## 2 Principle  
Quantitative spectral analysis is based on the Beer-Lambert law, which gives the relationship between the incident and the transmitted radiation through a gas cell filled with a molecular gas sample:  
\[I(\
u)=I_{0}(\
u)\times exp\{-\sigma(\
u)\times L\times C\}, \tag{1}\]  
where \(I_{0}\) and \(I\) are the incident and transmitted radiant powers, respectively; \(\sigma\) is absorption cross section of the molecule in cm\({}^{2}\)/molecule; \(L\) is absorption pathlength in cm; \(C\) is the density of the molecule in molecule/cm\({}^{3}\). Usually, the absorption cross section \(\sigma\) is also used to describe the absorption intensity. The line strength is retrieved by spectrally integrating the absorption line shape and applying the ideal gas law:  
\[S(T)=\frac{K_{B}TA}{X_{i}LP\;r_{iso}}, \tag{2}\]  
where \(K_{B}\), \(T\) (\(K\)), and \(P\) (\(Pa\)) are the Boltzmann constant, gas temperature, and total pressure of the gas sample, respectively; \(X_{i}\) is the amount fraction of \(i\) species; \(A\) (cm\({}^{-1}\)) is integral absorbance; \(r_{iso}\) is a correction factor for isotopic fractionation of the gas sample.  
WMS is commonly used to improve the sensitivity of gas sensing. The WMS theory and signal model have been detailed previously [28], and so are only briefly reviewed here. A periodic sawtooth ramp ridden by a high-frequency sinusoidal is applied to the laser injection current, thus the laser wavenumber \(\
u(t)=\
u_{c}+\
u_{a}\cos\omega t\) is scanned across the transition of gas to be detected, where \(\
u_{c}\) and \(\
u_{a}\), are the laser center wavenumber and modulation depth, respectively; \(\omega\) is the radian frequency. In case of ideal conditions, ignoring all kinds of interference, the modulated absorption signal is detected by a photodiode and then processed using a lock-in amplifier to demodulate the signal at the harmonics (1\(f\), 2\(f\), 3\(f\), etc.). The second harmonic component (WMS-2\(f\)) is commonly used for calculating the concentration of target gas. In the case of optically thin (\(\sigma(\
u)\cdot L\cdot C\leq 0.05\)), the ideal 2\(f\) signal is modeled as:  
\[A_{ideal\;2f}=\frac{2I_{0}CL}{\pi}\int_{0}^{\pi}-\alpha(\
u_{c}+\
u_{a}cos \theta)cos2\theta d\theta\propto I_{0}CL, \tag{3}\]  
where \(\alpha\) is the absorption coefficient and \(\theta=\omega t\) is the phase angle. When the incident laser intensity \(I_{0}\) and optical path \(L\) are constant, the amplitude of WMS-2\(f\) signal is proportional to the gas concentration. Practically, apart from the 2\(f\) signal descript in Equation (3), the detected signal consists of random noise and the derivation of optical fringes [29; 30]. The optical fringes appear as unpleasant spectral features that are usually mixed with the target absorption, and constitute one of the major obstacles in the gas detection. In a well-designed and -fabricated system, the optical fringes should be reduced, and only small residual fringes remain with sinusoidal waveforms, while random noise is seen as small time-varying wiggles superimposed on the true underlying signal, with small standard deviation. Thus, the detected signal could be described as:  
\[A_{\text{detected 2$f$}}=\text{e}_{n}+\sum a_{j}(t)\times\cos(\omega_{j}(t) \times t)+A_{\text{ideal 2$f$}}, \tag{4}\]  
where \(\alpha_{j}(t)\) and \(\omega_{j}(t)\) are the instantaneous amplitude and frequency of \(j\)th fringe component, respectively; \(A_{\text{ideal 2$f$}}\) is the WMS-2\(f\) signal modeled by Equation (3). The profiles of second harmonic of absorption, fringes, and noise will inherit the featuresof their origination. These profile differences among WMS-2\(f\), harmonic of optical fringes, and noise will be a novel breakthrough point to distinguish and eliminate the interference from the signal (details in Section 5.3).