## 4 Indoor Experiment
### System Calibration  
In order to accurately measure carbon isotopes, it is necessary to calibrate the sensing system using known concentrations of gas. The five different concentrations of CO\({}_{2}\) (20 ppm, 30 ppm, 40 ppm, 50 ppm, and 60 ppm) configured above were pumped into the MPGC with 5 min of measurement and a constant pressure of 20 Torr. The peak values of the second harmonic signal of \({}^{13}\)CO\({}_{2}\) and \({}^{12}\)CO\({}_{2}\) are shown in Figure 6.  
Figure 5: Second harmonic signals with the background absorption removed, obtained from five different concentrations (20, 30, 40, 50, and 60 ppm).  
The five datasets were averaged firstly, and then the relationship between the average voltage and calibration concentration was obtained, as shown in Figure 7.  
In Figure 7, the blue data points indicate the relationship between peak values of the second harmonic signal of \({}^{12}\)CO\({}_{2}\) and the corresponding concentration. On the other hand, the red data points indicate the relationship between peak values of the second harmonic signal of \({}^{13}\)CO\({}_{2}\) and the corresponding concentration. Using the acquired data, we can obtain the following formulas via zero-crossing linear fitting:  
\[{}^{12}\text{CO}_{2}=0.16\times\text{max}2f(^{12}\text{CO}_{2})-1.60, \tag{9}\]  
\[{}^{13}\text{CO}_{2}=0.002\times\text{max}2f(^{13}\text{CO}_{2})-0.128, \tag{10}\]  
Figure 6: Peak values of the second harmonic signal of \({}^{13}\)CO\({}_{2}\) and \({}^{12}\)CO\({}_{2}\) for five different concentration levels of 20 ppm, 30 ppm, 40 ppm, 50 ppm, and 60 ppm.  
Figure 7: Measured concentration data points and linear fitting curves of the CO\({}_{2}\) concentration versus 2f amplitude value.  
where \(max2f(^{12}C)\) and \(max2f(^{13}C)\) are the peak values of the second harmonic signals of \({}^{13}\)CO\({}_{2}\) and \({}^{12}\)CO\({}_{2}\), respectively. The values of zero crossing shown in Equations (9) and (10) were \(-\)1.60 for \({}^{12}\)CO\({}_{2}\) and \(-\)0.128 for \({}^{13}\)CO\({}_{2}\), respectively. As the output response of the sensor was calibrated by linear fitting, the response linearity could reach 0.9992 for \({}^{12}\)CO\({}_{2}\) and 0.9996 for \({}^{13}\)CO\({}_{2}\), as shown in Figure 7. Calibrated errors were inevitably introduced to the equations. For \({}^{12}\)CO\({}_{2}\), the range of calibrated error was between 12.5% and 4.94%. The calibration error decreased with the increase in concentration. For \({}^{13}\)CO\({}_{2}\), the range of calibrated error was between 10.3% and 8.25%. In addition, with the increase in concentration, the calibrated error decreased, and the trend was consistent with that of \({}^{12}\)CO\({}_{2}\). The above two formulas can calculate the \({}^{13}\)CO\({}_{2}\) and \({}^{12}\)CO\({}_{2}\) concentration by deriving the peak values of the second harmonic signal; thus, the carbon isotope ratio can be obtained accurately.  
The stability of the pressure control could reach \(-\)0.2%, and the precision could reach \(\pm\)0.04 Torr. The stability and precision of the pressure control affects the measurement performance of the sensing system. The static gas distribution mode, which closes the valves at the front and back ends of the MPGC (sealing the MPGC) when the gas pressure in the MPGC reaches the target value, was adopted to stabilize the gas pressure and reduce calibration error.