## 3 Comprehensive Comparison of Different Extraction Methods  
In this section, we make a comprehensive comparison between the proposed method and the other two methods, which are both based on the harmonic analysis. The analysis calculates the harmonic admittance, of which resonance and anti-resonance frequencies (\(f_{r},f_{a}\)) correspond to edges of the stopbands. Thus, finding the resonance and anti-resonance frequencies can determine the COM parameters. However, in some particular cases, the two required modes cannot be simultaneously excited, but it can be conquered by adding a tiny directivity [22]. To achieve this, there are two methods. One is adding a tiny directivity naturally, that is, slightly changing the Euler angle to make the substrate unidirectional. The other is adding a tiny directivity structurally, that is, the wavelength (\(\lambda\)) slightly differs from the period (2p). It can be realized by the Floquet boundary conditions with a tiny detuning: \(kx=(1+\Delta)k_{0}\), where the \(\Delta\) is a tiny proportion. For the sake of convenience, in the following text, we use method 1, 2 and 3 to denote our proposed method and the other two methods, respectively. The detailed settings of boundary conditions are listed in  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Boundary**} & **Mechanical Boundary** & **Electrical Boundary Conditions** \\  & **Conditions** & **Electrical Boundary Conditions** \\ \hline \(\Gamma_{T}\) & Free & Zero charge \\ \(\Gamma_{A}\) & Free & Continuity \\ \(\Gamma_{B}\) & Fixed & Ground \\ \(\Gamma_{+}\), \(\Gamma_{-}\) & Free & 1: \(V=0\) (SC); \(\int D_{i}\cdot n_{i}dS=0\) (OC) \\  & & 2 and 3: \(V_{T+}=IV\); \(V_{\Gamma-}=0V\) \\ \(\Gamma_{L}\), \(\Gamma_{R}\) & & 3: Floquet boundary condition: \(kx=(1+\Delta)k_{0}\) \\ \(\Gamma_{F}\), \(\Gamma_{E}\) & Periodic continuity boundary condition \\ \hline \hline \end{tabular}  
* Notice: 1, 2, 3 represent methods 1, 2 and 3.  
\end{table}
Table 2: The boundary conditions for the three different methods.  
Figure 2: Primitive cell of an infinitely periodic grating.  
Table 2 for the three different methods, where \(D_{i}\) is the electrical displacement and \(n_{i}\) is the normal vector of \(S\).  
We have extracted all COM parameters of the ST quartz substrate with the three different methods. We considered the same configuration as presented in [23], where the pitch of IDT was \(p\) = 3.932 \(\upmu\)m, the aluminum thin film thickness was 160 nm, and the finger width to pitch ratio was \(m/p\)= 0.5. The material constants were taken from ref. [24]. The sparse direct solver PARDISO was employed in the FEM calculation. The calculation was run on a 20 cores 2.2 GHz Xeon computer with 128 GB of RAM. The computation time of the method 1 was 30 s, while that of the others were both 1 h 3 min.  
After the calculation, three groups of COM parameters were obtained. Then, we calculated the relative admittances (Y11) by these COM parameters and COMSOL, respectively. As depicted in Figure 3, the Y11 calculated by the first group of COM parameters, extracted by the method 1, is in the closest correspondence to the COMSOL calculating Y11. The slight difference is caused by the PML, which introduces additional attenuation in the calculation [25]. Thus, it could be concluded that our proposed extraction method holds excellent characteristics of higher accuracy and more efficiency simultaneously.