## 2 Materials and Methods  
### 1-3 Piezolectric Composite Design  
Figure 1b illustrates a 1-3 composite transducer in the thickness mode. This model is symmetric of strain in the \(x\)-y direction. The transducer operates by using the \(E_{3}\) field. In addition, the model satisfies the identical vertical strain and electric field requirements for polymer and piezoelectric rods. The effective constitutive equation was derived as follows [19; 37]:  
\[T_{3,\text{eff}} =c_{33,\text{eff}}^{\text{E}}S_{3,\text{eff}}-e_{33,\text{eff}}E_{ 3,\text{eff}}, \tag{1}\] \[D_{3,\text{eff}} =e_{33,\text{eff}}S_{3,\text{eff}}+e_{3,\text{eff}}^{\text{S}}E_{ 3,\text{eff}}, \tag{2}\]  
where \(c^{\text{E}}\), \(e\), \(\varepsilon\), \(T\), \(S\), \(D\), and \(E\) are the stiffness under free electric field, piezoelectric constant, dielectric permittivity, stress, strain, electric displacement, and electric field, respectively. The superscript denotes the direction of each parameter. \(c^{\text{E}}\), \(e\), and \(\varepsilon\) of a 1-3 composite having the volume fractions \(V\) and \(\overline{V}\) in the active and the passive materials, were derived as follows [19; 37]:  
\[c_{33,\text{eff}}^{\text{E}}=V\left[c_{3,\text{eff}}^{\text{c,E}}-\frac{2 \overline{V}\left[\left(c_{13}^{\text{c,E}}\right)^{2}-\left(c_{12}^{\text{P} }\right)^{2}\right]}{V\left(c_{11}^{\text{P}}+c_{12}^{\text{P}}\right)+ \overline{V}\left(c_{11}^{\text{c,E}}+c_{12}^{\text{c,E}}\right)}\right]+ \overline{V}c_{11}^{\text{P}}, \tag{3}\]  
\[e_{33,\text{eff}}=V\left[c_{33}^{\text{c}}-\frac{2\overline{V}e_{31}^{\text{ c}}\left(c_{13}^{\text{c,E}}-c_{12}^{\text{P}}\right)}{V\left(c_{11}^{\text{P}}+c_{12}^{ \text{P}}\right)+\overline{V}\left(c_{11}^{\text{c,E}}+c_{12}^{\text{c,E}} \right)}\right], \tag{4}\]  
\[\varepsilon_{33,\text{eff}}^{\text{S}}=V\left[\varepsilon_{33}^{\text{c,S}}- \frac{2\overline{V}\left(\varepsilon_{31}^{\text{S}}\right)^{2}}{V\left(c_{11} ^{\text{P}}+c_{12}^{\text{P}}\right)+\overline{V}\left(c_{11}^{\text{c,E}}+c_ {12}^{\text{c,E}}\right)}\right]+\overline{V}\varepsilon_{11}^{\text{P}}, \tag{5}\]  
Figure 1: Schematic of the miniaturized, interstitial, focused ultrasound transducer, directed by the catheter (**a**), and the structure of 1–3 piezoelectric composite as an active layer (**b**). The resonance frequency (\(\
ot{f}_{\rm r}\)) of composite transducers is determined by the relation  
\[k_{\rm t,eff}=\sqrt{\frac{\pi}{2}\frac{f_{\rm r}}{f_{\rm a}}\tan\left(\frac{\pi}{ 2}\frac{f_{\rm a}-f_{\rm r}}{f_{\rm a}}\right)}, \tag{6}\]  
\[f_{\rm a}=\frac{v_{\rm L,eff}}{2h}, \tag{7}\]  
where \(f_{\rm r}\) and \(f_{\rm a}\) are the resonance and the anti-resonance frequencies, respectively, \(k_{\rm t,eff}\) is the effective electromechanical coupling, \(\rho_{\rm eff}\) is the effective density, and \(v_{\rm L,eff}\) is the speed of sound in the composite. Figure 2 shows the simulated properties of the 1-3 composite composed of PZT-4 and epoxy material. We chose PZT-4 as the active material as PZT-4 is mechanically robust while having a sufficiently high piezoelectric coefficient [38]. Its effective stiffness becomes lower than that of the single-phase material when a polymer (i.e., epoxy) with a relatively low elastic modulus is added to it (Figure 2a). The addition of the passive layer made the piezoelectric constant and the dielectric permittivity low, as shown in Figure 2b,c. By contrast, the effective coupling coefficient exhibits the maximum value in the 1-3 composite phase (Figure 2d). For example, the maximum effective coupling coefficient is approximately 0.46 at 64% of the PZT-4 volume fraction, whereas the coupling coefficient in the single-phase material is 0.38. The operation (or the resonance) frequency of the transducer is determined by the thickness of the transducer (Equations (6) and (7)). Figure 3 shows the predicted resonance frequency of the PZT-4/epoxy 1-3 composite transducer at various thicknesses of the transducer. Table 1 lists the material properties used for the analytical and numerical simulations.