## 2 Gas Sensor Mathematical Model  
The block diagram of the proposed hybrid gas sensor design is shown in Figure 1. Here, the source is a tunable fiber laser operating in line locked mode \(L\left(\lambda\right)\); ideally the beam is divided into two arms. In the first arm, which is called the measurement channel, the light passes through the gas cell before it arrives at the detector \(\left(P_{M}\right)\). In the second arm, which is called the reference channel, the beam arrives directly at the detector \(\left(P_{R}\right)\). In general terms, the measurement detector output can be described as:  
\[P_{M}\left(C,l\right)=\int_{\lambda_{1}}^{\lambda_{2}}L\left(\lambda\right)A \left(\lambda,C,l\right)R\left(\lambda\right)d\lambda, \tag{1}\]  
while the output of the reference channel is given by:  
\[P_{R}=\int_{\lambda_{1}}^{\lambda_{2}}L\left(\lambda\right)R\left(\lambda \right)d\lambda, \tag{2}\]  
where \(\lambda\) is the wavelength, \(A\left(\lambda,C,l\right)\) is the transmission through the gas cell, \(C\) is the gas concentration, \(l\) is the gas cell path length, and \(R\left(\lambda\right)\) is the detector responsivity. The depth of modulation \(\left(D_{M}\right)\) is given as the ratio between the difference of the two channels and their average, and it can be written as [1; 23]:  
\[D_{M}\left(C,l\right) =2\frac{P_{R}-kP_{M}\left(C,l\right)}{P_{R}+kP_{M}\left(C,l \right)}, \tag{3}\]  
In this way, \(D_{M}\) is unitless and in our case \(k\) is a proportionality constant to take into account losses in the measurement channel. In a practical way, \(k\) can be calculated as the ratio between the signals provided by the reference and the measurement detectors when the C\({}_{2}\)H\({}_{2}\) concentration is 0. In this form the depth of modulation will be 0 when the C\({}_{2}\)H\({}_{2}\) concentration is 0% and it will be increasing with the concentration. Now, let us represent the laser wavelength tuning position error as \(r\) and the laser intensity fluctuations by \(q\). Following, these parameters can be included in the depth of modulation equation and therefore this can be rewritten as:  
\[D_{M}\left(C,l\right) =2\frac{\int_{\lambda_{1}}^{\lambda_{2}}qL\left(\lambda\pm r,T \right)R\left(\lambda\right)d\lambda-k\int_{\lambda_{1}}^{\lambda_{2}}qL \left(\lambda\pm r,T\right)A\left(\lambda,C\right)R\left(\lambda\right)d \lambda}{\int_{\lambda_{1}}^{\lambda_{2}}qL\left(\lambda\pm r,T\right)R\left( \lambda\right)d\lambda+k\int_{\lambda_{1}}^{\lambda_{2}}qL\left(\lambda\pm r,T \right)A\left(\lambda,C\right)R\left(\lambda\right)d\lambda} \tag{4}\]  
Here, as the wavelength band \(\left(\lambda_{2}-\lambda_{1}\right)\), where the laser emitting is very narrow, it is possible to consider \(R\left(\lambda\right)\) as a constant and therefore all terms of Equation (4) can be canceled. Moreover, since the laser intensity fluctuation parameter \(q\) also can be treated as constant and it appears in all integrals of Equation (4), it can therefore be canceled. This issue demonstrates the advantage of using the two channels correlation spectroscopy method. However, if there is a tuning error in the laser wavelength, this will not be canceled by the correlation spectroscopy stage. Therefore, it is obligatory in this application that the laser has a high level of wavelength accuracy \(\left(r\approx 0\right)\).