# A Novel Love Wave Mode Sensor Waveguide Layer with Microphononic Crystals
## 2 Materials and Methods
### Transmission Parameters Analysis  
To further analyze the Love wave transmission in the new waveguide layer with micro PnCs, transmission parameters were analyzed. As the Love wave is a typical three-dimensional wave, its analysis cannot be simplified into a two-dimensional model as with the Rayleigh wave. Therefore, to reduce the calculation complexity, quasi-three-dimensional models were adopted. To guarantee that the excited wave had a Love wave mode, structures of the delay line in the theory were introduced in this paper according to the authors in [25]. The Love wave mode delay lines, as shown in Figure 5, consist of an emitting interdigital transducer (IDT), a receiving IDT and the transmitting path with or without Al/SiO\({}_{2}\)/36\({}^{\circ}\) LiTaO\({}_{3}\) piezoelectric PnCs, respectively. Quasi-three-dimensional models were applied to each to calculate their \(Y\) parameters. Figure 4: Band structure of Al/SiO\({}_{2}\)/36\({}^{\circ}\) LiTaO\({}_{3}\) and the typical eigen mode. In the models, one row was extracted in the direction parallel to the IDT and the periodic boundary condition was adopted in this direction. Perpendicular to the IDT, there were two pairs of the transmitting and receiving IDT and 10 periods of pillars on the transmission path. On both sides of the edge of the transmission path, PML was adopted. On the bottom of the models, PML was also used to suppress the strong reflection of the boundary. With the models above, FEM in frequency domain analysis, combined with the electric terminals applying on the IDT, was used to calculate the Y parameters of both delay line structures. In the model, transmitting IDT was set as terminal 1 and ground, and receiving IDT was set as terminal 2 and ground. For terminal 1, 1 V voltage was applied. With the algorithm in the FEM software, the Y parameters can be derived. The results of the \(Y\) parameters are as shown in Figure 6.  
Furthermore, to analyze the transmission of the piezoelectric PnCs, a P matrix as shown in Figure 7 was introduced. According to the theory of the \(P\) matrix [29], IDT can be described as a three-port network (Figure 4) with two acoustical ports and one electrical port, as shown in the following equation:  
\[\left[ \begin{array}{c}b_{1}\\ b_{2}\\ i\end{array} \right]=\left[ \begin{array}{ccc}P_{11}{}^{IDT}&P_{12}{}^{IDT}&P_ {13}{}^{IDT}\\ P_{21}{}^{IDT}&P_{22}{}^{IDT}&P_{23}{}^{IDT}\\ P_{31}{}^{IDT}&P_{32}{}^{IDT}&P_{33}{}^{IDT}\end{array} \right]\left[ \begin{array} []{c}a_{1}\\ a_{2}\\ u\end{array} \right] \tag{5}\]  
where \(a_{i}\) is the input of the acoustical ports, \(b_{i}\) is the output of the acoustical ports, \(u\) is the voltage of the electrical port, \(i\) is the electrical current of the electrical port and \(P_{ij}^{IDT}\) is the \(P\) matrix element of the IDTs. Figure 5: The quasi-three-dimensional models of delay line structures with and without micro PnCs. Figure 6: Transmission of the Love wave mode delay line. Differing from IDT, the \(P\) matrices of the free waveguide layer and waveguide layer with micro piezoelectric PnCs transmitting paths can be reduced to two acoustical ports, as shown in the following equation:  
\[\left[ \begin{array}{c}b_{1}\\ b_{2}\end{array} \right]=\left[ \begin{array}{cc}P_{11}&P_{12}\\ P_{21}&P_{22}\end{array} \right]\left[ \begin{array}{c}a_{1}\\ a_{2}\end{array} \right] \tag{6}\]  
Based on the \(P\) matrices of IDT and transmitting paths, the equivalent models of the two SAW delay lines can be derived as shown in Figure 8. We assume that the arrangement of delay line structures is perfectly symmetric and the emitting IDT and the receiving IDT are the same. Thus, the \(P\) matrices of these IDTs are the same.