# Analysis and Design of Single-Phase Unidirectional Transducers with High Directivity
## 2 Design and Optimization
The four independent parameters of the model are the effective SAW velocity \(\
u\), the reflection coefficient per unit length \(\kappa\), the transduction coefficient per unit length \(\alpha\), and the static capacitance per unit length \(C\). The four COM parameters can be calculated by FEM [20; 21]. This approach allows accurate calculation of the transduction amplitude, reflectivity, and the unidirectional angle for any cell geometry. There are five parameters, including the widths of the three fingers and the distances between adjacent fingers, which need to be determined in the EWC-SPUDT cell. It would be a huge undertaking to optimize all of these five parameters at the same time, and the physical meaning of the geometry would be unclear. Therefore, it is necessary to understand the influence of each finger on reflectivity and other COM parameters. In this paper, calculations were conducted by FEM with the commercial software package COMSOL Multiphysics. Al was selected as the electrode material and 128\({}^{\circ}\) YX LiNbO\({}_{3}\) was chosen as the substrate due to its high coupling coefficients (K\({}^{2}\cong\) 5.6%) [22]. Our previous studies demonstrated that the reflectivity is strongly dependent on the normalized electrode thickness \(h/\lambda\), which first increases and then decreases with an increase in electrode thickness [23]. Accordingly, a thin Al electrode with \(h/\lambda=0.6\%\) is chosen to provide adequate reflectivity. Compared with widely used FEM/BEM hybrid methods [24; 25], FEM has obvious advantages, especially its remarkable generality. It can be used to analyze a SPUDT composed of arbitrary materials, crystal cuts, as well as different electrode shapes [26; 27; 28; 29; 30; 31]. Therefore, we use FEM to calculate the parameters needed for simulation and then to optimize the design of the transducer structure. Firstly, the structural model of the transducer unit is established as shown in Figure 2, and the influence of the electrode width on the reflectivity is discussed by taking the electrode width as a variable. To suppress reflections of SAW from the bottom, an additional 128\({}^{\circ}\) YX LiNbO\({}_{3}\) layer of 1\(\lambda\) is attached at the bottom as a perfect matching layer (PML) [32; 33]. The top surface is mechanically free, while the bottom is fixed. Mechanical and electrical periodic conditions are applied on both sides of the unit model [34; 35]. The left electrode is assigned as the electrical terminal with \(V=1\), while the right one is grounded. The reflectivity can be calculated by Equation (7):  
\[|\kappa|\lambda=2\pi\frac{f_{s+}-f_{s-}}{f_{s+}+f_{s-}} \tag{7}\]  
where \(f_{s+}\) and \(f_{s-}\) stand for the anti-symmetric and symmetric frequencies, respectively. The relationships between the reflectivity and the width of two fingers \(w\) and their distance \(g\) are depicted in Figure 3, from which two conclusions can be drawn: (i) The reflectivity reaches its maximum value when \(g\) approaches and this is because the phase difference between the SAW reflected by the second finger and that reflected by the first finger is \(2\pi\), resulting in in-phase superposition; (ii) When \(g=\lambda/2\), the reflectivity increases at first and then decreases with the increase in electrode width, and the maximum value of 5.56% occurs at about \(w=3\lambda/32\). A measured electrode reflectivity given in [36] confirms our conclusion. Figure 3: Reflectivity per period of Al electrodes on 128\({}^{\circ}\) YX LiNbO\({}_{3}\) substrate. Figure 2: FEM model of an IDT unit cell with boundary conditions. As shown in Figure 1, the EWC-SPUDT cell consists of three fingers. The first finger plays the role of generating surface acoustic waves; however, it also reflects the incident acoustic waves, while the reflection can be cancelled out with the entry of the second finger.