## 4 Relative Uncertainty Analysis for Scattering-Rate Measurement System
### Relative Uncertainty Analysis
In addition, it could be seen from Figure 13b that there was an obvious overall offset between the experimental results and the simulation results. There were two main reasons for this phenomenon: On the one hand, it was assumed that the sample plate and the sample opening were parallel to each other in the experiments. However, in reality the absolute parallelism between the sample plate and the sample opening could not be achieved, resulting in the obvious offset for the measured results. On the other hand, due to the imperfection of the model itself used in this paper, it was different from the real situation, which could not perfectly reflect the real measurement situation. In order to increase the measurement accuracy and reduce the measurement uncertainty, the working distance in the experiment is set to 500 mm. Since the actual sample opening size of the integrating sphere is 12 mm, the maximum allowable laser incidence angle is 0.57\({}^{\circ}\). According to the curves in Figure 13a,b, the relative uncertainties of the received luminous fluxes obtained by simulation and experiment could be calculated under different laser incident angles and sample gap distances. The results showed that when the laser incidence angle changed from 0\({}^{\circ}\) to 0.57\({}^{\circ}\), the relative uncertainties of the received luminous fluxes obtained by simulation and experiment were \(\pm\)0.65% and \(\pm\)1.77%, respectively. When the gap distance changed from the range 0 to 1 mm, the relative uncertainties of the luminous fluxes obtained by simulation and experiment were \(\pm\)2.02% and \(\pm\)1.06%, respectively. The above analysis results indicated that the main factors affecting the measuring relative uncertainty were the laser incidence angle and the sample gap distance. Figure 13: Influences of laser incidence angle and gap between sample and integrating sphere on measurement results: (**a**) laser incident angle and (**b**) gap between sample and integrating sphere. Therefore, in order to effectively reduce the experimental relative uncertainty, it is necessary to ensure that the laser beam vertically enters the integrating sphere, and keep the sample and integrating sphere close together. Since the same measurement system is used, it can be assumed that the measurement relative uncertainty of the detector output photocurrent for the reference sample is equal to that of the measured sample. Then, according to Equation (13), and combined with the error propagation theory [34], the quantitative relationship between the reflectivity relative uncertainty \(u_{s}\) of the coating to be measured and the measured photocurrent relative uncertainty \(u_{I}\) of the detector can be deduced as follows:  
\[u_{s}=\pm\sqrt{2}\rho_{ref}u_{I} \tag{17}\]  
Because there is a relationship between the measured photocurrent and the received luminous flux, as shown in Equation (14), Equation (17) can be converted into the following:  
\[u_{s}=\pm 1.06u_{\Phi} \tag{18}\]  
where \(u_{\Phi}\) is the relative uncertainty of the luminous flux received by detector. It can be seen from Equation (18) that there is a positive proportional relationship between the measuring relative uncertainty of the scattering rate and the measuring relative uncertainty of the luminous flux received by detector.