# Acoustic Equivalent Lasing and Coherent Perfect Absorption Based on a Conjugate Metamaterial Sphere
## 2 Scattering Model of an ACM Sphere  
For an acoustic scattering system, where a scatterer couples to a discrete set of scattering channels (denoted as \(l\)), the harmonic pressure field outside the scatterer can be expressed as  
\[p\left(\overset{\rightarrow}{r},t\right)=\sum_{l}\left[a_{l}p_{l}^{\text{in}} \left(\overset{\rightarrow}{r}\right)+b_{l}p_{l}^{\text{out}}\left(\overset{ \rightarrow}{r}\right)\right]e^{-i\omega t}. \tag{1}\]  
Here, \(\omega\) is the angular frequency, \(p_{l}^{\text{in}}\left(\overset{\rightarrow}{r}\right)\) and \(p_{l}^{\text{out}}\left(\overset{\rightarrow}{r}\right)\) indicate the acoustic input and output \(l\)-th channel mode with proper analytic form. For a plane scatterer, the channel modes are plane waves, while for a cylindrical scatterer, they are cylindrical waves. The coefficients \(a_{l}\) and \(b_{l}\) are the complex amplitudes of the incoming and outgoing channel waves, which are further related by the scattering matrix \(S\) as  
\[b_{l}=\sum_{m}S_{l,m}a_{m}, \tag{2}\]  
where \(l\) and \(m\) denote scattering channels. It is noted that the scattering matrix \(S\) is symmetric, namely, \(S_{l,m}=S_{m,l}\). The scattering matrix could describe the acoustic response of the scatterer under incidence. For example, by fine tailoring the acoustic parameters (including gain) and geometry of the scatterer, an eigenvalue of the scattering matrix could diverge at a real frequency, and the corresponding eigenstate indicates a solution with only outgoing waves, namely, a lasing mode [37]. Thus, the scattering system works as an acoustic equivalent laser at this frequency, which produces coherent outgoing waves with extensive intensities. The time-reversed counterpart to this lasing mode is a CPA mode [38; 39; 40; 41; 42; 43; 44; 45; 46], where the acoustic and geometric parameters of the scatterer are the same as those in the lasing mode, except that the gain is replaced with loss with equal magnitude. Here, an eigenvalue of the scattering matrix tends to zero at the same frequency for the lasing mode, and the corresponding eigenstate indicates a solution without outgoing waves. Now, the scattering system works as an acoustic coherent perfect absorber at this frequency, which could totally absorb a specific coherent incident wave pattern determined by the eigenstate. An acoustic system with effective loss (gain) could realize only the CPA (AEL) effect [38], while the simultaneous realization of AEL and CPA effects requires a system consisting of both gain and loss, which has been proposed based on an acoustic parity-time symmetric system [39] and an anti-parity-time symmetric system [40]. Besides, previous studies focused on one-dimensional (1D) and two-dimensional (2D) systems. In 1D systems, the AEL and CPA effects were demonstrated for plane normal incidence [39; 40]. In a 2D panel system, such effects could be realized for plane oblique incidence [37]. In a 2D cylindrical system, the CPA effect was realized for cylindrical incidence with angular momentum [38]. In this work, we propose a multilayered ACM sphere to realize the AEL and CPA effects individually/simultaneously for spherical incidence. Figure 1a,b show the diagram of the proposed multilayered ACM sphere in a 3D view and a cross section in the \(yz\)-plane, respectively. The multilayered ACM sphere is immersed in a conventional background medium (labeled as the zeroth region), whose mass density \(\rho_{0}\) and bulk compressibility \(\beta_{0}\) are both real and positive. The outer radius of the \(q\)-th layer (\(q=1,\,2,\,\ldots,\,Q\)) of the ACM sphere is \(R_{q}\), and each layer is characterized by the refractive index \(n_{q}\) and effective mass density \(\rho_{q}\). The effective mass density \(\rho_{q}\) and bulk compressibility \(\beta_{q}\) for the \(q\)-th ACM layer are expressed as \(\rho_{q}=\big{|}\rho_{q}\big{|}e^{ia_{q}}\) and \(\beta_{q}=\big{|}\beta_{q}\big{|}e^{-ia_{q}}\), respectively, whose phases are complex conjugates of each other.