# Acoustic Negative Refraction and Planar Focusing Based on Purely Imaginary Metamaterials
## 2 Scattering Properties of a PIM Slab  
We first focus on the scattering properties of a purely imaginary medium (PIM) slab with a thickness of \(d\) in air, as shown in Figure 1. The bulk compressibility and the mass density are given as \(\beta_{s}=i\beta_{r}\beta_{0}\) and \(\rho=-i\rho_{r}\rho_{0}\), respectively. Here, \(\beta_{0}\) and \(\rho_{0}\) are the corresponding parameters of air, and \(\beta_{r},\rho_{r}\) are real numbers. The density and velocity of air are \(\rho_{0}=1.2\) kg/m\({}^{3}\), and \(c_{0}=343\) m/s, respectively. When the imaginary part of the parameter is positive, it represents the gain, and when the imaginary part of the parameter is negative, it represents the loss. Although the PIMs contain both gain and loss elements, the phases of the bulk compressibility \(\beta_{s}\) and the mass density \(\rho\) are conjugates of each other, so the acoustic velocity \(c=\pm\sqrt{\left(\rho\beta_{s}\right)^{-1}}\) and the refractive index \(n=\pm\sqrt{\rho\beta_{s}/\rho_{0}\beta_{0}}\) are both real numbers.  
Consider monochromatic acoustic waves with frequency \(\omega\) and incident angle \(\theta\). Under the \(e^{i\omega t}\) convention, the acoustic fields \(p\) satisfy the Helmholtz equation \((d^{2}/dx^{2})p+\omega^{2}\beta_{s}\rho p=0\). The equation admits a solution  
\[\left\{\begin{array}{l}p(x,y)=p_{f}e^{-jk_{x}x}e^{-k_{y}y}+p_{bl}e^{jk_{x}x}e^ {-k_{y}y},\quad(x<0)\\ p(x,y)=p_{fr}e^{-jk_{x}(x-d)}e^{-k_{x}y}+p_{br}e^{jk_{x}(x-d)}e^{-k_{x}y},\ (x>d) \end{array}\right. \tag{1}\]  
where, \(k_{y}=k_{0}\sin\theta\) are the wave vector along the \(y\) directions, \(k_{0}=2\pi/\lambda\) is the wave vector in air, \(k_{x}=\sqrt{{k_{0}}^{2}-{k_{y}}^{2}}\) and \({k_{x}}^{\prime}=\sqrt{{n^{\prime}}{2k_{0}}^{2}-{k_{y}}^{2}}\) are the \(x\) wave vector components in the air and PIM media, respectively. Due to the boundary conditions, \(k_{y}\) remains constant in all layers. The coefficients \(p_{fl}\), \(p_{bl}\), \(p_{fr}\) and \(p_{br}\) are the complex amplitudes of the incoming or outgoing waves, with the subscript \(f(b)\) for forward (backward) acoustic waves, and the subscript \(l(r)\) for the left \(x<0\) (right \(x>d\)).  
In terms of the scattering matrix \(S\), the relationship between the amplitudes of the incident and outgoing waves is  
\[\begin{pmatrix}p_{bl}\\ p_{fr}\end{pmatrix}=\begin{pmatrix}r&t\\ t&r\end{pmatrix}\begin{pmatrix}p_{fl}\\ p_{br}\end{pmatrix},\ S=\begin{pmatrix}\text{S}_{11}&\text{S}_{12}\\ \text{S}_{21}&\text{S}_{22}\end{pmatrix}=\begin{pmatrix}r&t\\ t&r\end{pmatrix}. \tag{2}\]  
Figure 1: Schematic graph of a PIM slab in air.  
In the above equations, \(r\) and \(t\) represent the reflection and transmission coefficients of the slab, respectively. By calculating the elements of the scattering matrix, we obtain the transmittance and reflection coefficients as (see Appendix A)  
\[r=\text{S}_{11}=\text{S}_{22}=\frac{(\eta^{2}-1)2i\sin\phi}{(\eta+1)^{2}e^{i \phi}-(\eta-1)^{2}e^{-i\phi}},\;t=\text{S}_{12}=\text{S}_{21}=\frac{4\eta}{( \eta+1)^{2}e^{i\phi}-(\eta-1)^{2}e^{-i\phi}}. \tag{3}\]  
Here, \(\phi=k_{x}{{}^{\prime}}d\) is the propagation phase in the slab, and \(\eta=\rho{{}^{\prime}}k_{x}/\rho_{0}k_{x}{{}^{\prime}}\) is the relative impedance of the PIM-air interface. The equalities of \(\text{S}_{11}=\text{S}_{22}=r\) and \(\text{S}_{12}=\text{S}_{21}=t\) is observed due to the symmetry of the system.