## 2 TDLAS-WMS Detection Theory
### Single Optical Path Time Division Multiplexing Differential Modulation  
We can tune the laser output wavelength by modulating the drive current based on WMS [22]. A stair-stepping segmented low-frequency signal (10 Hz) was used to modulate the laser's injection current to further eliminate the fluctuation of laser light intensity on the single-light path detection system. As shown in Figure 2, the stair-stepping segmented low-frequency signal (10 Hz) is divided into L, M, and H segments. The M segment is a sawtooth signal. This segment is used for C\({}_{2}\)H\({}_{2}\) concentration detection. The laser injection current is modulated by superimposing a high-frequency sine signal (5 kHz). Thus, the laser's output wavelength sweeps the absorption peak of C\({}_{2}\)H\({}_{2}\). The currents of L and H segments are constant, and the laser's output wavelength driven by the currents is in the non-absorption region of the C\({}_{2}\)H\({}_{2}\) molecule. The laser's mean value distribution can be considered in the short single period detection time of 100 ms at 10 Hz because its intensity fluctuation is mainly derived from its own background noise. Therefore, the average light intensity corresponding to the L and H segment waveforms is obtained, and the data processing is performed  
Figure 1: C\({}_{2}\)H\({}_{2}\) (blue) absorption line in the range of 1520.05–1520.15 nm. The absorption line strength of CO\({}_{2}\), H\({}_{2}\)O, and CH\({}_{4}\) is much smaller than that of C\({}_{2}\)H\({}_{2}\). as the inversion basis of the light intensity \(I_{0M}\) of the M segment signal, which further eliminates the interference caused by the laser light intensity fluctuation in the single optical path detection system. After modulation, the output frequency \(v\) and the output intensity \(I_{0}\)\({}^{\prime}\) are obtained as follows. \[v =v_{0}+v_{m}\cos\omega t \tag{3}\] \[I_{0}\)\({}^{\prime} =I_{0}(1+\eta\cos\omega t) \tag{4}\]  
where \(v_{0}\) is the center frequency of the laser's output, \(v_{m}\) is the modulated SA, \(\eta\) is the light intensity modulation index, and \(f=\omega/2\pi\) is the modulation frequency. According to Equation (4) and Equation (1), the light intensity with gas absorption is the following. \[I(v,t)=I_{0}(1+\eta\cos\omega t)e^{[-\alpha(v)CL]} \tag{5}\]  
The actual gas absorption and light intensity modulation are extremely small, i.e., \(\alpha(v)<<1\) and \(\eta<<1\). The Taylor series expansion is performed, according to Equation (5). The light intensity is obtained, ignoring the high-order terms, as follows. \[I(v,t)=I_{0}[1+\eta\cos\omega t-\alpha(v_{0}+v_{m}\cos\omega t)]CL \tag{6}\]  
From Equation (2), the center frequency \(v_{0}\) of the laser is adjusted to coincide with the gas absorption peak, i.e., \(v_{0}=v_{cm}\). Then, the light intensity with gas absorption is obtained as follows. \[I(v,t)=I_{0}(1+\eta\cos\omega t-\frac{\alpha_{0}CL}{1+m^{2}\cos^{2}\omega t}) \tag{7}\]  
\[m=\frac{v_{m}}{\Delta v} \tag{8}\]  
The detection signals \(H_{L}(t)\), \(H_{M}(t)\), and \(H_{H}(t)\) of the photodetector can be expressed as follows. \[H_{L}(t)=spl_{0L}(1+\eta\cos\omega t) \tag{9}\]  
\[H_{M}(t)=spl_{0M}(1+\eta\cos\omega t-\frac{\alpha_{0}CL}{1+m^{2}\cos^{2}\omega t}) \tag{10}\]  
\[H_{H}(t)=spl_{0H}(1+\eta\cos\omega t) \tag{11}\]  
where \(s\) is the conversion factor of the photodetector, \(p\) is the amplification factor of the detection circuit, and \(I_{0L}\), \(I_{0M}\), \(I_{0H}\), and are the incident light intensity corresponding to the L, M, and H segments of  
\[v =v_{0}+v_{m}\cos\omega t \tag{3}\]0 Waveform of stair-stepping segmented low-frequency signal. the low-frequency signal that satisfy \(I_{0M}=\frac{I_{0}+I_{0M}}{2}\). Differential signal \(H(t)=\frac{H_{L}(t)+H_{H}(t)}{2}-H_{M}(t)\) is obtained as follows. \[v =v_{0}+v_{m}\cos\omega t \tag{3}\]1  
The Fourier series expansion is performed on Equation (12) to obtain the second harmonic component coefficient \(H_{2f}\) as follows.