## 5 Results and Discussion  
Simulation results demonstrate that the measurement accuracy of overlapping absorption line measurement is greatly affected by laser parameters. The maximum measurement errors caused by changes in linear parameters, including \(\varphi_{1}\) and \(K_{1}\), were 0.18% and 0.17%, respectively. The maximum measurement errors caused by changes in nonlinear modulation parameters, including \(\varphi_{2}\) and \(K_{2}\), were 38.8% and 25.2%, respectively. The nonlinear parameters of the laser have a much greater impact on the measurement results than the linear parameters. In order to verify the validation of 2f-SR, a CH\({}_{4}\)-sensing system based on the overlapping absorption lines was developed using a 2.3 \(\upmu\)m distributed feedback (DFB) laser (Canada, Norcade, NL2327) and a self-developed multi-pass cell (MPC) with a 41.5 m effective absorption path length to measure trace CH\({}_{4}\) concentration, as shown in Figure 5. The DFB is driven using a temperature controller (USA, Thorlabs, TED200C) and a current controller (USA, Thorlabs, LDC20C). A fiber coupler is utilized to split the light from the laser output into the MPC, to measure CH\({}_{4}\) concentration, and into Fabry-Perot etalons, to measure the laser parameters. The MPC is our self-developed three-lumen MPC, which has an effective optical range of 41.5 m. The anterior chamber mirror consists of two semicircular reflector mirrors, and the posterior chamber contains a circular mirror. The parameters of the Fabry-Perot etalon used in the experiment are described in Table 2. The signals from the etalon and the MPC are focused on two photodetectors (USA, Thorlabs, PDA10D2), with wavelength range 900-2600 nm. A lock-in amplifier (FOMTO, LIA-MV-150-S) is used to extract the 2f signal. The outputs are collected by 16-bit analog-to-digital converters (ADC). Finally, the signals, including detection and laser parameter signals, are processed using 2f-SR to eliminate absorption line interference and improve the measurement accuracy. To verify the effectiveness of 2f-SR, a 105 ppmv CH\({}_{4}\) standard gas was measured using the selected overlapping absorption lines at room temperature. The measurement 2f data and 2f data corrected by 2f-SR are depicted in Figure 6. The result based on the overlapping absorption lines is 95.69 ppmv. The 2f-SR result based on the absorption line at 4294.570 cm\({}^{-1}\) is 104.11 ppmv, and the 2f-SR result based on the absorption line at 4294.557 cm\({}^{-1}\) is 31.81 ppmv. The ratio of 2f between the target absorption line and the interference absorption line is 3, which is consistent with their line intensity ratio. This result provides early evidence for the accuracy of 2f-SR. The CH\({}_{4}\) concentration was reduced by 8.02% due to the influence of the interference absorption line. Figure 4: Influence of \(K_{1}\) and \(K_{2}\) change on the amplitude of 2f from the overlapping absorption lines. 0.005 ppmv was achieved when the corresponding average time was 350 s. The results demonstrate that the detection method of overlapping absorption lines based on 2f-SR can achieve ppmv-level detection performance. 2f-SR has significant advantages over spectral line separation methods based solely on absorption lines. The response of the sensing system to a 50 ppmv CH\({}_{4}\) standard gas was measured independently six times in a laboratory environment to verify the effect of 2f-SR on the measurement accuracy. Raw data and 2f-SR data are presented in Figure 8. The raw measurement error was 4%, and the measurement error after 2f-SR was 0.8%. The measurement error was reduced by a factor of 5 using 2f-SR. Thus, the experimental results validate 2f-SR. 2f-SR can improve the measurement accuracy of overlapping absorption lines, and reduce the MDL of sensing systems using overlapping absorption lines. Compared with classical direct absorption spectral line separation, 2f-SR has greater accuracy and is more suitable for trace gas detection; however, the system structure becomes more complex.