# MRI Compatible Planar Material Acoustic Lenses
## 2 Methodology and Theoretical Analysis  
Fresnel zone plates are circular concentric structures, which are known as Fresnel regions. Every consecutive region has a \(\pi\) phase shift between them. This fact makes a coherent contribution to obtain high intensity levels at focal length (\(F_{L}\)), which is the location in the axial coordinate where the focus is placed. The number of Fresnel regions is defined as \(N\), this includes both opaque and transparent acoustic sections. The working frequency is defined as \(f_{0}\) and radial distances (\(r_{n}\)) of each Fresnel zone can be obtained using Equation (1) valid for plane wave incidence.  
\[r_{n}=\sqrt{n\lambda F_{L}+\left(\frac{n\lambda}{2}\right)^{2}}\ \ \ \ \ \ n=1,2,...,N \tag{1}\]  
In this work, underwater transmission is considered and lenses are designed for ultrasound applications. Therefore, piston sources have to be considered when FZPs are implemented. Due to spherical wave incidence consideration, Equation (2) has been used where \(d\) is the separation between the point source and the lens.  
\[d+F_{L}+\frac{n\lambda}{2}=\sqrt{d^{2}+r_{n}^{2}}+\sqrt{F_{L}^{2}+r_{n}^{2}} \tag{2}\]The acoustic wave has to propagate through the host medium, then cross Fresnel regions and afterwards continue through the host medium. A three-layer configuration has to be considered (Figure 1). Acoustic impedance (\(Z\)) is defined as the product of the medium density (\(\rho\)) and the sound propagation velocity (\(c\)) in it. Therefore, it is necessary to consider the input (\(Z_{in}\)) and output (\(Z_{out}\)) acoustic impedance and the transmission pressure coefficients must be calculated (\(t\)). This coefficient is a clear indicator of the blocking capacity of the elements of the FZP. Hence, \(t\) is defined as the relation between the transmitted field and the incident field. Density (\(\rho\)), sound propagation velocity (\(c\)) and acoustic impedance (\(Z_{mat}\)) values have been shown in Table 1. Using these values in Equation (3), \(Z_{in}\) could be obtained [17].  
\[Z_{in}=Z_{mat}\frac{Z_{out}+jZ_{mat}\tan(k_{mat}d)}{Z_{mat}+jZ_{out}\tan(k_{ mat}d)} \tag{3}\]  
where \(k_{m}\) is the wave number, defined as \(k_{m}=\omega/c\). Considering \(\omega=2\pi f_{0}\). Once \(Z_{in}\) is obtained, reflection coefficient is defined in Equation (4).  
\[r_{in}=\frac{Z_{in}-Z_{water}}{Z_{in}+Z_{water}} \tag{4}\]  
The equation that relates the field balance as a function of the impedance and reflection coefficient of the system is defined in Equation (5) and gives \(t\) values depending on the material.  
\[|t|=\frac{|p_{t}^{+}|}{|p_{in}^{+}|}=\sqrt{(1-|r_{in}|^{2})} \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(\rho\) **(kg/m\({}^{3}\))** & \(c\) **(m/s)** & \(Z_{mat}(Rayls)\) & \(Z_{mat}/Z_{water}\) \\ \hline ABS & 1050 & 2250 & 2.4\(\cdot\)10\({}^{6}\) & 1.58 \\ PLA & 1240 & 2220 & 2.8\(\cdot\)10\({}^{6}\) & 1.84 \\ PMMA & 2690 & 1191 & 3.2\(\cdot\)10\({}^{6}\) & 2.14 \\ Brass & 8400 & 4700 & 39.5\(\cdot\)10\({}^{6}\) & 26.32 \\ PLA-Air-PLA & 398 & 944 & 3.6\(\cdot\)10\({}^{5}\) & 0.25 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Density and sound speed values. Acrylonitrile butadiene styrene (ABS), polylactic acid (PLA), and polymethyl methacrylate (PMMA).  
Figure 1: Transmission diagram of the implemented lenses.  
Considering the transmission coefficient values obtained (0.23 for brass, 0.51 for PLA-Air-PLA and more than 0.95 for ABS, PLA and PMMA), it can be affirmed that full implemented MRI compatible material lenses will focus less energy at the \(F_{L}\) if it is compared to brass FZP or ideal SZP. Therefore, one solution is proposed to obtain the desired impedance contrast. A FZP that includes an air chamber inside the structure has been implemented by using a 3D-printer. Thus, both lenses, full-PLA and air-chamber, have been compared.