# Trace CH\({}_{4}\) Gas Detection Based on an Integrated Spherical Photoacoustic Cell
## 2 SPAC Design and Theoretical Analysis
\(j_{l}(r)\) is the first-order spherical Bessel function, \(P_{l}^{m}(x)\) is a joint Legendre function, \(k_{l,n}\) is the n-th root to satisfy the boundary conditions, which can be expressed as:  
\[\frac{\partial j_{l}(kr)}{\partial r}\Big{|}_{r=R}=0, \tag{5}\]  
\(P_{j}(r)\) describes the spatial distribution of acoustic pressure in the spherical PA cavity of mode \(j\). The Fourier transform of Equation (1) can be obtained:  
\[\left(\
abla^{2}+\frac{\omega^{2}}{c^{2}}\right)P(r,\omega)=\frac{\gamma-1}{ c^{2}}i\omega H(r,\omega), \tag{6}\]  
where \(\omega\) is the frequency of the modulated excitation light source, and the acoustic pressure changes are expanded according to each vibration mode:  
\[P(r,\omega)=\sum_{j}A_{j}(\omega)P_{j}(r), \tag{7}\]  
put in Equation (5) to get the expression of \(A_{j}(\omega)\) as:  
Figure 1:0  
In Equation (8), \(\omega_{j}\) is the resonant angular frequency, \(Q\) is the quality factor, \(V_{c}\) is the volume of the PA cavity, \(P_{L}\) is the optical power of the incident laser, the overlap integral in the equation represents the coupling degree of the light intensity distribution and the normal mode, and \(g\) is the light intensity distribution function. According to Equations (3) and (4), the spatial distribution and resonance frequency of the mode (100) and the mode (120) can be obtained as follows [39]:  
Figure 1:1  
Figure 1:2Figure 1:3  
Figure 1:4  
From Equation (9), the acoustic wave in mode (100) only has a radial distribution, and is called the first-order radial resonant mode. This mode is characterized by the acoustic wave propagating along the radial direction, with the maximum acoustic pressure at the center of the cell and a node between the center of the cell and the edge. In addition to the component of radial propagation, the acoustic wave in the SPAS also has an angular component. The mode of (120) is called the second-order angular resonant mode. As shown in Equation (11), in the mode (120), the acoustic wave propagates along the sphere at the boundary of the cell. The acoustic pressure in the radial part is zero at the center of the cell and reaches its maximum at the edge. In the angular part, the acoustic pressure reaches its maximum at 0\({}^{\circ}\), 90\({}^{\circ}\) and 180\({}^{\circ}\), and there is a node in the middle. For our PA cell, according to Equations (10) and (12), it can be easily calculated that the first-order radial resonant frequency and the second-order angular resonant frequency are 24,480 Hz and 18,020 Hz, respectively. A 3D model based on the finite element method was created to make an assay of the PA field distributions of the SPAC. In our numerical calculations, the heat source domain feature was used to replace the heat generated by the high-frequency pulsed laser. The user control mesh was selected. The maximum cell size was 0.8 mm, the minimum cell size was 0.375 mm, the maximum cell growth rate was 1.3, the curvature factor was 0.3, and the narrow area resolution was 5. The results converged completely. The acoustic field characteristics inside the PA cell are the key to the sensitivity and the detection limit of the PAS trace gas detection system, so 16,000-28,000 Hz with steps of 10 Hz were taken as the research range of the simulation frequency, which is shown in Figure 1:5 There are two peaks of 18,250 Hz and 24,540 Hz, which correspond to mode (120) and mode (100), and are in good agreement with the formula analysis results. Figure 4 presents the acoustic pressure distribution of the SPAC at the second-order angular resonance frequency of 18,250 Hz. As can be seen in Figure 4, the acoustic pressure is zero at the center of the cell and reaches its maximum at the edge. In the angular part, the acoustic pressure reaches its maximum at 0\({}^{\circ}\), 90\({}^{\circ}\) and 180\({}^{\circ}\), and there is a node in the middle. Figure 5 shows the acoustic pressure distribution of the SPAC at the first-order radial resonant frequency of 24,540 Hz.