## 4 Relative Uncertainty Analysis for Scattering-Rate Measurement System
### Relative Uncertainty Analysis  
For the high-precision scattering-rate measurement system in this paper, the analysis shows that the following five factors could be the main relative uncertainty sources for measurement: the laser power stability, the laser incidence angle, the gap distance between the sample plate and the integrating sphere, the ambient light angle and the off-center position of the incident laser beam. The relative uncertainty is determined by the following expression:  
\[u_{\text{rel}}=\frac{u}{\overline{x}} \tag{15}\]  
where \(u_{\text{rel}}\) represents the relative uncertainty of measurement results; \(\overline{x}\) is the arithmetic mean of statistics; and \(u\) represents the uncertainty of measurement results, which can be expressed by the following equation. \[u=\text{max}\{|x_{i}-\overline{x}|,i=1,2,\dots,n\} \tag{16}\]  
where \(n\) is the number of measured data. Since the response band of the detector used in this paper lies in the infrared band, the angle and intensity of the ambient light have little effect on the detector under the condition of laboratory environment. The experimental result also show that the relative uncertainty of infrared detection caused by different incident angles of ambient light is less than \(\pm\)0.018%, as shown in Figure 12a. Therefore, this relative uncertainty can be ignored in the laboratory environment. Due to the use of aiming-and-positioning system, the off-center position of the incident laser beam in the experiments is estimated to be less than 3 mm. As can be seen from Figure 12b, the fluctuation of experimental luminous flux received by the detector is rather small in the range of off-center position \(-\)3 mm to 3 mm, with an average value of 75.24 mW and a measured uncertainty of \(\pm\)0.229 mW. Therefore, the maximum measurement uncertainty due to the off-center position of the incident laser beam is \(\pm\)0.30%. The maximum output power of the laser used in this experiment is 1000 mW with the measured uncertainty of 1.9 mW. Therefore, the relative stability of the laser power is \(\pm\)0.19%. The laser incidence angle is different due to the working distance, that is, the longer the working distance, the smaller the allowable laser incidence angle. Therefore, a long working distance is conducive to improve the reflectivity measurement accuracy. Restricted by the opening size of the integrating sphere, the influences of the laser incidence angle in the range of 0-1.95\({}^{\circ}\) on the received luminous fluxes of the detector are simulated and experimentally measured. The gap distance between the sample plate and the integrating  
Figure 12: Influence of ambient light angle and off-center position of incident beam on experimental luminous fluxes: (**a**) ambient light angle and (**b**) off-center position of incident beam. sphere is found to be in the range of 0-1 mm. The influences of laser incidence angle and gap distance on the received luminous flux has been simulated and measured respectively, and they are shown in Figure 13.  
It can be seen from Figure 13a that the simulated luminous flux basically showed a slight downward trend with the increase of laser incidence angle. However, the experimental results showed a limited fluctuation in a certain range with the increase of laser incidence angle. The reason was that the change of laser incidence angle would affect the spatial distribution of scattered light energy in the integrating sphere at a certain extent. Because the surface of the standard sample could be considered as a Lambert diffuse reflection surface, the scattering brought a weak randomness. Figure 13b indicated that the luminous flux received by the detector decreased slowly with the increase of the sample gap distance. This phenomenon could be easily understood: the expanding gap distance between the sample and the integrating sphere would lead to the increasing amount of beam leakage at the sample region. Therefore, the luminous flux received by the detector also decreased.