## 4 Discussion  
The numerical simulation results revealed the performances of the single-phase and the 1-3 composite transducers. The single-phase transducer could not produce a higher acoustic pressure output than the 1-3 composite one, as shown in Figures 6 and 7. This is because the thickness of the active layer, which determines the operation frequency, approaches the lateral size of the active element in relatively low-frequency operation conditions (e.g., \(<\)5 MHz). The close resonance frequencies of the vertical and the lateral vibrations cause coupling of the vibration modes. Figure 9 shows the vibration coupling in the electric impedance response. For example, for the single-phase transducers (i.e., dashed lines), the lateral vibration modes are found in many different frequency bands, which  
Figure 12: Simulated performance of the 1–3 composite focused ultrasound transducer; (**a**) the acoustic pressure field with respect to the number of the transducer elements, (**b**) the acoustic pressure output and the acoustic intensity with respect to the number of arrays, and (**c**) electric impedance responses with the varying number of active elements in the water media. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Specification** & **Unit** & **Value** & **Specification** & **Unit** & **Value** \\ \hline Num. of elements & - & 3 by 1 & Acoustic intensity & W/cm\({}^{2}\) & 188.2 \\ \hline Aperture size & mm\({}^{2}\) & 2 \(\times\) 2 & Acoustic pressure & MPa & 23.6 \\ \hline Operation freq. & MHz & 5 MHz & Electric impedance & Ohm & 7.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulated specification of the designed focused ultrasound transducer. interfere with the thickness vibration mode. Consequently, the high impedance due to the mode coupling has an adverse effect on the generation of the acoustic pressure output from the FUS transducer. However, the influence of the vibration coupling in the miniaturized transducer design is difficult to eliminate in the typical single-phase transducer as the lateral dimension of the active layer must be relatively small in accordance with the size of the lumen where the FUS device would travel in the human body. By contrast, the electric impedance curve in the 1-3 composite design shows a clear resonance with a relatively broad frequency band (Figure 9). The high electromechanical coupling (e.g., \(\sim\)46% in the 1-3 composite and \(\sim\)38% in the single-phase) and the predominant thickness vibration mode could produce a relatively high acoustic pressure output, as shown in Figure 6. The lateral dimension of the small piece of the piezoelectric rods was differentiated from the thickness size by employing the 1-3 composite design, which resulted in the de-coupling of the lateral and the thickness vibration modes. Nonetheless, the width of the single piezoelectric rod in the 1-3 composite could still affect the performance of the transducer in the case where the width was not small enough compared to the thickness. Specifically, even though there was no significant change in the electric impedance with respect to the lateral dimension (Figure 11), the acoustic pressure output was affected by the lateral size of the single piezoelectric rods in the 1-3 composite (Figure 8). For a relatively large lateral dimension, the dimensional ratio became small, causing interference of the lateral vibration mode and the corresponding low acoustic pressure output along the thickness direction. Therefore, the dimension ratio of the piezoelectric rod in the 1-3 composite needs to be small enough (\(<\sim\)1:6) to isolate the mode coupling effect. In other words, according to the simulation results, it was suggested that the width of the piezoelectric rod should be approximately six times smaller than the thickness. However, a too-small size of the piezoelectric rods may compromise the feasibility of fabrication.