## 2 Carbon Isotope Detection Principle and Absorption Line Selection  
### Carbon Isotope Detection Principle  
The basic principle of the TDLAS technique for carbon isotope detection is the Beer-Lambert law [17]. After passing through the MPGC filled with the measured gas, the laser measures the absorption of specific gas molecules with a given optical path length to obtain the concentration characteristics of the gas molecules. The Beer-Lambert law can be described by the following formula:\[I_{1}(v)=I_{0}(v)e^{-\alpha(v)PCL}, \tag{1}\]  
where \(I_{0}(v)\) is the emitting light intensity of the laser, \(I_{1}(v)\) is the light intensity after passing the measured gas, \(L\) is the effective length of the absorption optical path, \(P\) is the pressure in the cell, \(C\) is the gas concentration, and \(\alpha(v)\) is the molecular absorption coefficient. Then, \(\alpha(v)\) can be expressed as follows:  
\[\alpha(v)=S(T)\cdot g(v-v^{\prime})\cdot C, \tag{2}\]  
where \(S(T)\) is the absorption intensity of gas at temperature \(T\), \(g(v-v^{\prime})\) is the absorption line function of measured gas, which relates to the concentration \(C\) and pressure \(P\) of the measured gas, and \(v^{\prime}\) is the initial frequency of the energy level transition of gas molecule. To improve the minimum detection limit (MDL) performance of the sensor, tunable diode laser absorption spectroscopy-wavelength modulation spectroscopy (TDLAS-WMS) was adopted to eliminate the 1/f noise caused by the ICL or external environmental disturbances [18]. The time-dependent wavelength of the ICL can be described as follows:  
\[v=v_{0}+A\cos(\alpha t), \tag{3}\]  
where \(v_{0}\) is the central frequency of the emitting light, which is determined by the low-frequency component of the driving signal, and \(A\) and \(\omega\) are the amplitude and frequency of the high-frequency component of the driving signal, respectively. By substituting Equation (3) into Equation (1) and expanding it in the form of a cosine Fourier series, we get  
\[I_{1}(v)=\sum_{n=0}^{\infty}A_{n}(v_{0})\cos(n\omega t), \tag{4}\]  
where \(A_{n}\) is the amplitude of each harmonic component, which can be expressed as follows [19]:  
\[A_{n}(v_{0})=\frac{I_{0}(v)\cdot 2_{n-1}\cdot C\cdot L}{n!}\cdot A^{n}\cdot \frac{d^{n}\alpha(v)}{dv^{n}}_{|v=v_{0}}. \tag{5}\]  
According to Equation (5), the amplitude of the second harmonic component is  
\[A_{2}(v_{0})=\frac{I_{0}(v)\cdot C\cdot L}{4}\cdot A^{2}\cdot\frac{d^{2} \alpha}{dv^{2}}_{|v=v_{0}}. \tag{6}\]  
Based on the above formulas, the second harmonic components at the center frequency reach maximum values, which are positively proportional to the gas concentration. In summary, TDLAS-WMS is the optimum choice to analyze the measured gas concentration, which can effectively reduce the 1/f noise, increase the signal-to-noise ratio, and improve the MDL performance of the sensor [18]. For carbon isotope detection, \(\delta^{13}C\) is defined to represent the carbon isotope ratio, shown as follows [20]:  
\[\delta^{13}C=1000\times(\frac{[^{13}CO_{2}]/[^{12}CO_{2}]}{RPDB}-1)\,\% \omega, \tag{7}\]  
where RPDB is the \({}^{13}C\)/\({}^{12}C\) ratio of Pee Dee Bellemite (PDB), assumed to 0.01124 [21]. The parameters of spectral line absorption intensity are a function of temperature; therefore, when the temperature of the measured gas changes, it will bring uncertainty to the measurement results. The reason is that the temperature has different influences on the absorption intensity of the two pair lines of the carbon isotope. In the detection of isotope \(\delta^{13}C\) values, the effect of temperature on abundance values can be described by  
\[\Delta\delta\approx\frac{\Delta T\cdot\Delta E}{k\cdot B\cdot T^{2}}\times 1000 \%\omega, \tag{8}\]where \(\Delta\delta\) is the isotope ratio change, \(\Delta T\) is the gas temperature change, \(B\) is the Boltzmann constant, \(k\) is a constant, \(T\) is the absolute temperature, and \(\Delta E\) is the low energy level difference between the two absorption lines.