# Acoustic Equivalent Lasing and Coherent Perfect Absorption Based on a Conjugate Metamaterial Sphere
## 5 Simultaneous AEL and CPA Based on a Huge Monolayer Sphere  
In the previous sections, it was demonstrated that by fine modulating the acoustic and geometric parameters of a small monolayer ACM sphere, an AEL or CPA mode with order \(l\) could be realized at a working frequency. In this section, we will show that a huge monolayer ACM sphere can support the AEL and CPA modes simultaneously. When \(R_{1}\) is far beyond the incidence wavelength \(\lambda_{0}\), i.e., \(R_{1}\gg\lambda_{0}\), the phases \(\phi_{l}\) of \(B_{l}\) for all orders \(l\) approach \(-\pi/2\) (see Figure 2a), and the amplitudes \(\xi_{l}\) of \(B_{l}\) approach 1 (see Figure 2b). Therefore, when \(R_{1}\gg\lambda_{0}\), together with \(|\rho_{1}|=|n_{1}|\rho_{0}\), the condition to realize the AEL mode (see Equations (15) and (16)) can be written as  
\[\alpha_{1}^{\rm L}=-\pi/2,\ A_{l}\Big{(}\Big{|}n_{1}^{\rm L}\Big{|}\Big{)}=1, \tag{19}\]  
\[\alpha_{1}^{\rm L}=\pi/2,\ A_{l}\Big{(}\Big{|}n_{1}^{\rm L}\Big{|}\Big{)}=-1. \tag{20}\]  
Furthermore, when \(R_{1}\gg\lambda_{0}\), the function \(A_{l}(|n_{1}|)\) will approach  
\[A_{l}(|n_{1}|)\rightarrow\tan(|n_{1}|k_{0}R_{1})\ (\text{for even $l$}), \tag{21}\]  
\[A_{l}(|n_{1}|)\rightarrow-\cot(|n_{1}|k_{0}R_{1})\ (\text{for odd $l$}). \tag{22}\]  
Therefore, for even-order \(l\), the condition to realize the AEL mode can be expressed as  
\[\alpha_{1}^{\rm L}=-\pi/2,\ \tan\Bigl{(}\Big{|}n_{1}^{\rm L}\Big{|}k_{0}R_{1} \Bigr{)}=1, \tag{23}\]  
\[\alpha_{1}^{\rm L}=\pi/2,\ \tan\Bigl{(}\Big{|}n_{1}^{\rm L}\Big{|}k_{0}R_{1} \Bigr{)}=-1, \tag{24}\]  
Figure 4: Simulated total acoustic pressure fields of the CPA modes in the \(yz\)-plane for spherical wave incidence with order \(l\). (**a**) \(l=0\), \(a_{1}^{\rm C}=0.6358\pi\), and \(\big{|}n_{1}^{\rm C}\big{|}=1.7163\). (**b**) \(l=1\), \(a_{1}^{\rm C}=0.6818\pi\), and \(\big{|}n_{1}^{\rm C}\big{|}=2.3674\). (**c**) \(l=2\), \(a_{1}^{\rm C}=0.8170\pi\), and \(\big{|}n_{1}^{\rm C}\big{|}=2.9715\). In all simulations, \(R_{1}=0.35\lambda_{0}\), \(\lambda_{0}=1\) m, and the distribution of the incident pressure field at the outer simulation boundary (\(r=\lambda_{0}\)) is \(P_{l}(\cos\theta)\).  
while that for odd order \(l\) is  
\[\alpha_{1}^{\rm L}=-\pi/2,\ \cot\Bigl{(}\Bigl{|}n_{1}^{\rm L}\Bigr{|}k_{0}R_{1} \Bigr{)}=-1, \tag{25}\]  
\[\alpha_{1}^{\rm L}=\pi/2,\ \cot\Bigl{(}\Bigl{|}n_{1}^{\rm L}\Bigr{|}k_{0}R_{1} \Bigr{)}=1. \tag{26}\]  
According to Equation (18), for even-order \(l\), the corresponding CPA mode condition can be expressed as  
\[\alpha_{1}^{\rm C}=\pi/2,\ \tan\Bigl{(}\Bigl{|}n_{1}^{\rm C}\Bigr{|}k_{0}R_{1} \Bigr{)}=1, \tag{27}\]  
\[\alpha_{1}^{\rm C}=-\pi/2,\ \tan\Bigl{(}\Bigl{|}n_{1}^{\rm C}\Bigr{|}k_{0}R_{1} \Bigr{)}=-1, \tag{28}\]  
while that for odd order \(l\) is  
\[\alpha_{1}^{\rm C}=\pi/2,\ \cot\Bigl{(}\Bigl{|}n_{1}^{\rm C}\Bigr{|}k_{0}R_{1} \Bigr{)}=-1, \tag{29}\]  
\[\alpha_{1}^{\rm C}=-\pi/2,\ \cot\Bigl{(}\Bigl{|}n_{1}^{\rm C}\Bigr{|}k_{0}R_{1} \Bigr{)}=1. \tag{30}\]  
Comparing Equations (23)-(26) and (27)-(30), when \(\alpha_{1}=-\pi/2\) (\(\alpha_{1}=\pi/2\)) and \(\tan(|n_{1}|k_{0}R_{1})=\cot(|n_{1}|k_{0}R_{1})=1\) [\(\tan(|n_{1}|k_{0}R_{1})=\cot(|n_{1}|k_{0}R_{1})=-1\)], the large ACM sphere will extremely amplify spherical waves with all even orders \(l\) and totally absorb spherical waves with all odd orders \(l\) simultaneously. In contrast, when \(\alpha_{1}=-\pi/2\) (\(\alpha_{1}=\pi/2\)) and \(\tan(|n_{1}|k_{0}R_{1})=\cot(|n_{1}|k_{0}R_{1})=-1\) [\(\tan(|n_{1}|k_{0}R_{1})=\cot(|n_{1}|k_{0}R_{1})=1\)], the large ACM sphere will extremely amplify spherical waves with all odd orders \(l\) and perfectly absorb spherical waves with all even orders \(l\).  
Figure 5a,b show \(A_{1}(|n_{1}|)\) with \(l\) = even and \(l\) = odd, respectively, in which \(R_{1}=100\lambda_{0}\). Some values of \(|n_{1}|\) can be found to realize the coexistence of the AEL and CPA modes. For example, if \(|n_{1}|=1.0112\), \(A_{l}=1\) for even \