## 4 Results and Discussion  
Figure 3 demonstrates the signal of 70% oxygen concentration in the 35 \(\times\) 35 \(\times\) 35 cm diffuse integrating cavity. The red curve in Figure 3a denotes the detected signal, \(\Phi_{\text{out}}\). The black baseline is the incident light radiation, \(\Phi_{0}\), which was obtained by linear fitting based on the data points besides the absorption signal in the red curve. Here, the data points in the range of 250-350 and 700-800 in Figure 3a are used to linearly fit the baseline. Thus, the absorbance can be calculated as shown in Figure 3b. Figure 2: (**a**) Structure of a cubic diffuse integrating cavity; (**b**) schematic of the gas-sampling system and TDLAS (tunable diode laser absorption spectroscopy) for measuring the absorption of oxygen with a cubic cavity as the gas cell. Here, the peak value of the absorbance was defined as the optical parameter (OP), i.e., the maximum value of \(\ln(\Phi_{0}/\Phi_{\mathrm{out}})\). According to Equation (14), the OP is proportional to the optical path length, once the oxygen concentration is constant. Thus, because the oxygen concentration is stable in air, the EOPL of the diffuse cavity was calibrated by comparing the oxygen absorption signal in the cavity and in the air [14; 15; 16]. Based on this method, each optical path length was measured 15 times, and the error bars are shown in Figure 4. The EOPL of the 35 \(\times\) 35 \(\times\) 35 cm diffuse integrating cavity was 590 \(\pm\) 9 cm, i.e., the average value was 590 cm and the detection error was 9 cm. The detection limit of this system was \(\sim\)0.6% for oxygen measurement. Detailed continuous detection results and Allen Variance are shown in Appendix B. Then, different oxygen concentrations were measured with the cubic diffuse cavity. The relationship between the OP and oxygen concentration is shown in Figure 5a. Figure 4: The EOPL (effective optical path length) calibration curve for the cubic integrating cavity between the OP (optical parameter) and optical path lengths in the air. Figure 3: (**a**) Direct scan PMT (photomultiplier tube) signal for 70% oxygen concentration in the 35 \(\times\) 35 \(\times\) 35 cm diffuse integrating cavity; (**b**) calculated result of the absorbance from the PMT signal. The black points show the OP values calculated using the detected spectra. It was found that the OP values depended linearly on the concentrations in the low concentration range and were non-linear when the concentration was relatively high. The blue line shows a linear fitting result using the OP data in the 0-18% oxygen concentration range. Obviously, the data gradually diverge from the blue line with an increasing oxygen concentration. The non-linear fitting result (the red curve) was obtained for all absorption data using Equation (5) with the fitted \(L_{0}\) = 39 \(\pm\) 5 cm, \(\rho\) = 0.98 \(\pm\) 1, \(L_{\text{ave}}\) = 22 \(\pm\) 2 cm, and \(f\) = 0.004 \(\pm\) 1. As shown in Figure 5b, the residual error between the non-linear fitting result and the experimental data was lower than 2%. Thus, the Budget error of the gas sampling system was lower than 2%, including errors of gas flow-rate mixing and gas cylinders. The error of the detection system is analyzed in Appendix B. It indicates that the theoretical predictions by the beam reflection analysis agreed with the experiments. In fact, the theoretical value of \(L_{0}\) and \(L_{\text{ave}}\) were 35 and 23.3 cm respectively for this cavity [21, 24]. The fitting results agree with the theoretical values within the measured error ranges. The fitting reflectivity of the inner coating was slightly lower than the value provided by the supplier, which was caused by imperfections in the coating (pollution during drying and a humidity difference). To determine the non-linear transition point, the deviation between the linear (blue curve) and nonlinear (red curve) fitting results was calculated and is shown as the green curve in Figure 5c. The deviation increased with increasing oxygen concentration.