of the PIM slab does not affect the  
Figure 1: Sketch of the scattering process of (**a**) laser and (**b**) CPA modes in PIM slabs. The blue rays indicate the incident waves, while the black and red rays indicate the scattered waves in the PIM slab and background medium, respectively. acoustic field distribution, which is set to \(4\lambda\) in the following simulations. A plane wave radiation boundary is imposed on the incident and transmitted boundaries. The incident angle is set to be \(20^{\circ}\) and the black arrows in Figure 2 indicate the direction of energy flow in each region. According to Equation (3), when the m is taken as zero, the density obtained is \(0.13\) and the bulk compressibility is \(1.00\). The left panel of Figure 2a shows the total acoustic field pattern when a plane wave is incident from the left side at an incident angle of \(20^{\circ}\), as shown by the red arrow. The total field far exceeds the intensity of the incident field (refer to the color bar) and the distribution is symmetric about the line \(x=0\). Thus, this is consistent with the distribution pattern of the even acoustic laser mode. The right panel of Figure 2a gives the acoustic field distribution when two waves of opposite phase are incident from the left and right sides with the identical angle indicated by the red and yellow arrows, respectively. The absence of any acoustic scattering in the background media indicates that the coherent incident waves are completely absorbed in the PIM slab structure by the effect of interference as well as material absorption. Additionally, the results show that the amplitude distribution is asymmetric with respect to the line \(x=0\). Therefore, an odd CPA mode can also be excited with the same structure and material parameters. Similarly, when taking m as one in Equation (4), we can obtain a density of \(0.40\) and a bulk compressibility of \(0.65\). The two panels of Figure 2b show the total acoustic pressure field when one wave is incident from the left and when two waves are incident in phase from both sides, respectively. By observing the field patterns and the direction of the energy flow indicated by the black arrows, it could be found that odd laser modes and even CPA modes can be excited. To investigate the role played by the parameter \(m\) in  
Figure 2: The simulated acoustic fields for laser modes (**left** panels) and CPA modes (**right** panels). The parameters in (**a**) are \(\rho_{r}=0.13\) and \(\beta_{r}=1.00\), in (**b**) are \(\rho_{r}=0.40\) and \(\beta_{r}=0.65\), and in (**c**) are \(\rho_{r}=1.99\) and \(\beta_{r}=1.82\). In all simulations, the incident wavelength is \(\lambda\), the width of the PIM slab is \(d=2\lambda\), the incident amplitude is \(1\) Pa, and the incident angle is set to be \(20^{\circ}\). the equations, the other terms are kept the same and the parameter \(m\) in Equation (4) is changed from one to four. The coefficients obtained are \(\rho_{r}=1.99\) and \(\beta_{r}=1.82\). Similar to the cases of Figure 2b, the odd laser mode and the even CPA mode are successfully excited as shown in Figure 2c. The difference is that the mode of the PIM slab in Figure 2c is three orders higher due to the change of m from one to four. The simulation results in Figure 2 confirm that even laser modes and odd CPA modes are supported when the acoustic parameters conform to Equation (3), while odd laser modes and even CPA modes can be achieved when the acoustic parameters conform to Equation (4). The integer \(m\) determines the order of the modes in the PIM slab and can be used to adjust the material parameters to a range suitable for experimental implementation.