## 3 Design of the SOE
### Ray Tracing through SOE  
With the known POE prismatic geometry, the design process for the hemispherical-domed SOE could start by solar ray tracing at the iterated SOE radius. Note that the absorption of light within the SOE was also ignored for this process. Similar to the POE optimization, the spectral segmentation of light [23] was considered while tracing rays through the SOE material, and the optimized SOE parameters were averaged according to the segments' weight factors. In a simplified 2D consideration of the flat-spot Fresnel lens, the light incident at an angle of \(+\alpha\) required two sets of equations while ray tracing, one for each side of the symmetrical lens (\(L\) for left-side and \(R\) for right-side). All used terms are explained in Figures 1 and 2. While the angles of incidence and the angles of refraction at the upper POE surface, \(+\alpha\) and \(\beta\) respectively, are the same for both sides of the lens, the angles of incidence at the lower \(i\)th groove surface, \(\gamma_{i,R/L}\), for the right and left sides of the lens, can be calculated in reference to Equation (2) as  
\[\gamma_{i,R} =\theta_{i,optimum}+\beta \tag{10}\] \[\gamma_{i,L} =\theta_{i,optimum}-\beta \tag{11}\]  
Figure 1: Schematic of the refraction geometry for the primary optical element (POE) design. (Half-lens shown is not to scale). Rays were then refracted at an angle of \(\delta_{i,R/L}\) from the lower POE surface, calculated by invoking Equation (3) for both sides of the lens as  
\[\delta_{i,R/L}=\sin^{-1}\!\left(\sin\gamma_{i,R/L}n_{lens}\right) \tag{12}\]  
The refracted ray's path length inside the \(i\)th groove \(h^{\prime}{}_{i,R/L}\) was found by referring to Equation (4):  
\[h^{\prime}{}_{i,R}=\left(L+\frac{w\tan\theta_{i,optimum}}{2}\right)\frac{\sin (90+\gamma_{i,R}-\beta)}{\cos\gamma_{i,R}} \tag{13}\]  
\[h^{\prime}{}_{i,L}=\left(L+\frac{w\tan\theta_{i,optimum}}{2}\right)\frac{\sin (90-\gamma_{i,L}-\beta)}{\cos\gamma_{i,L}} \tag{14}\]  
The angled rays making contact with the horizontal, negative \(x\)-axis after this second refraction, \(\xi^{\prime}{}_{i,R/L}\), can be calculated by  
\[\xi^{\prime}{}_{i,R}=90+\left(\delta_{i,R}-\gamma_{i,R}+\beta\right) \tag{15}\]  
\[\xi^{\prime}{}_{i,L}=90-\left(\delta_{i,L}-\gamma_{i,L}-\beta\right) \tag{16}\]  
The linear vertical distance, \(A_{i,R/L}\), for the \(i\)th groove on either side can be obtained using the corresponding value from Equation (5):  
\[A_{i,R/L}=F-h^{\prime}{}_{i,R/L}\cos\beta+L \tag{17}\]  
Figure 2: Refraction schematic through the secondary optical element (SOE). Rays from the left side of the lens are in green, and those from the right side are in orange—only four rays are shown for simplicity. (All are to scale except the incident light at the POE surface.) (The black short-dashed line is the projection of the left center ray without the SOE.) Similarly, the reference linear horizontal distance, \(B_{i,0,R/L}\), is given by  
\[\gamma_{i,L} =\theta_{i,optimum}-\beta \tag{11}\]0 \[\gamma_{i,L} =\theta_{i,optimum}-\beta \tag{11}\]1  
Without any SOE, when each refracted ray reaches the focal plane, it will be at a horizontal distance of \(CD_{i,noSOE,R/L}\) from the center, which can be calculated by  
\[\gamma_{i,L} =\theta_{i,optimum}-\beta \tag{11}\]2 \[\gamma_{i,L} =\theta_{i,optimum}-\beta \tag{11}\]3  
When installing the SOE of radius \(r\), each ray that hits it will have an angle \(\xi_{i,R/L}\) with the normal to its curved surface, depending on the lens side it refracts from.