## 2 Module Design and Simulation
### Simulation: Illumination Performance and Tolerance Analyses
In this research, the selected collimator provided a beam with a divergence of 1.2\({}^{\circ}\), so there were some staggered rays that could not reach the receiver. The loss of the double Fresnel lenses was 44.26 Lux. The double Fresnel lenses are denoted by \(L_{2}\). Using LighTools(tm) software, \(L_{2}\) can be calculated by Equation (4):  
\[L_{2}=\frac{\textit{Flux on Receiver 1-Flux on Receiver 2}}{\textit{Flux on Receiver 1}} \tag{4}\]  
By calculating Equations (3) and (4), respectively, we found L1 = 0.7% and L2 = 22.3%. The total loss is the sum of \(L_{1}\) and \(L_{2}\) and can be calculated by Equation (5):  
\[L_{\text{total}}=\frac{\textit{Flux of LED source-Flux on Receiver 2}}{\textit{Flux of LED source}} \tag{5}\]  
With the parameters given above, the efficiency was found to be 77%, including the material absorption, the Fresnel losses, and the geometrical loss of the Fresnel lens. Tolerance is an important issue for free-form lenses. In this study, the effects of the angle between the collimator lens and double linear Fresnel lens were analyzed in terms of uniformity and efficiency as shown in Figure 6:0 Figure 10a-c shows the effects of the line distributions along the X axis and Y axis across the center of the illuminated area for \(dT\) angles of 2\({}^{\circ}\), 3\({}^{\circ}\), and 4\({}^{\circ}\). Figure 10d shows the results of simulations of the efficiency and the uniformity as a function of angle tolerance. When the deflection angle \(dT\) of the elements increased from 0\({}^{\circ}\) to 4\({}^{\circ}\), the efficiency and the uniformity were reduced accordingly. Because the change in angle between optical devices changes the incident angle of the collimated light beam to the linear Fresnel lens, the distribution will be skewed to the corresponding angle, reducing uniformity. The permissible deviation of \(dT\) is 1.8\({}^{\circ}\) as long as the uniformity and the efficiency are kept at 90%. Figure 6:1 Illustration of the simulation structure for loss analysis and ray-tracing analysis. In addition, another tolerance was also considered in this study, which is the angle of the collimated beam. The simulation results of the efficiency and the uniformity as a function of beam angle of the collimated beam \(d\theta\) of \(2^{\circ}\), \(4^{\circ}\), and \(6^{\circ}\) are shown in Figure 11a-c. When the beam angle was wider, the square illumination shape changed more, and the uniformity and efficiency also decreased. This is explained by the beam angle related to incident angle of the light to the Fresnel lens. When the beam angle is wider, the output light does not focus on the focal line, so it is impossible to control the shape of distribution in two directions. The maximum beam angle was \(3^{\circ}\), and \(90\%\) efficiency and uniformity was obtained. Figure 6:2 Light distribution on a receiver with (**a**) \(d\theta=2^{\circ}\); (**b**) \(d\theta=4^{\circ}\); and (**c**) \(d\theta=6^{\circ}\). (**d**) Efficiency and uniformity depend on the angle of the collimated beam. Figure 6:3 Light distribution on a receiver with (**a**) \(dT=2^{\circ}\); (**b**) \(dT\)=\(3^{\circ}\); and (**c**) \(dT\)=\(4^{\circ}\). (**d**) Efficiency and uniformity depend on deflection angle \(dT\).