## 3 Integration of an Acoustic Resonator with a Multi-Pass Cell  
Different shapes of resonant cavities exhibit different distributions of sound pressure. In this work, considering the integration of the acoustic resonator and the multi-pass cell, we chose to construct a relatively simple cylindrical resonant cavity (with two window elements at the two ends in Figure 3 below). From Formulas (2) and (3), it can be concluded that both light power and cavity size can affect the amplitude of sound pressure. For a photoacoustic system based on a multi-pass cell, the light power is determined by factors such as the laser power and the total number of reflections of the laser beam. Increasing the total number of reflections will lead to higher optical power. Regarding the dimensions of the resonant cavity, from Formula (3) one sees that the cell constant of a cylindrical resonant cavity is inversely proportional to its cross-sectional area. To obtain stronger photoacoustic signals, it is essential to maximize the total number of passes within a smaller resonant cavity. The total number of reflections and the radius of the photoacoustic cell should be optimized when designing a multi-pass photoacoustic system. Figure 1 shows the relationship between the total light power and the total number of reflections of the laser beam in a multi-pass cell when the loss of each window element is 0.3% (i.e., the transmission coefficient is 99.7% for each of the two window elements). As the total number of reflections increases, the total light power continues to rise. However, due to the presence of losses, the rate of the increase of the total light power progressively slows down. Excessive reflection counts will diminish the gain achieved with each reflection. Figure 1 indicates that the reflectivity of the mirror has a significant impact on the system performance. Higher reflectivity means less loss and stronger gain. In the experiment, considering the cost factor, we chose silver-coated mirrors to build the multi-pass cell (the reflectivity of the silver-coated mirrors is about 95%). If a narrowband mirror of higher reflectivity is chosen, the system performance can be further improved. Certainly, the ability of higher reflectivity to enhance the system performance is limited by the losses of window elements and scattering. A multi-pass cell composed of two silver-coated mirrors (M1 and M2) with a diameter of 25.4 mm and a radius of curvature of 50.8 mm was designed to form a single-line spot pattern. The incident position was set near the edge of the mirror M1. With the distance of the two mirrors and the incident angle setting as 102 mm and 14.6\({}^{\circ}\), the laser beam was reflected 21 times in a limited area at the center of the multi-pass cell. To visually demonstrate the differences between the designed multi-pass cell and the traditional Herriot cell, we also designed a traditional Herriot cell which has the same size as the designed multi-pass cell. Simulations for both designs were then conducted using ZEMAX (R1.03). Figure 2 shows the optical path diagram of the designed multi-pass cell and the traditional Herriot cell. In the structure shown in Figure 2a, the light spots are distributed in a linear pattern, whereas in the traditional Herriot cell shown in Figure 2b, the light spots exhibit a ring-shaped distribution. As can be seen from Figure 2, compared to the traditional Herriot cell, our multi-pass cell has a more concentrated light distribution. This allows us to make the diameter of the resonant cavity smaller, often implying better performance. The proposed configuration of MPC combined with a compact acoustic resonator with a radius of 8 mm and a length of 36 mm results in high instrument performance. Figure 3a shows the structure and light path of the multi-pass cell and the corresponding photoacoustic cell.