## 4 Validation of Homogenization Principle
### Simulation  
The optical components used in the demonstrator are designed and simulated in Zemax Opticstudio. The laser beams from the RGB laser diodes are firstly collimated, and the beam diameters are expanded to approximately 20 mm. After that, a dichroic prism combines these RGB beams into one white beam. Therefore, this combined incident beam has a quasi-gaussian distribution. It is simulated in Zemax as Gaussian beams with the beam size 6.5 mm at \(1/e^{2}\) intensity and the divergency of \(3^{\circ}\) on both horizontal and vertical axes. The light source is set to 20 mm before the LA. Table 1 lists the parameters of the used LCoS.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Resolution** & **Active Area** & **Aspect Ratio** & **Aperture Ratio** & **Reflectance** & **Waveband** \\ \hline
1280 \(\times\) 768 & 12.29 \(\times\) 7.37 mm & 5: 3 & 89\% & 66\% & 420–700 nm \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter list of the liquid crystal on silicon (LCoS), which is used as the lighting modulator in the demonstrator.  
Figure 4: Expansion how the second LA revise the initial divergent angle from the light source, \(d\), \(\alpha\) and \(f_{1}\) are the same as in Figure 3.  
In order to have a good illumination and projection performance from the LCoS, the incident angle on the LCoS surface is intentionally designed to less than 10\({}^{\circ}\) on each direction according to [21]. The principle and geometrical relationships explained in Section 3 are used as the design approach for this demonstrator. The material of the optical components in the simulation model is uncoated PMMA (Polymethyl methacrylate). The clear aperture of each LA is 30 \(\times\) 30 mm\({}^{2}\), on which 60 single-side lenslets are arranged side by side. The simulation model is shown in Figure 5. This figure also shows the light distributions of the incident beam and the detector surface, acting as the LCoS display surface. The combined circular incident beam is shaped into a rectangular shape with the same dimension as the active area of the modulator. The quasi-Gaussian distribution of the incident beam is transformed into a homogeneous distribution eventually. Besides, the incident angles on both horizontal and vertical directions are proven to be less than 10\({}^{\circ}\). Since the requirements on both homogeneity and image sharpness for headlamp systems are not as high as for video projectors, this angle can be slightly larger to have a shorter optical length [10].  
According to the simulation result, the optical efficiency of this homogenizer composed of two LAs and a relay lens is 74.59%. This efficiency represents the part of the light collected within the target illumination area. This collection efficiency goes slightly upward as the initial beam divergency decreases, reaching 76.24% when the initial incident beam is ideally collimated. Once the second LA is removed, the collection efficiencies drop to 46.95% and 69.73% in both cases. Therefore, the second LA makes a considerable contribution to the collection efficiency when the beam from the light source is divergent.