## 2 Theoretical Model and Simulation  
### Numerical Simulation  
Both the finite element method (FEM) and finite-difference time-domain (FDTD) method can be used to model the process of light-excited acoustic waves. In terms of computational speed, FDTD has a great advantage. However, considering the complexity of photoacoustic imaging, the FEM method was selected as it is more versatile and rigorous. Three configurations for exciting photoacoustic waves were simulated, namely single-pulse single-point excitation (Model 1), single-pulse laser array excitation (Model 2), and laser array with time-delayed excitation between spots (Model 3). Except for the number of excitation light sources and the time delay between the spots, other parameters in the simulation remained unchanged, including the sample size, material, energy density, and pulse duration. The modeled sample was a homogeneous, isotropic cylindrical aluminum block with a height of 4 mm and a radius of 4 mm. The laser array was incident perpendicularly to the top surface of the cylinder. Material parameters of aluminum from reference [27] were used for the simulation, with a surface wave sound velocity of 2940 m/s, a transverse wave sound velocity of 3080 m/s, and a longitudinal wave sound velocity of 6320 m/s. A laser source was used to generate the ultrasound, with a pulse width of 20 ns, pulse energy of 160 uJ, and energy density of 510 mJ/cm\({}^{2}\). The laser spot radius on the sample surface was 100 um, and it was assumed that 80% of the light was absorbed. Figure 1 shows the aluminum cylinder in the Cartesian coordinate system, with three laser spots A, B, and C focused on the top surface. The simulation used the solid heat transfer module and the solid mechanics module of COMSOL Multiphysics to simulate in the transient field  
Figure 1: Cylindrical aluminum block and the three laser illumination spots A, B, and C on its top surface. Model 1 and Model 2 were used as reference simulations to compare with Model 3. In Model 1, a single excitation pulse was used to illuminate the single spot \"B\". Model 2 was based on Model 1, with excitation spots \"A\" and \"C\" added on either side of the single laser spot B. The size and energy density of these three laser spots were the same as for Model 1. As shown in Figure 1, through pre-simulation results, the distance between adjacent laser sources was selected as 0.9 mm, which is about one wavelength of the excited acoustic wave. Model 3 was based on Model 2 with a time delay imposed on the central spot B. It can be seen from the geometric properties of the propagation of acoustic waves in Figure 2a that as the waves propagate, the difference between d\({}_{1}\) and d\({}_{2}\) becomes smaller. When the distance between adjacent sources is much smaller than d\({}_{1}\), and the wavefront has traveled sufficiently far, \(\delta_{\text{d}}=\text{d}_{2}-\text{d}_{1}<\epsilon\times\text{v}\). \(\epsilon\) is defined as the allowable time error, which is chosen to be as same as the simulation time step, i.e., 0.01 us, v is the velocity of the excited acoustic waves. During this time, the acoustic wave travels about one-tenth of the acoustic wavelength. The new merged wavefront due to superposition can be considered as a continuous wavefront that can propagate in an infinite medium. The distance Y at which the superposed acoustic waves add constructively on the \(y\)-axis can be calculated from  
\[\text{Y}^{2}+\text{L}^{2}=(\text{Y}+\epsilon\times\text{v})^{2} \tag{2}\]  
where L is the distance between adjacent sources, and \(\epsilon\times\text{v}\) is the maximum acceptable error. In Model 2, the distance Y for acoustic waves without a time delay was calculated to be 13.5 mm, which is larger than the size of the sample. The corresponding traveling time is about 4.5 us, calculated from the velocity of the surface acoustic wave. Furthermore, using this arrangement in practical applications would make it difficult to image features such as defects or impurities within the sample.