# Acoustic Negative Refraction and Planar Focusing Based on Purely Imaginary Metamaterials
## 4 Negative Refraction and Plane Focusing  
It is worth pointing out that Equations (4) and (5) have the same form but with opposite signs. Therefore, if the conjugate operation is performed on the constitutive parameters, the acoustic laser mode can be converted to CPA mode and vice versa. In other words, if we obtain the CPA (acoustic laser) mode in a PIM slab with parameters \(\beta_{s}=i\beta_{r}\beta_{0}\) and \(\rho=-i\rho_{r}\rho_{0}\), then the acoustic laser (CPA) mode will be realized in a PIM* slab by conjugating the parameters, such that \(\beta_{s}=-i\beta_{r}\rho_{0}\), \(\rho=i\rho_{r}\rho_{0}\). The coexistence of CPA and acoustic laser modes offers the possibility of negative refraction.  
As sketched in Figure 4, the structure devised consists of three blocks, in which the PIM slab and PIM* slab, both with a thickness of \(d\), are separated by an air region of a width of \(L\). The left PIM slab is designed to operate as a CPA, while the right PIM* slab with conjugate parameters is operated as an acoustic laser.  
The thickness \(d\) of both the PIM slab is \(0.3\lambda\), and the length of the air gap \(L\) is \(12\lambda\). Based on the above theoretical derivation (see Appendix B), the absolute values of refraction coefficient \(|r|\) and transmission coefficient \(|t|\) (in dB) are related to the incident angle \(\theta\) and the refraction index \(n\), are shown in Figure 5a,b. It can be seen that there is a dark blue curve extending from coordinate [0, 0.8] to coordinate [90, 1.2] in Figure 5a. On this curve, the reflection coefficient is almost zero, while the corresponding transmission is approximately one, as shown in Figure 5b. To verify whether such a response is a negative refraction or a resonance phenomenon, the scattering coefficients of the PIM* are calculated as shown in Figure 5c.d. It can be seen that there is a dark red curve with the same shape. Since the absolute values of the refraction coefficients \(|r|\) and transmission coefficients \(|t|\) tend to infinity (refer to the color bar), this curve corresponds to acoustic laser modes of the PIM* slab. As can been seen from the black dotted and dash lines in Figure 6a, the corresponding overall transmission coefficients \(|t|\) tend to unity, while the reflection coefficients \(|r|\) are almost zero. Thus, such a response is consistent with negative refraction in the PIM-air-PIM* structure, with one side working as a coherent power sink and the other as a power amplifier, while the overall transmission is unitary.  
Figure 4: Schematic graph of the PIM-air-PIM* structure.  
Figure 5: The absolute amplitude of the reflection \(|r|\) (**a**) and transmission coefficient \(|t|\) (**b**) as a function of the variables \(\theta\) and refractive index \(n\) of the PIM-air-PIM* structure. The absolute amplitude of the reflection \(|r|\) (**c**) and transmission coefficient \(|t|\) (**d**) as a function of the variables \(\theta\) and refractive index \(n\) of the PIM* slab.  
Figure 6: (**a**) The absolute value of reflection \(|r|\), transmission \(|t|\), and the refractive index \(|n|\) for the co-existence of CPA and Laser mode as a function of the incident angle \(\theta\). (**b–d**) The pressure field distribution at incident angles of \(\theta=0^{0}\), \(\theta=30^{0}\) and \(\theta=55^{0}\), respectively.