# Acoustic Equivalent Lasing and Coherent Perfect Absorption Based on a Conjugate Metamaterial Sphere
## 3 AEL Based on a Small Monolayer Sphere  
Let us start with the simplest case of a small monolayer ACM sphere, i.e., \(Q=1\). Then, the scattering coefficient of the monolayer ACM sphere can be obtained by taking \(Q=1\) into Equation (8), i.e.,  
\[s_{0}^{l}=\frac{j_{l}^{\prime}(k_{1}R_{1})h_{l}^{(2)}(k_{0}R_{1})-\eta_{1}j_{l }(k_{1}R_{1})h_{l}^{(2)^{\prime}}(k_{0}R_{1})}{\eta_{1}j_{l}(k_{1}R_{1})h_{l}^ {(1)^{\prime}}(k_{0}R_{1})-j_{l}^{\prime}(k_{1}R_{1})h_{l}^{(1)}(k_{0}R_{1})}. \tag{9}\]  
The AEL mode corresponds to a divergent eigenvalue of the scattering matrix, namely, a divergence scattering coefficient \(s_{0}^{l}\) for the proposed system. Therefore, the condition for the AEL mode, which could generate a strong spherical wave with the order \(l\), can be concluded from the zero of the denominator of Equation (9), namely  
\[\frac{h_{l}^{(1)}(k_{0}R_{1})}{h_{l}^{(1)^{\prime}}(k_{0}R_{1})}=\exp(i\alpha_ {1})\frac{|\rho_{1}|}{|n_{1}|\rho_{0}}\frac{j_{l}(|n_{1}|k_{0}R_{1})}{j_{l}^{ \prime}(|n_{1}|k_{0}R_{1})}. \tag{10}\]  
For simplicity, we define \(A_{l}(|n_{1}|)=j_{l}(|n_{1}|k_{0}R_{1})/j_{l}^{\prime}(|n_{1}|k_{0}R_{1})\) and \(B_{l}=h_{l}^{(1)}(k_{0}R_{1})/\)\(h_{l}^{(1)^{\prime}}(k_{0}R_{1})\), so Equation (10) can be simplified to  
\[B_{l}=\exp(i\alpha_{1})|\rho_{1}|A_{l}(|n_{1}|)/(\rho_{0}|n_{1}|). \tag{11}\]  
The function \(A_{l}\) is the ratio of the spherical Bessel function of the first kind to its derivative and therefore is real, while \(B_{l}\) is the ratio of the spherical Hankel function of the first kind to its derivative, which is complex. Thus, Equation (11) can be further expressed as  
\[\exp(i\phi_{l})\xi_{l}=\exp(i\alpha_{1})|\rho_{1}|A_{l}(|n_{1}|)/(\rho_{0}|n_ {1}|), \tag{12}\]  
where \(\xi_{l}\) and \(\phi_{l}\) are the amplitude and phase of \(B_{l}\), respectively. There are two kinds of solutions for Equation (12), denoted as:  
\[\alpha_{1}=\phi_{l},\ A_{l}(|n_{1}|)=\rho_{0}|n_{1}|\xi_{l}/|\rho_{1}|, \tag{13}\]  
\[\alpha_{1}=\phi_{l}+\pi,\ A_{l}(|n_{1}|)=-\rho_{0}|n_{1}|\xi_{l}/|\rho_{1}|. \tag{14}\]  
If the amplitude of the mass density \(|\rho_{1}|=|n_{1}|\rho_{0}\) is taken for the ACM sphere, the acoustic parameters of the ACM sphere are simply determined by the phase angle \(\alpha_{1}\) and \(n_{1}\), and Equations (13) and (14) can be rewritten as  
\[\alpha_{1}^{\rm L}=\phi_{l},\ A_{l}\Big{(}\Big{|}n_{1}^{\rm L}\Big{|}\Big{)}= \xi_{l}, \tag{15}\]  
\[\alpha_{1}^{\rm L}=\phi_{l}+\pi,\ A_{l}\Big{(}\Big{|}n_{1}^{\rm L}\Big{|}\Big{)} =-\xi_{l}. \tag{16}\]  
Here, \(\alpha_{1}^{\rm L}\) and \(n_{1}^{\rm L}\) satisfying Equations (15) and (16) represent the particular phase angle and refractive index of the monolayer ACM sphere, respectively, leading to the AEL mode,where the ACM sphere could generate a strong spherical wave with the order \(l\) at a given working frequency. Figure 2a,b show the phase \(\phi_{l}\) and amplitude \(\xi_{l}\) of \(B_{l}\) (see Equation (12)) versus \(R_{1}\), in which black solid, red dashed, and blue dotted curves correspond to \(l=0\), \(1\), and \(2\), respectively. For \(R_{1}=0.35\lambda_{0}\), \(\phi_{0}=-0.6358\pi\), \(\phi_{1}=-0.6818\pi\), and \(\phi_{2}=-0.8170\pi\) (see Figure 2a); \(\xi_{0}=0.9103\), \(\xi_{1}=1.0151\), and \(\xi_{2}=1.0901\) (see Figure 2b). Therefore, some values of \(a_{1}^{\rm L}\) and \(n_{1}^{\rm L}\) can be found to realize the AEL mode when \(R_{1}=0.35\lambda_{0}\). Taking \(l=0\) as an example, Figure 2c shows the \(A_{l}\) for \(l=0\) versus the refractive index \(|n_{1}|\). The values of \(\left|n_{1}^{\rm L}\right|\) that lead to the AEL mode for \(l=0\) can be found to search the intersections of \(\xi_{0}\) (or \(-\xi_{0}\)) (black dashed lines) and \(A_{0}\) (black solid curves). For example, for \(l=0\), \(a_{1}^{\rm L}=-0.6358\pi\) and \(\left|n_{1}^{\rm L}\right|=1.7163\) satisfy Equation (15); \(a_{1}^{\rm L}=0.3642\pi\) and \(\left|n_{1}^{\rm L}\right|=0.9698\) satisfy Equation (16).