# Effective Optical Scattering Range Determination Based on Ray Tracing
## 2 Theoretical Background
### Ray Transfer in Optical Systems  
The scattering reaches the detector and becomes stray light through three processes. First, the considered surface is illuminated and such a surface are often called an illuminated surface. Then, the incident light is scattered by the imperfections of the surface. Finally, the scattered rays are selected by the layout of the optical system according to its position and  
Figure 2: Relation between roughness and scattering parameters.  
directions. In other words, the illuminated surfaces need to be viewed by the detector [8]. For an infrared system, an additional source exists which is the internal emission of the sensor itself, which is often called thermal self-emission. Detailed analysis of such radiation is not the goal here. This paper focuses on relating the classical external stray light to the surface roughness, which is a concern for most optical systems. Such paths can be verified experimentally and precisely identified with the time of flight method which allows the determination of the origin of stray light by temporally discriminating the light at the detector during stray light testing [13]. The point spread function under the effect of surface error can be calculated as [14]:  
\[\text{PSF}_{\text{surface error}}=\text{PSF}_{0}*[\text{SR}\cdot\delta( \alpha,\beta)+(\frac{\text{TIS}}{\sigma_{\text{rms}}{}^{2}})\cdot\text{PSD}] \tag{12}\]  
where \(\text{PSF}_{0}\) is the ideal point spread function, SR is the Strehl ratio, \(\alpha,\beta\) are the direction cosines. In this way, the influence of surface error can be related to the image quality. In the optical design phase, only a few rays are considered to estimate the effect of aberration. However, due the stochastic nature of roughness, ray tracing is a more flexible method which directly reflects the acceptance-rejection nature of an optical system. Since the scattered rays are missed during the propagation, the ray transfer efficiency \(\eta_{\text{r}}\) can be defined as:  
\[\eta_{\text{r}}=\frac{\sum\limits_{i=1}^{N}V_{i}}{N_{0}} \tag{13}\]  
In Equation (13), \(N_{0}\) is the total number of considered rays. \(V_{i}\) is the visibility term which equals 1 if the scattered ray reaches the detector or otherwise is 0. Because each scattered ray is uncorrelated, the process can be taken as a Bernoulli experiment. When \(N_{0}\) approaches infinity, the ray transfer efficiency can be interpreted as the possibility that a scattered ray reaches the detector, which is:  
\[\lim_{N_{0}\rightarrow\infty}\mathbb{P}\left(\left|\frac{\sum\limits_{i=1}^{N }V_{i}}{N_{0}}-P_{0N}\right|\right)=0 \tag{14}\]  
\(P_{0N}\) is the possibility of the scattered beam reaching the detector. Equations (13) and (14) reveal the selection of the scattering by the optical system. Moreover, with ray tracing, a preferred angular distribution of the scattering can be derived. In this way, the frequency range of each optical element can be derived. The cutoff frequency is an essential reference for figuring and polishing. For example, if a spectrum peak is not in the range of the optical response range of the system, the periodic structure most likely can be controlled. On the other hand, the ray distribution can be used to analyze the effect of a local defect. For example, the optical scattering of a scratch relies highly on the angle of incidence and the scattering angle. Measuring all of the situation can be difficult and unnecessary. Therefore, it is reasonable to determine the potential geometry and then measure only a few configurations, which will be discussed in the following section.