# Effective Optical Scattering Range Determination Based on Ray Tracing
## 3 Application
### Sample  
The sample used in this paper is made by Zerodur with magnetorheological finishing. A reflective silver-enhanced coating is used to achieve the high reflective performance in the wavelength of 1064 nm. The photograph of the superpolished sample is shown in Figure 3.  
As mentioned before, there is no unique rms roughness for a surface. The measured results can be different for different instruments, sampling or frequency ranges. Therefore, a rigorous roughness characterization requires a combination of a variety of techniques. In order to get the topographic information, the sample is measured with an atomic force microscope (AFM, a field of 0.02 mm \(\times\) 0.02 mm, 256 \(\times\) 256 pixels). It works by scanning with a probe very close to the sample surface and measuring attractive or repulsive forces. The measurement result is shown in Figure 4a where the rms roughness equals 0.3 nm.  
White-light Interferometry (WLI) is another typical method in microscopic profilometry. It is also used to measure the surface (10\(\times\) and 50\(\times\) objective). Figure 4b shows the profile measurement of the sample with 10\(\times\) objective with WLI. Due to the short coherence length of the white light, only the area whose optical path is close to the interferometer arm length can be detected. The rms roughness in this case is 0.5 nm.  
The angle-resolved scattering measurement is a typical non-contact method. It has proven to be an effective method to determine surface roughness from the above discussion. An important condition is that, for roughness determination, the scattering should be solely caused by roughness. It should not be due to particulate contamination or local defects such as scratches, skeks or digs. The sample is measured with a table-top system for light scatter measurement (Albatross-TT) [15]. The angles of incidence are 5\({}^{\circ}\), 45\({}^{\circ}\) and 75\({}^{\circ}\), respectively.  
Figure 4: Topography measurement result of the superpolished sample shown in Figure 3. (**a**) The AFM measurement result, (**b**) the WLI measurement result with 10\(\times\) objective.  
Figure 3: The superpolished sample with reflective silver-enhanced coating.  
The scattered angle ranges from \(-85^{\circ}\) to \(85^{\circ}\) and the wavelength is 1064 nm. The results are presented in Figure 5.  
The scattered measurement can be converted into the PSD data with Equation (7). Figure 6 is plotted considering all of the above-mentioned techniques. It can be seen that the PSDs under different techniques are in different frequency ranges. The relevant frequency ranges are shown in Table 1. The PSDs are in good agreement which will later be used for ray tracing.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Technique** & **Frequency Range (\(\mu\)m\({}^{-1}\))** \\ \hline Angle-resolved scattering & 0.016–1.022 \\ WLI 10\(\times\) & 0.0206–2.85 \\ WLI 50\(\times\) & 0.0298–14.17 \\ AFM & 1.5–398 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Frequency range for different techniques.  
Figure 5: The BRDF measurement results under different angles of incidence.  
Figure 6: PSD measurement results with different techniques including WLI.