# Effective Optical Scattering Range Determination Based on Ray Tracing
## 3 Application
### Ray Tracing Result
The maximum spatial frequency for the primary is 0.007 \(\upmu\)m\({}^{-1}\) which is shown in the orange area in Figure 10. The green area is the spatial frequency range for the secondary with the maximum 0.02 \(\upmu\)m\({}^{-1}\). Both Figures 9 and 10 indicate the different characteristics in stray light contribution for the primary and the secondary even through the roughness and the scattering properties are assumed. Figure 8: The paths of scattering that are from the primary (**a**) and the secondary (**b**) of the Cassegrain telescope. Figure 9: The probability distribution of the scattered ray deviation from the specular reflected direction. Determining the spatial frequency range through ray tracing can also be related to the fabrication. For example, with a hypothetical periodic structure with the spacing \(1/\mathrm{f_{p}}\) on the surface, which is usually due to processing, a peak can be witnessed from the PSD curve which is shown as the red curve in Figure 10. Even though the periodic imperfection can cause small-angle scattering, it will not contribute to the in-field scattering because fp is neither in the frequency range of the primary nor that of the secondary. In other words, the scattering caused by such a kind of structure can be blocked or absorbed with proper mechanical design or black surface treatment. A single local defect such as small particles scatters the incident beam differently and such a surface usually suffers from a higher scattering loss. For such a case, the clean surface assumption is not fulfilled and the Rayleigh-Rice vector perturbation theory is no longer valid. Figure 11a shows the scattering image of a particle on the considered surface. In comparison, Figure 11b is the instrument signature where the roughness is the only source of scattering. The measurement is conducted with a highly sensitive optical roughness sensor (Horos) [17; 18] which can record the scattering image and the BRDF value simultaneously. The central peak is the specular reflected beam and the surroundings are the scattered light. The diffraction caused by the particle is also considered. Typically, the scattering caused by particles is usually anisotropic and a thorough measurement can be difficult and time-consuming. On the other hand, the scattering caused by the same particle might be different for optically different elements and positions. Analytical calculation can be difficult, making ray tracing an extremely flexible and practical method. In order to analyze the effect of the local particles, three particles are introduced on the primary with the X-Y plane coordinates of \((-2,\,4)\), \((1,\,-3)\), \((-1.7,\,1)\) for particle 1, particle 2 and particle 3. The particles are assumed to be circular and produce a scattering pattern the same as that shown in Figure 11a. The scattering paths for the three particles are shown in Figure 12a. The deviation of scattering angle from the specular path is shown in Figure 12b. The red area is the scattering distribution of the entire surface and the others are for the scattering distribution of the local particles. It shows that for a rotationally symmetrical surface, the local defects have a similar distribution to the overall scattering distribution. Therefore, with the directional statistics of the scattering that contributes the stray light, a BRDF measurement aiming at the scattering of a single particle can be arranged. By inserting the measured data, the scattering caused by defects and reaching the detector is \(1.34\,\times\,10^{-10}\) W, \(3.54\,\times\,10^{-10}\) W, \(1.09\,\times\,10^{-9}\) W, respectively, for unit incident power. Compared with \(1.05\,\times\,10^{-5}\) W caused by roughness scattering, the stray light produced  
Figure 10: The effective frequency range of the PSD for the primary and the secondary. by such three particles is much smaller than the roughness scattering and can be ignored. However, with an accumulation of such particles, the image quality can be reduced.