## 2 Methods  
### Solutions of COM Equations  
Figure 1 shows the schematic of COM model for the IDT, in which \(W\) is the aperture length, \(p\) is the pitch of the IDT, the wavelength (\(\lambda\)) is the twice of the pitch (\(\lambda=2p\)). A drive voltage \(V\) connect over the bus bars of the IDT excites waves. In reverse, the waves propagating under the electrodes cause flow of the current \(I\). The \(R(x)\) and \(S(x)\) represent the slowly varying amplitudes of the modes propagating in the positive and negative \(x\)-directions, respectively.  
The presence of the electrodes on the surface introduces a coupling between the modes. Assuming a linear coupling between the amplitudes due to reflectivity, and a linear transduction coefficient due to piezoelectricity, the final COM equation can be obtained [7]:  
\[\left\{\begin{array}{l}\frac{\mathrm{d}R(x)}{\delta x}=-j\delta R(x)+j \kappa S(x)+j\alpha V,\\ \frac{\mathrm{d}S(x)}{\delta x}=j\delta S(x)-j\kappa^{*}S(x)-j\kappa^{*}V,\\ \frac{\mathrm{d}f(x)}{\delta x}=-2j\kappa^{*}R(x)-2j\kappa S(x)+j\omega CV, \end{array}\right. \tag{1}\]  
where \(\delta\) is the detuning parameter:  
\[\delta=k-k_{0}=k_{r}-k_{0}-j\gamma=\frac{\omega}{v}-k_{0}-j\gamma,\]  
Figure 1: The schematic of COM model for the IDT.  
\(k\) is the complex wavenumber, \(k=k_{r}-j\gamma\), \(k_{r}\) is the resonance wavenumber and \(k_{0}\) is the center wavenumber, \(k_{0}=2\pi/\lambda=\pi/p\). The COM parameters of the model are velocity \(v\), reflectivity \(\kappa\), transduction coefficient \(\alpha\), static capacitance per unit length \(C\) and propagation attenuation \(\gamma\) (superscript \({}^{*}\) denotes the complex conjugate). These parameters are determined from the simulation of an infinite periodic grating under different boundary conditions [19].  
When considering no attenuation (\(\gamma\) = 0), the equation system (1) under SC conditions (i.e., \(V\) = 0) gives the relationships between stopband edges and velocity \(v\), reflectivity \(\kappa\):  
\[\left\{\begin{array}{l}\
u=\lambda\frac{(f_{sc}+f_{sc-})}{2}\\ |\kappa|\lambda=2\pi\frac{f_{sc}+-f_{sc-}}{f_{sc+}+f_{sc-}}\end{array}\right. \tag{2}\]  
Under OC conditions (i.e., \(I\) = 0), the equation system (1) gives the relationships between stopband edges and transduction coefficient \(\alpha\), directivity phase \(\varphi\):  
\[\left\{\begin{array}{l}|\alpha_{n}|=\sqrt{\omega C_{n}\lambda\pi\Big{(}\frac {f_{sc}+f_{sc-}}{f_{sc+}+f_{sc-}}-1\Big{)}}\\ \cos(\varphi)=\cos\big{(}\alpha^{2}/\kappa\big{)}=\frac{(f_{sc+}-f_{sc-})^{2} -(f_{sc+}-f_{sc-})^{2}-[(f_{sc+}+f_{sc-})-(f_{sc+}+f_{sc-})]^{2}}{2(f_{sc+}+f_ {sc-})](f_{sc+}+f_{sc-})-(f_{sc+}+f_{sc-})]}\end{array}\right. \tag{3}\]  
where \(\alpha_{n}\) is the normalized transduction coefficient, namely, \(\alpha_{n}=\alpha\lambda/\surd(W/\lambda)\). \(C_{n}\) is the normalized static capacitance, \(C_{n}\) = \(C\lambda/W\). The \(\varphi\) is the directivity phase, namely, \(\varphi\) = \(2\varphi_{\alpha}\) - \(\varphi_{\kappa}\), where \(\varphi_{\alpha}\) and \(\varphi_{\kappa}\) are the phases of \(\alpha\) and \(\kappa\). Equations (2) and (3) imply that all COM parameters, except the attenuation \(\gamma\), can be obtained from the lower and upper edges of the SC and OC stopbands (\(f_{SC-},f_{SC+},f_{OC-},f_{OC+}\)), if the \(C\) was previously extracted or given.