## 3 Bidirectional Negative Refraction  
In the previous section, we showed that the coexistence of CPA mode and laser mode can be achieved using a PIM slab with specific acoustic parameters. When the thickness and acoustic material parameters of the PIM slab satisfy Equation (3), even laser mode and odd CPA mode can be supported; when the thickness and acoustic material parameters of the PIM slab satisfy Equation (4), even CPA mode and odd laser mode can be supported. The coexistence of modes provides the possibility of negative refraction with a pair of identical PIM slabs. As sketched in Figure 3a, the structure consists of two PIM slabs separated by a distance \(L\) in air. The relationship between the transmission and reflection coefficients of the system with the incident angle \(\theta\) and spacing \(L\) can be obtained by calculating the elements of the scatting matrix of the system [28]. When the acoustic material parameters of the PIM slab satisfy Equation (3) or Equation (4), the calculated results are shown in Figure 3b,c, respectively. It can be found that in this matched case, the amplitude of reflectance is in the order of \(10^{-15}\) and the logarithm of the amplitude of the transmittance is in the order of \(10^{-16}\). Therefore, the reflection coefficient is almost zero, while the transmission coefficient is very close to the unity. Both are essentially independent of the incident angle and the spacing. Therefore, omnidirectional negative refraction can be achieved and is independent of the spacing between the two PIM slabs. Numerical simulations were performed using the finite element method to verify the theoretical analysis results. As shown in Figure 4, two identical PIM slabs with width \(2\lambda\) are immersed in the air and the four black vertical lines indicate the outer boundaries of the left and right PIM slabs, respectively. The incidence angle is arbitrarily taken as \(30^{\circ}\). The black arrows indicate the direction of the energy flow in each region. Figure 3: (**a**) Schematic diagram of negative refraction based on two identical PIM slabs. (**b**)The magnitude of the transmission coefficient and (**c**) the logarithm of the amplitude of the transmittance coefficient as a function of the angle of incidence \(\theta\) and the spacing \(L\) between the two slabs. In Figure 4a,b, the density and the bulk compressibility of the PIM slab are \(\rho_{r}=0.14\) and \(\beta_{r}=1.84\), respectively. The material parameters are calculated using Equation (3) with m taken as zero, which supports the coexistence of even laser mode and odd CPA mode. While in Figure 4c,d, the density \(\rho_{r}=1.01\) and the bulk compressibility \(\beta_{r}=1.01\) are calculated using Equation (4) with m taken as two and the coexistence of odd laser mode and even CPA mode are supported. The simulation results in Figure 4 show that in all four cases, the amplitude of total acoustic pressure field in the background medium is equal to that of the incident field (see color bar) and there is no significant scattering. When the acoustic wave is incident from the left shown by the blue arrows, it can be seen from Figure 4a,c that a backward energy flow occurs in the air between the two slabs, forming a negative refraction. The backward wave in the air gap forms a CPA mode with the incident wave on the first PIM slab and a laser mode with the transmitted wave on the second PIM slab. Similarly, as shown in Figure 4b,d, the incident acoustic wave from the right side indicated by the blue arrows also generates a reverse energy flow in the air between the two slabs, which together with the incident acoustic wave constitutes the CPA mode and together with the transmitted acoustic wave constitutes the laser mode. The discussion of mode polarity in the PIM slab is the same as in the previous section.