# Acoustic Negative Refraction and Planar Focusing Based on Purely Imaginary Metamaterials
## 3 Acoustic Laser Mode and CPA Mode  
Since the material of PIM contains both gain and loss elements, some special wave modes, including acoustic laser and CPA, may appear depending on which element dominates. In the following, we explore the excitation conditions and mode characteristics of the CPA and laser modes of acoustic PIM by analyzing the scattering coefficients. For an acoustic laser oscillator, the incident wave might induce quite intense transmission and reflection so strongly that the injected wave can be ignored. This means that the reflection and transmission coefficients tend to infinity. Thus, the condition for the acoustic laser mode can be obtained by setting the denominator of Equation (3) to zero, i.e., \((\eta+1)^{2}e^{i\phi}-(\eta-1)^{2}e^{-i\phi}=0\). Thus, the corresponding dispersion relations can be solved as  
\[\left\{ \begin{array}{ll}\eta&=-i\tan\frac{k_{x}{{}^{\prime}}}{2}d\;(\text{ even mode})\\ \eta&=i\cot\frac{k_{x}{{}^{\prime}}}{2}d\;(\text{odd mode})\end{array} \right. \tag{4}\]  
In contrast, for a perfect absorber, the in-phase coherent waves incident from both sides could be fully absorbed without any scattered waves. The boundary condition can be expressed as \(p_{bl}=p_{fr}=0\). From Equation (2), we obtain \(r\cdot p_{fl}+t\cdot p_{br}=0\) and \(t\cdot p_{fl}+r\cdot p_{br}=0\). This requires either \(p_{fl}=p_{br}\) (even mode) and finding the condition for \(r+t=0\), or \(p_{fl}=-p_{br}\) (odd mode) and finding the condition for \(r-t=0\). Therefore, the condition for the CPA mode can be calculated as  
\[\left\{ \begin{array}{ll}\eta&=i\tan\frac{k_{x}{{}^{\prime}}d}{2},\;(\text{ even mode})\\ \eta&=-i\cot\frac{k_{x}{{}^{\prime}}}{2}d,(\text{odd mode})\end{array} \right. \tag{5}\]  
The parity of the mode indicates whether the field distribution is symmetric or anti-symmetric in the x-direction around the center of the PIM layer. For a PIM slab, the scattering relations can be satisfied for both \(\beta_{r}=\rho_{r}\) and \(\beta_{r}\
eq\rho_{r}\). Based on Equation (4), the dispersion relationships between the wave vector \(k_{y}\) and the refractive index \(n\) of the acoustic laser modes is shown in Figure 2a, where the red solid and dashed curves correspond to the even and odd modes, respectively. Take \(n=\beta_{r}=\rho_{r}=1.2\) as an example, the wave vectors in the \(y\) direction wave vector for the even acoustic laser mode and odd acoustic laser mode are \(0.42k_{0}\) and \(0.84k_{0}\), respectively. To match the tangential momentum, the even and odd acoustic laser mode approximately to occur at the incident angles of \(25.12^{0}\) and \(57.58^{0}\), respectively. To verify the analytical results, numerical simulations are performed using COMSOL Multiphysics software. A plane wave radiation boundary is imposed on the incident and transmitted boundaries, and a Floquet periodic boundary condition is applied in the y direction. The width of the PIM slab is set to \(2\lambda\). Because a periodic boundary condition is set in the y-direction, the height of the PIM slab does not affect the acoustic field distribution, which is set to \(4\lambda\) in the following simulations. For plane acoustic waves with an amplitude of \(1.0\) Pa incident from the left side with the corresponding incident angles, the results are displayed in Figure 2b,c. By observing the field patterns and the direction of the energy flow (black arrows) in Figure 2b,c, it can be found that quite intense outgoing waves are generated on both sides of the PIM slabs. In both cases, the total acoustic pressure fields are very strong with respect to the incidence, although the amplification is different (see color bars). The incoming acoustic waves oscillate within the slab to accumulate energy, and the additional energy is provided by the gain portion of the PIM, which is similar to the mechanism of the laser mode in a cavity system with a gain medium.