# Acoustic Equivalent Lasing and Coherent Perfect Absorption Based on a Conjugate Metamaterial Sphere
## 2 Scattering Model of an ACM Sphere
Here, \(\big{|}\rho_{q}\big{|}\) (\(\big{|}\beta_{q}\big{|}\)) denotes the amplitude of the effective mass density (bulk compressibility), \(\alpha_{q}\) denotes a phase angle, and \(n_{q}=\pm\sqrt{\rho_{q}\beta_{q}/\rho_{0}\beta_{0}}\) is the refractive index of the \(q\)-th ACM layer. For \(0<\alpha_{q}<\pi/2\) and \(3\pi/2<\alpha_{q}<2\pi\), the sign \"\(+\)\" is taken, and the ACM features right-handedness (positive refractive index), while for \(\pi/2<\alpha_{q}<3\pi/2\), the sign \"\(-\)\" is taken, and the ACM features left-handedness (negative refractive index). In particular, when \(\alpha_{q}=\pi/2\) and \(\alpha_{q}=3\pi/2\), the ACM is called a pure imaginary metamaterial, and the material exhibits ambiguous handedness. When \(\alpha_{q}=0\) and \(\alpha_{q}=\pi\), the ACM degenerates into double-positive and double-negative materials, respectively. Due to the spherical symmetry of the proposed ACM sphere, the channel modes used to describe the acoustic field distribution (see Equation (1)) are naturally spherical waves, which are a set of orthogonal separate variable solutions of the acoustic Helmholtz equation in spherical coordinates (\(r,\,\theta,\,\varphi\)). Here, \(r,\,\theta\) and \(\varphi\) denote the radius, angular and azimuthal angles in spherical coordinates, respectively. The case of axial symmetry is considered here,  
Figure 1: The diagram of a multilayered ACM sphere system: (**a**) a 3D view; (**b**) a cross section in the \(yz\)-plane. where the acoustic field is independent of coordinate \(\varphi\). Therefore, for \(0\leq q\leq Q-1\), the total acoustic pressure in the \(q\)-th region can be expressed as  
\[p_{q}(r,\theta)=\sum_{l=0}^{\infty}\left[a_{q}^{l}h_{l}^{(2)}\left(k_{q}r\right) +b_{q}^{l}h_{l}^{(1)}\left(k_{q}r\right)\right]P_{l}(\cos\theta). \tag{3}\]  
Here, \(h_{l}^{(1)}\) and \(h_{l}^{(2)}\) are the spherical Hankel functions of the first and second kinds with the order \(l\), and \(P_{l}(\cos\theta)\) is the Legendre polynomial of order \(l\), respectively. The parameter \(k_{q}=n_{q}k_{0}\) denotes the wavenumber for the \(q\)-th region, in which \(k_{0}=2\pi/\lambda_{0}\) is the wavenumber for the background medium and \(\lambda_{0}\) is the wavelength in the background medium. The expressions \(a_{q}^{l}h_{l}^{(2)}(k_{q}r)P_{l}(\cos\theta)\) and \(b_{q}^{l}h_{l}^{(1)}(k_{q}r)P_{l}(\cos\theta)\) represent the incoming and outgoing channel spherical waves of order \(l\) in the \(q\)-th region, with \(a_{q}^{l}\) and \(b_{q}^{l}\) being the corresponding complex amplitudes. Here, the time variant item \(e^{-i\omega t}\) is omitted for simplicity. The radial evolution of the spherical waves is given by the spherical Hankel functions, and the field evolution over \(\theta\) is given by \(P_{l}(\cos\theta)\). Because of the orthogonality of the spherical waves, the continuity of acoustic pressure at the interface \(r=R_{q}\) reads  
\[h_{l}^{(1)}\left(k_{q-1}R_{q}\right)s_{q-1}^{l}-C_{q}^{l}a_{q}^{l}/a_{q-1}^{l} =-h_{l}^{(2)}\left(k_{q-1}R_{q}\right), \tag{4}\]  
while the continuity of radial acoustic velocity at \(r=R_{q}\) leads to  
\[\eta_{q}h_{l}^{(1)^{\prime}}\left(k_{q-1}R_{q}\right)s_{q-1}^{l}-\eta_{q-1}D_{ q}^{l}a_{q}^{l}/a_{q-1}^{l}=-\eta_{q}h_{l}^{(2)^{\prime}}\left(k_{q-1}R_{q} \right). \tag{5}\]  
Here, \(\eta_{q}=\rho_{q}/\left(n_{q}\rho_{0}\right)\) denotes the acoustic impedance of the \(q\)-th region relative to the background medium, the scattering coefficient \(s_{q-1}^{l}=b_{q-1}^{l}/a_{q-1}^{l}\) is the ratio between the complex amplitudes of the outgoing and incoming spherical waves in the (\(q\)-1)-th region, \(h_{l}^{(1)^{\prime}}\) and \(h_{l}^{(2)^{\prime}}\) are the first-order derivatives of the corresponding spherical functions, \(C_{q}^{l}=h_{l}^{(2)}\left(k_{q}R_{q}\right)+s_{q}^{l}h_{l}^{(1)}\left(k_{q}R _{q}\right)\), and \(D_{q}^{l}=h_{l}^{(2)^{\prime}}\left(k_{q}R_{q}\right)+s_{q}^{l}h_{l}^{(1)^{ \prime}}\left(k_{q}R_{q}\right)\), respectively.