# Effective Optical Scattering Range Determination Based on Ray Tracing
## 1 Introduction  
Micro-irregularities are distributed over the entire optical surface and scatter the incident light, which is inevitable even for high-quality optics. The scattering caused by surface profile height is well discussed in [1; 2; 3; 4]. The statistical randomness of an optical surface can be expressed by the power spectral density function (PSD). It gives the information on the surface error as a function of the spatial frequency and relies highly on the manufacturing of the optical surface. For example, for a periodic irregularity produced by single-point diamond turning, the PSD gives the information about the tool feed rate, depth of cut, and vibration of the machine [5]. Furthermore, the scattering of an optical smooth surface can be predicted with the Rayleigh-Rice vector perturbation theory which relates the scattered power to the surface power spectral density [6].  
However, in real scenarios, the optical system selects the scattered rays according to their positions and directions. The position information can allow the estimation of the scattering flux in combination with the incident radiance and scattering response of the surface [7]. The direction information can be used to determine the spatial frequency with the hemispherical grating equations. The scattering that is able to reach the detector becomes stray light and limits the resolution of the image [8; 9]. The pattern is usually caused by the scattering near the specular direction which is called small-angle scattering. In comparison, the scattering that cannot reach the detector of the system results in energy loss and reduces the contrast of the image [10]. This process is related to the wide-angle scattering which corresponds to the high spatial frequency region of PSD [10]. The band of delivered scattering differs for different optical elements even though the optical surfaces have the same topography. Therefore, due to the stochastic nature of the scattering and the diversity of the optical systems, the numerically intensive ray tracing method is a perfect method to determine the cutoff frequency. However, the combination of the surface features, optical scattering and ray tracing theory is not yet well discussed. In this paper,the classic optical scattering is combined with the ray tracing theory to determine both the system reaction of scattering and band of surface imperfections. Relevant definitions and the thermotical background are introduced in Section 2. In Section 3, two typical surface features are traced in order to determine the effective range of the surface imperfections. The methods can determine the critical spatial frequency range of roughness which can instruct the manufacturing processes.  
Moreover, local defects such as contamination are also considered. Traditionally, the local defects are characterized by the cleanness level discussed in [11] where the considered particles are idealized and scattering loss is statistically related to the cleanness. However, the effect of a single defect is usually hard to measure or quantify. Based on the ray tracing method discussed in this paper, the influence of a single imperfection is also analyzed which can inversely instruct the configuration of scattering measurement. The method reveals the selective nature of an optical system from the statistical view and its great flexibility.