## 2 Theory  
To fundamentally understand the reason for the non-linear relationship between the absorption signal and gas concentration, the reflection process for a light beam passing through a diffuse integrating cavity was analyzed theoretically. In general, an ideal arbitrary integrating cavity is defined as a cavity in which any two points on the inner wall can be connected to a segment inside the cavity. The inner wall of an integrating cavity is covered with a highly diffuse reflective coating, which can be considered as an ideal Lambertian radiation source. Thus, an incident beam could be diffused continuously. A simplified model of the beam transmission process is shown in Figure 1. The input light radiation flux is denoted by \(\Phi_{0}\), and the first light path, \(L_{0}\), is the distance between the entrance aperture and the first reflection spot. Neglecting the radiation loss through the entrance and detection windows, the radiant flux after the first reflection can be expressed by Equation (1), according to the Beer-Lambert law:  
\[\Phi_{1}=\rho\Phi_{0}\exp(-\alpha L_{0}). \tag{1}\]  
Here, \(\rho\) is the reflectivity of the inner surface, \(\alpha\) is the absorption coefficient, which is equal to the product of the particle number density \(N\) and the absorption cross section \(\sigma\). A single-pass average path length of \(L_{\text{ave}}\) was introduced to describe the distance between successive light reflections [21]. If the port fraction of the cavity is \(f\), then light travels a distance \(L_{\text{ave}}\) and a portion exits from the output aperture. Light exiting the cavity is received by a detector:  
\[\Phi_{\text{1 out}}=f\rho\Phi_{0}\exp(-\alpha(L_{0}+L_{\text{ave}})). \tag{2}\]  
The remaining radiation flux is  
\[{\Phi_{1}}^{\prime}=(1-f)\rho\Phi_{0}\exp(-\alpha(L_{0}+L_{\text{ave}})). \tag{3}\]  
This light radiation is still transmitted in the integrating cavity and part of this light outputs again after traveling \(L_{\text{ave}}\). The rest is reflected by the internal wall again. This process repeats continuously. Table 1 shows the output radiation after each reflection from an arbitrary integrating cavity. The total light radiation, \(\Phi_{\text{out}}\), received by the detector is proportional to the sum of the output radiation:  
\[\Phi_{\text{out}}=\sum_{1}^{n}\Phi_{\text{nout}}=\ f\Phi_{0}\exp(-\alpha(L_{0}+ L_{\text{ave}}))\frac{1}{1/\rho-(1-f)\exp(-\alpha L_{\text{ave}})}. \tag{4}\]  
The absorbance, \(A\), can be expressed as:  
\[A=-\ln(\frac{\Phi_{\text{out}}}{\Phi_{0}})=-\ln[f\exp(-\alpha(L_{0}+L_{\text{ave }}))\frac{1}{1/\rho-(1-f)\exp(-\alpha L_{\text{ave}})}]. \tag{5}\]  
We found that the absorbance, \(A\), is a complicated expression here. However, a linear relationship between absorbance and gas concentration was expected in gas detection for easy calibration. Thus, some approximations should be introduced to express \(\Phi_{\text{out}}\) in the form of the Beer-Lambert law:  
\[\alpha L_{\text{ave}}<<1,\ \alpha L_{0}<<1 \tag{6}\]  
\[\alpha\bigg{(}L_{0}+\frac{1}{1-\rho(1-f)}L_{\text{ave}}\bigg{)}<<1. \tag{7}\] \begin{table}
\begin{tabular}{c c} \hline
**Reflection Multiplier** & **Output Radiation from the Aperture** \\ \hline
1 & \(f\rho\Phi_{0}\exp(-\alpha(L_{0}+L_{\text{ave}}))\) \\
2 & \(f(1-f)\rho^{2}\Phi_{0}\exp(-\alpha(L_{0}+2L_{\text{ave}}))\) \\
3 & \(f(1-f)^{2}\rho^{2}\Phi_{0}\exp(-\alpha(L_{0}+3L_{\text{ave}}))\) \\... &... \\ \(n\) & \(f(1-f)^{n-1}\rho^{n}\Phi_{0}\exp(-\alpha(L_{0}+nL_{\text{ave}}))\) \\ \hline \end{tabular}
\end{table} 
Table 1: Output light radiation after each reflection from an arbitrary diffuse integrating cavity. Figure 1: Simplified model of the radiation transmission process in an arbitrary integrating cavity. It was found that Equation (7) is a stricter condition compared to Equation (6). Once Equation (7) is satisfied, Equation (6) must be followed.