# Effective Optical Scattering Range Determination Based on Ray Tracing
## 3 Application
### Ray Tracing Result  
Optical scattering is a critical origin and is usually hard to remove when it is in the path of a specular beam. In this section, the stray light from scattering occurring in a typical Cassegrain telescope is considered. The optical configuration is listed in Table 2. The specular optical path is shown in Figure 7. The roughness and the scattering properties are given in Section 3.1. The simulation is achieved with numerically intensive software ASAP [16]. In the beginning, the roughness scattering is considered. With the directional information of the scattering, the effective frequency range can be derived. On the other hand, the scattering of local defects is also demonstrated. Such scattering is usually more critical and depends highly on the optical configuration such as incident angle, scattered angle and wavelength. When considering the scattering caused by roughness, the BRDF was introduced in the ray-tracing model and first-order scattering was computed for each mirror separately. The scattering of the primary and the secondary are separately considered. Figure 8a shows the scattering from the primary and Figure 8b shows the scattering path from the secondary. As can be seen, the scattering reaches the detector only when a certain distribution of direction and position is fulfilled. On the other hand, the roughness scattering is usually isotropic due to the random nature of the surface texture and the axis-symmetry of the Cassegrain telescope. Because the local surface normal of a curved surface is different from different positions, it is reasonable to consider the angular deviation of the scattering path from the specular path. It makes sense because the BRDF of an isotropic surface is shift-invariant. Then, the BRDF curve should be shifted so that that the peak or the zero-deviation position is always in the direction of the specular path. A total of 30,000 scattered rays are picked out which originate from the primary and the secondary and reach the detector. For unit incident power, the scattering flux from the primary that reaches the detector is 1.05 \(\times\) 10\({}^{-5}\) W and for the secondary is 8.5 \(\times\) 10\({}^{-5}\) W. The deviation of each ray is recorded and plotted in Figure 9. which is the probability distribution indicating the angular distribution of the scattering that hits the detector. The green curve represents the deviation of the scattering from the primary while the orange represents the scattering from the secondary. It can be seen from Figure 9 that the scattering is evenly distributed around the specular beam for a rotationally symmetrical system. Furthermore, the histogram clearly shows that the scattering from the primary is more concentrated. The standard deviation of the scattering is 0.13\({}^{\circ}\) for the primary while for the secondary, the standard deviation is 0.56\({}^{\circ}\). According to Equations (13) and (14), the ray transfer efficiency for the scattering of the primary is 2 \(\times\) 10\({}^{-5}\) while that for the secondary is 4 \(\times\) 10\({}^{-4}\). It shows that the scattering from the \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Optical Element**} & **Radius of Curvature** & **Semi-Diameter** & **Spacing** \\  & **(mm)** & **(mm)** & **(mm)** \\ \hline Primary & \(-\)40 & 5 & \(-\)15 \\ Secondary & \(-\)12.5 & 1.5 & 25 \\ Detector & \(\infty\) & 0.5 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optical configuration of a typical Cassegrain system. Figure 7: The optical path of the specular beam of a typical Cassegrain telescope. secondary has a larger possibility to reach the detector. The scattering from the primary is selected and only the scattering that is close to the specular light can reach the detector. On the other hand, according to Equation (8), the scattered angle and the incident angle are converted into the spatial frequency which is shown in Figure 10. The dots are for the measurement data from Section 3.1 and the blue curve represents the fitted curve.