## 3 Method  
The 2f-SR method includes three steps: gas temperature measurement based on the temperature characteristics and strength of lines, 2f signal recovery, and fast multi-line fitting. The principle of line-of-sight gas temperature measurement is now introduced in detail. The underlying principle of gas concentration measurement based on infrared absorption spectroscopy is the Beer-Lambert law [11], which can be expressed as follows:  
\[I(v)=I_{0}(v)\exp\left[a(v)LC\right] \tag{1}\]  
where \(I_{0}\) is the initial light intensity; \(I\) is the transmitted light intensity after it is absorbed by the gas; \(a(v)\) is the absorption coefficient, which is the product of absorption line strength \(S\) and lineshape function \(g(v)\); \(C\) is the molecule number density; and \(L\) is the effective absorption path length. The line strength and full width at half-maximum (FWHM) of an absorption line is greatly affected by gas temperature. Different absorption lines have different temperature characteristics, and gas temperature can be accurately measured based on this principle. Given that WMS can reduce system noise by two to three orders of magnitude, 2f signal can be used to improve system performance. Gas temperature measurement is expressed as follows [22]:  
\[\delta T=(\Delta E\cdot\mathrm{hc}/\mathrm{k})/[\mathrm{ln}R_{2f}+\mathrm{ln}R (T_{0})+\mathrm{ln}G] \tag{2}\]  
where \(G\) is the constant related to the system, \(R(T_{0})\) is the ratio of the absorption line strength at \(T_{0}\), \(R_{2f}\) is the amplitude ratio of 2f signals, and \(\Delta\)E is the ground state energy level difference. A pair of well-isolated absorption lines is selected to enable real-time measurement of the gas temperature. The absorption spectra of 100 ppmv CH\({}_{4}\) and 1% H\({}_{2}\)O at a 40 Torr gas pressure, a 298 K gas temperature, and a 41.5 m effective optical path length is presented in Figure 1. The selected absorption lines are not disturbed by other gases and can be well separated. The relationship between gas temperature and the ratio of line strength of the selected absorption lines is expressed as:  
\[T=-3\cdot R_{2f}+116.61\]  
Figure 1: Overlapping absorption lines at 4294.570 cm\({}^{-1}\) used in the experiment and temperature measurement absorption line. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Absorption Line (cm\({}^{-1}\))** & **Percentage at 298 K (\%)** & **Percentage at 297.5 K (\%)** \\ \hline
4294.570 & 98.525 & 96.015 \\
4294.556 & 1.470 & 3.858 \\
4294.591 & 0.004 & 0.125 \\
4294.595 & 0.001 & 0.002 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The effect of interference absorption lines at 297.5 K and 298 K.  
The principle of 2f signal recovery is now introduced. WMS adheres to the following principle: a low-frequency triangle signal used to scan over the target absorption line is modulated by a high-frequency sinusoidal signal. The 2f signal containing the gas concentration is extracted using a lock-in amplifier. WMS can suppress 1/f noise and improve the gas concentration measurement accuracy. Given the effects of laser characteristics on the 2f signal, the 2f signal is analyzed in detail in combination with the laser modulation parameters. The luminous intensity of the laser is proportional to the driving current and can be expressed as follows:  
\[I(t)=I_{0}(v_{1})\left[1+K_{1}\delta_{v}\text{cos}\left(2\pi f_{\text{m}}t+ \varphi_{1}\right)+K_{2}[\delta_{v}]^{2}\text{cos}\left(2\pi\cdot 2f_{\text{m}}t+ \varphi_{2}\right)\right] \tag{4}\]  
where \(I_{0}(v_{1})\) is the average laser strength at \(v_{1}\),\(f_{\text{m}}\) is the modulation frequency, \(K_{1}\) and \(K_{2}\) are the linear and nonlinear modulation coefficients, respectively, \(\varphi_{1}\) and \(\varphi_{2}\) are the linear and nonlinear phase shifts of intensity modulation, respectively, and \(\delta_{v}\) is the amplitude modulation. In any optical system, there is an etalon caused by the reflection of the optical surface.