## 2 Materials and Methods
### Measurement Setup  
Three different sets of measurements were carried out for this work using the setup shown in Figure 1. For the first set, the curvature of the FUS transducer was determined by taking advantage of its exposed negative terminal on the radiating surface; for this, a multimeter (Fluke 289, Fluke Corporation, Everett, WA, USA) and a 3D positioning system (Onda Corporation, Sunnyvale, CA, USA) driven in manual mode were used. The transducer was placed at the final position for acoustic field measurements (to be made after) in which the transducer will emit in positive \(z\)-direction; this first measurement was made in the air. Then, the multimeter was set to measure electric continuity, and one of its probes was attached to the positioning system while the other was connected to the negative terminal of the transducer. The measurement probe was moved step-by-step (step resolution of 6.36 \(\upmu\)m) towards the transducer surface, down in the \(z\)-direction (relative to the direction of the ultrasound emission), for a fixed \(x\)-coordinate until the probe touched the transducer conductive surface, indicated by a \"beep\" from the multimeter. This procedure was repeated diametrically for every \(x\)-coordinate from \(-14.573\) mm to 14.573 mm with a step resolution of 0.3312 mm, covering the transducer case and the curved piezoelectric plate. The used step-resolution was smaller than a half of a wavelength of 2 MHz ultrasound in water at 20 \({}^{\circ}\)C (approx. 0.38 mm). The determined \(z\)-coordinates for all the \"\(x\)\" were saved for further use. These final coordinates were used for the second below explained measurement. For the second set, the transducer was immersed in a tank filled with degassed distilled water carefully keeping intact the transducer setup used for the first set (see Figure 1). This second set of measurements was carried out using the previous obtained coordinates for the curvature to determine the radiation pattern \"very close to the transducer surface\". The radiation measured at this region would closely represent the vibration distribution of the curved piezoelectric disk, assuming the effect of other regions of the radiating surface is negligible. For this, a wideband needle hydrophone PZTZ44-0400 (Onda Corporation, Sunnyvale, CA, USA) with a 40 \(\upmu\)m aperture and sensitivity of \(-260\) dB referred to 1 V/\(\upmu\)Pa was mounted on the positioning system, replacing the multimeter probe. The measurements were made at the positions determined previously but 1 mm in front of the transducer surface (in \(z\)-direction). The transducer was driven with a wave generator (Array 3400, Array Electronic Co., Taiwan, China) using s 20 Vpp sine tone-burst; the received data were recorded in a PC. The ultrasound signals were recorded for 1, 5, 10, and 20 sine cycles to determine the radiation pattern under different excitation conditions but, more specifically, to know in which number of cycles the vibration presents a quasi-stationary pattern. The data were post-processed in MATLAB (R2017a, MathWorks, Natick, MA, USA) to determine the peak-to-peak values at each spatial point and the full radiation distribution for each excitation condition. Figure 1: The setup for the curvature/acoustic pressure measurements. The third set of measurements was performed to determine the full acoustic fields of the focused transducer. This was measured using the previously described setup. The acoustic fields were obtained by saving only the peak-to-peak acoustic pressure at each point in the radiated volume. It was recorded on an XZ plane covering the transducer dimensions starting at \(z=2\) mm and finishing at \(z=50\) mm from the transducer case, with a step resolution of 0.3312 mm in the \(x\)-direction and 0.5000 mm in the \(z\)-direction. The XY planes were measured at two depths, 0.2 cm and 1.7 cm (at the focus), using a step resolution of 0.3312 mm in both directions; these planes were obtained to verify the symmetry of the acoustic radiation.