# Acoustic Equivalent Lasing and Coherent Perfect Absorption Based on a Conjugate Metamaterial Sphere
## 4 CPA Based on a Small Monolayer Sphere  
In contrast to the AEL mode, the CPA mode corresponds to a vanishing eigenvalue of the scattering matrix, namely, a vanishing scattering coefficient \(s_{0}^{l}\) for the proposed system. Then, the incident spherical wave with the order \(l\) could be fully absorbed without any scattered waves. Therefore, the condition for the CPA mode can be concluded from the zero of the numerator of Equation (9), namely  
\[\frac{h_{l}^{(2)}(k_{0}R_{1})}{h_{l}^{(2)\prime}(k_{0}R_{1})}=\exp(i\alpha_{1}) \frac{|\rho_{1}|}{|n_{1}|\rho_{0}}\frac{\dot{j}_{l}(|n_{1}|k_{0}R_{1})}{\dot{j}_ {l}^{\prime}(|n_{1}|k_{0}R_{1})}. \tag{17}\]  
By numerically solving Equation (17), based on a similar procedure for solving Equation (10), one can obtain a series phase angle (\(\alpha_{1}^{\rm C}\)) and refractive index (\(n_{1}^{\rm C}\)) solutions for realizing the CPA mode with the desired order \(l\) at a given working frequency. It is worth pointing out that Equations (10) and (17) have the same form but different kinds of spherical Hankel functions. Due to the complex conjugate nature between the first and second kinds of spherical Hankel functions, for the same radius \(R_{1}\), the solutions (\(\alpha_{1}^{\rm C}\), \(n_{1}^{\rm C}\)) of Equation (17) have a special relationship with that (\(\alpha_{1}^{\rm L}\), \(n_{1}^{\rm L}\)) of Equation (10), namely  
\[\alpha_{1}^{\rm C}=-\alpha_{1}^{\rm L},\ \left|n_{1}^{\rm C}\right|=\left|n_{1}^{ \rm L}\right|. \tag{18}\]  
On the other hand, the CPA mode could be regarded as a time reversal counterpart of the lasing [40; 41; 42; 43; 44; 45]. Therefore, an AEL mode always corresponds to a CPA mode where the acoustic constitutive parameters of the two modes are complex conjugates. For the proposed ACM sphere, the complex conjugate acoustic parameters directly lead to Equation (18). In the following, instead of directly solving Equation (17), we employ the particular parameters (\(\alpha_{1}^{\rm L}\), \(n_{1}^{\rm L}\)) for achieving the AEL (see previous section), together with Equation (18) to obtain the parameter solutions (\(\alpha_{1}^{\rm C}\), \(n_{1}^{\rm C}\)) for achieving the CPA. Figure 4 shows the simulated total acoustic pressure fields in the \(yz\)-plane where spherical waves with orders \(l=0\), \(1\), and \(2\) are incident on ACM spheres with particular acoustic parameters, respectively. The radii (\(R_{1}\)) for all ACM spheres are \(0.35\lambda_{0}\), with \(\lambda_{0}=1\) m as the incidence wavelength, and the distribution of the incident pressure field at the outer simulation boundary (\(r=\lambda_{0}\)) is set as \(P_{l}(\cos\theta)\). In the case of \(l=0\) (Figure 4a), \(\alpha_{1}^{\rm C}=0.6358\pi\) and \(\left|n_{1}^{\rm C}\right|=1.7163\) are taken to realize the CPA mode. Similarly, for \(l=1\) (Figure 4b), \(\alpha_{1}^{\rm C}=0.6818\pi\) and \(\left|n_{1}^{\rm C}\right|=2.3674\); for \(l=2\) (Figure 4c), \(\alpha_{1}^{\rm C}=0.8170\pi\) and \(\left|n_{1}^{\rm C}\right|=2.9715\). In other words, all the simulation parameters in Figures 3 and 4 are the same, except that the acoustic parameters of the ACM spheres in Figure 4 are complex conjugates with those in Figure 3.  
In contrast to the AEL modes (Figure 3), the acoustic fields in Figure 4 are weak (see color bars). Quantitatively, the values of the total fields at the outer simulation boundary are very close to \(P_{l}(\cos\theta)\), respectively, which are equal to the incident pressure. Meanwhile, when \(l=0\), the acoustic field presents the shape of concentric circles. When \(l=1\) (2), the acoustic field is symmetric about the \(z\)-axis (\(y\)-axis and \(z\)-axis). The field profiles outside the spheres maintain the shapes of 0 th-order, 1 st-order, and 2 nd-order spherical waves, respectively, and no obvious interference pattern is observed. The incident spherical waves with desired order \(l\) are thus totally absorbed without any scattering.