# Trace CH\({}_{4}\) Gas Detection Based on an Integrated Spherical Photoacoustic Cell
## 2 SPAC Design and Theoretical Analysis  
The schematic diagram of the integrated SPAC is shown in Figure 1. The whole device is divided into two parts. The first part of the SPAC includes a hemispheric PA cavity, two vents and two air taps. The second part consists of the other hemispheric PA cavity, a collimator, a ceramic ferrule, a cantilever and two single-mode fibers. The two parts are coupled to each other through an intermediate thread to form the spherical PA cavity. The inner diameter of the PA cell is 20 mm. The outer diameter of the cylindrical brass shell is 30 mm. The single-mode fiber connected with a DBF laser is coupled to the SPAC by a collimator. A ceramic ferrule located in the center of the brass shell is used to transmit the detection light of the fiber-optic cantilever-based acoustic sensor. The length and width of the cantilever beam are 2.8 mm and 1.0 mm. Two small vents with a radius of 0.25 mm, which are connected to two air taps, are opened inside the brass shell for the exchange of measured gas. In order to ensure the air tightness of the device in the experiment, the joint of the two parts is sealed using a silastic sealed ring. The frequency response curve of the cantilever-based acoustic sensor is tested based on the WLI demodulation algorithm. As shown in Figure 2, the sensor has the highest sensitivity at a resonant frequency of 1290 Hz, which is the first-order resonant frequency of the cantilever beam. In a PAS system, when the gas molecules absorb light energy transitioning from the ground state to the excited state, part of it is converted into translational energy of the gas, and then expressed as heat energy to increase the temperature of the gas in the cell. Since the volume of the PA cell is fixed, the increased temperature will lead to an increase in pressure, and then acoustic signals will be generated under the action of periodic excitation of the light source. In the case of a weak absorption, the thermal power density generated by the incident light in the gas is \(H\left(\overset{\rightarrow}{r},t\right)=\tau/\tau_{c}\left[\beta I\left( \overset{\rightarrow}{r},t\right)\right]\), where \(\beta\) is the absorption coefficient of the gas, \(I(r,t)\) is the intensity distribution of the incident light, \(\tau\) is the vibrational relaxation time of gas molecules in the PA cell, \(\tau_{c}\) is the vibration-translational relaxation time of gas molecules in the PA cell, and \(H\) has a linear relationship with \(\beta\) and \(I\). Figure 1: The structure of the SPAC. Figure 2: Frequency response of the cantilever-based acoustic sensor. For a SPAC, the change in the gas acoustic pressure in the cell \(p\left(\overset{\rightarrow}{r},t\right)\) is described by the acoustic equation [39]:  
\[\
abla^{2}p\left(\overset{\rightarrow}{r},t\right)-\frac{1}{c^{2}}\frac{\partial ^{2}p\left(\overset{\rightarrow}{r},t\right)}{\partial t^{2}}=-\frac{\gamma-1} {c^{2}}\frac{\partial H\left(\overset{\rightarrow}{r},t\right)}{\partial t}, \tag{1}\]  
where \(\overset{\rightarrow}{r}\) is the position vector, \(c\) is the acoustic velocity in the PA cell, \(p\) is the acoustic pressure in the cavity, and \(\gamma\) is the specific heat capacity ratio (the ratio of specific heat capacity at constant pressure to specific heat capacity at constant volume). The wave equation satisfies the boundary condition on the cell wall:  
\[\frac{\partial p\left(\overset{\rightarrow}{r},t\right)}{\partial n}=0, \tag{2}\]  
where \(n\) is the normal vector at the boundary of the spherical cavity. For a standard spherical cavity, the resonant acoustic mode \(p_{j}\left(\overset{\rightarrow}{r}\right)\) (expressed in spherical coordinates) and the resonant frequency \(f_{j}\) can be solved as:  
\[p_{j}\left(\overset{\rightarrow}{r}\right)=p_{nlm}(r,\theta,\varphi)=j_{l}(k _{l,n}r)P_{l}^{m}(cos\theta)e^{im\varphi}, \tag{3}\]  
\[f_{j}=k_{l,n}c/2\pi R, \tag{4}\]  
where \(j=(nlm)\), \(n\), \(l\), \(m\) are the order of radial, theta angle and \(\varphi\) angle direction acoustic modes, respectively.