# Acoustic Equivalent Lasing and Coherent Perfect Absorption Based on a Conjugate Metamaterial Sphere
## 6 AEL and CPA for Multiple Orders Based on a Small Multilayered Sphere  
Based on a small monolayer ACM sphere, the AEL or CPA modes are realized, but it can only amplify or absorb spherical wave with a single order \(l\). A monolayer ACM sphere with a radius much larger than the wavelength could simultaneously realize even (or odd) order AEL modes and odd (or even) order CPA modes, but the bulky geometry may limit its potential applications. In order to manipulate spherical waves with multiple channel orders simultaneously, a multilayered ACM sphere is considered because it contains material parameters with more degrees of freedom. Generally, a multilayered structure consisting of \(Q\) layers could simultaneously manipulate spherical waves with total \(Q\) orders [46]. Here, a three-layered ACM sphere (\(Q=3\)) is proposed as an example, where the outer radius of each layer is \(R_{1}=0.35\) m, \(R_{2}=0.25\) m, and \(R_{3}=0.15\) m, the incident wavelength is \(\lambda_{0}=1\) m, and the refractive indexes of three layers \(\left|n_{q}\right|\) are all set to 2 for simplicity, unless otherwise stated. Then, the scattering coefficient \(s_{0}^{l}\) can be obtained by iterating Equations (6) and (8). We first aim to realize the AEL mode for order \(l=1\) and the CPA modes for orders \(l=0\) and 2, which requires \(s_{0}^{0}=0\), \(s_{0}^{1}\rightarrow\infty\), and \(s_{0}^{2}=0\), simultaneously. By numerically solving these three equations, one can finally find the required acoustic parameters of the three-layered ACM sphere as \(\rho_{1}=(1.5075+0.0089i)\rho_{0}\), \(\rho_{2}=(-0.1038+0.855i)\rho_{0}\), and \(\rho_{3}=(-0.0537+0.2748i)\rho_{0}\), respectively. For a three-layered ACM sphere with such acoustic parameters, Figure 6a-c show the simulated total pressure fields in the \(yz\)-plane where spherical waves with orders \(l=0\), 1 and 2, respectively, are incident on the sphere. For \(l=0\) (see Figure 6a) and \(l=2\) (see Figure 6c), the values of the total fields at the outermost simulation boundary \(r=\lambda_{0}\) are close to the corresponding incident pressure \(P_{l}(\cos\theta)\), and no obvious interference pattern is observed in the background medium, indicating the realization of CPA modes for orders \(l=0\) and 2. In contrast, for \(l=1\) (see Figure 6b), the total acoustic field in the air is far beyond the incidence field (see color bar), verifying the realization of the AEL mode for order \(l=1\). In addition, the multilayered structure could be designed to manipulate spherical waves with multiple orders at different frequencies \(f\). For the second example, a three-layered ACM sphere is designed to realize a CPA mode with order \(l=0\) at 300 Hz, an AEL mode at 343 Hz for \(l=1\), and a CPA mode at 400 Hz for \(l=2\). By solving \(s_{0}^{0}=0\) at 300 Hz, \(s_{0}^{1}\rightarrow\infty\) at 343 Hz, and \(s_{0}^{2}=0\) at 400 Hz simultaneously, it can be obtained that \(\rho_{1}=(1.0810-1.7525i)\rho_{0}\), \(\rho_{2}=(0.1611-2.5259i)\rho_{0}\), and \(\rho_{3}=(-0.1425+0.2342i)\rho_{0}\), respectively. The corresponding simulated results are shown in the upper panels of Figure 7a-c. For \(l=0\) (see Figure 7a), the acoustic field is independent of \(\theta\), while for \(l=2\) (see Figure 7c), the acoustic intensity is strong along the \(y\) and \(z\)-axes. In both cases, the values at the outermost simulation boundary \(r=1\) m are very close to the corresponding setting incidence pressure \(P_{l}(\cos\theta)\), and no obvious interference pattern is observed. Therefore, the CPA modes are realized for two orders at two frequencies. For \(l=1\) (see Figure 7b), the acoustic intensity is maximal along the \(z\)-axis and minimal along the \(y\)-axis. In addition, the total field far exceeds the incidence, i.e., the AEL mode occurs. Figure 6: Simulated total acoustic pressure fields of the AEL and CPA modes in the \(yz\)-plane for spherical wave incidence with different orders \(l\) based on a three-layered ACM sphere: (**a**) \(l=0\); (**b**) \(l=1\); (**c**) \(l=2\).