## 3 Simulation Study
Therefore, it can be inferred from Figure 5c that, when the thermal effect propagated to these points, the thermally induced displacement far exceeded the displacement caused by photoacoustic pressure, within 1 us vibrations, which was caused by the photoacoustic pressure occurring in the curves by first spreading to the negative direction of the \(Z\)-axis and then to the positive direction. However, after 1 us, the displacement increased in the positive direction, because the thermal effect spread to this point, and the displacement disturbance caused by the thermal effect was greater than that caused by sound pressure, though the thermal diffusivity in silicon was 8.8 \(\times\) 10\({}^{-5}\) slower than the propagation speed of photoacoustic waves. Since the spot size at the 1/e\({}^{2}\) was 100 um, as can be seen from Figure 4b, in the 1/e\({}^{2}\) area of the laser spot range, the temperature changing trend at each point was similar to that of the center point (0, 0, 1 mm), while at the edge of the laser spot, the laser-induced heat was relatively low, so the temperature change range was small. Within the radius of 0.3 mm from the center point (0, 0, 1 mm), the overall trend of temperature change was usually unchanged, that is, the temperature reached the maximum value within a certain time delay when the laser pulse reached the maximum value in the time domain, and then slowly decayed. When  
Figure 4: The temperature changes vs. time in different points on the silicon surface in model 1.  
it exceeded the area outside the laser spot range, the temperature change was relatively small, \(<\) 1 K, because the thermal effect propagation was relatively slow, as can be seen from Table 1; the thermal conductivity of silicon at 293 K was 1.38 W/(m \(\cdot\) K). From Equation (5), it can be seen that the temperature or thermal effect caused a corresponding change in displacement, meaning that they had a proportional relationship. Although photoacoustic signals also caused a corresponding displacement, the displacement caused by the thermal effect was far greater than that caused by photoacoustic pressure within the size range of the laser spot. Therefore, the changing trend of displacement was mainly dominated by the thermal effect within the range of point 0.1 mm. Thus, when using solid mechanics to observe photoacoustic signals in COMSOL, the observation points were all outside the range of the laser spot, and therefore, the influence of temperature on displacement was excluded. However, because of their fast propagation speed, sound waves were observed in a small range of 0.2 mm, but as time passed, the heat also reached this point, indicating that the displacement of the initial part of the curve in Figure 5c strengthened instead. Figure 5: The displacement changes vs. time in different points on the silicon surface on model 1. However, the incident intensity was too high for silicon, and the maximum temperature of the center point (0, 0, 1 mm) reached 1050 K, which is close to 1273.15 K, the melting point of silicon. Therefore, based on model 1, the incident intensity was reduced by 10 times, and then model 2 was constructed. The results of the distribution of temperature and displacement at each point are shown in Figures 6 and 7, respectively. By comparing Figures 4 and 6, and Figures 5 and 7, it can be found that the overall changing trend of temperature or displacement at each point is similar. As shown in Figure 6a, the maximum temperature at the center point was 375 K, which is much lower than the melting point of silicon and, therefore, suitable for real-life applications. The displacement value is reflected in the simulation, but it is characterized by the light intensity value in the experiment. To achieve consistency, the relationship between the measured reflected intensity according to the displacement is deduced below. Figure 6: The temperature changes vs. time in different points on the silicon surface in model 2.