## 2 Methods
### Multiple Delay and Sum with Enveloping
Notably, due to the characteristic time of thermal diffusion being much longer than the laser pulse width, the heat is almost completely confined within the absorber during the expansion stage. This confined thermal elastic expansion generates a significant initial pressure rise within the absorber, and the resulting pressure disturbance propagates to the surrounding medium in the form of ultrasound. As time evolves (usually in the nanosecond-to-millisecond range), the heat begins to transfer to the surrounding medium through the thermal diffusion mechanism, which follows [PERSON]'s law of heat conduction. As heat is lost, the absorber temperature gradually decreases and triggers a contraction effect, thereby generating a negative pressure signal with opposite polarity to the initial expansion stage. This dynamic process of first expansion and then contraction ultimately produces characteristic N-shaped bipolar pressure pulses. Moreover, the distance between the signal peak and the signal valley is proportional to the size of the absorber. Shown in Figure 2 are the theoretical data of the photoacoustic signal of a spherical absorber. Based on the aforementioned descriptions, it is evident that the photoacoustic signal generated by an absorber is not a single-valued signal. If one focuses only on the instantaneous data received by the sensor at a certain moment during the imaging process, this is obviously not comprehensive enough. Given the impact of factors such as the sampling frequency of the sensor data, treating them as a single-valued signal during analysis may potentially introduce certain errors. Therefore, we need to analyze the complete signal of each imaging point throughout the whole period. Figure 1: This picture illustrates the interaction between the target point and its neighbors when using the DAS algorithm for imaging. By incorporating a time index into Equation (1), we derive Equation (3). It is defined as follows:  
\[Y_{multi-DAS}(x,y;t)=\sum_{i=1}^{N}S(i,t+\tau(x,y;i)) \tag{3}\]  
where \(Y_{multi-DAS}(x,y;t)\) represents the sum of the data returned by each sensor to the pixel \((x,y)\) at time \(t\). Equation (3) demonstrates that for any position \((x,y)\) within the imaging region, the complete beamforming signal across the entire imaging period can be reconstructed through multiple DAS operations. \(S(i;t)\) is the signal received by the \(i\)-th sensor at time \(t\), \(\tau(x,y;i)\) is defined in the same way as in Equation (2). From Equation (3), we can see that when \(t=0\), the output of \(Y_{multi-DAS}(x,y;t)\) is the same as that of \(Y_{DAS}(x,y)\). This also indicates that the DAS algorithm only computes the signal at time \(t=0\). \[Y_{multi-DAS}(x,y;0)=\sum_{i=1}^{N}S(i,\tau(x,y;i))=Y_{DAS}(x,y). \tag{4}\]  
According to Equation (3), the complete signal for each pixel during the whole period can be acquired. This signal corresponds to the photoacoustic signal emitted by the pixel point throughout the imaging period, and the photoacoustic signals generated at different positions are different. We set the following experimental conditions within the imaging area: Firstly, at the center position of the 21.6 mm \(\times\) 21.6 mm square imaging area, a uniform circular sound source model with a radius of 0.2 mm was placed, as shown in Figure 3a. We selected two pixel points for comparison analysis--the first point was located at the center of the sound source (the true source point), and the second point was located 1.2 mm to the right of the center, outside the sound source (a non-source point). Through Equation (3), the complete signals of the two points were obtained, respectively, as shown by the black line in Figure 3b,c. Figure 2: Theoretical photoacoustic signal of spherical absorbers. The complete photoacoustic signals acquired from the two pixel points both exhibit nearly N-shaped bipolar characteristics. However, there are also some obvious differences in signal amplitude and phase between the two.