## 3 Design of the SOE
### SOE Geometry Optimization  
With the purpose of maximizing the CAP value while minimizing both the SOE and the receiver geometry, the following three design parameters, illustrated in Figure 3a, should be optimized:  
1. The SOE radius, \(r\). Minimizing this parameter will save fabrication material and cost and decrease the transmission losses through the SOE;The focus size, \(FS\), which is the horizontal distance from the vertical symmetry axis to the farthest refracted ray, at the position where all refracted rays are closest to that axis. This also represents the recommended solar receiver radius. While it depends on the SOE size, the smaller this parameter is for the same acceptance angle, the higher the geometrical concentration ratio, \(CG\), and the CAP values;
3. The SOE-receiver spacing, \(SRS_{opt}\), which represents the vertical distance from the \(FS\) to the dome's bottom surface--taking that surface at a vertical distance from the Fresnel lens equal to the focal length [5; 11; 15]. \(SRS_{opt}\), as it also depends on the SOE size, can be further optimized by simulation for either a higher concentration of rays or a better focal irradiance homogeneity. The value obtained from the algorithm is the one achieving the highest CAP at the given SOE and \(FS\) values. The proposed optimization process accounts for rays refracted from the outermost (edge: \(g\)) grooves along with the innermost (1) ones and is represented by \((i,side)=[(1,R),(g,R),(1,L),(g,L)]\), taking all other rays to be refracted within those extremes. The procedure, illustrated in Figures 3b and 4, is as follows:  
1. The initial guess of the SOE radius, \(r_{0}\), is set to the maximum value of \(CD_{i,noSOE,side}\) among all four refracted rays as \[r_{0}=\max\{CD_{i,noSOE,side}\ :\ (i,side)=[(1,R),(g,R),(1,L),(g,L)]\}\] (28) The ray tracing analysis through the SOE is then carried out as per the equations of the previous sections. Noting that each \(j\)th iteration of the SOE radius, \(r_{j}\), will be equally incremented by a user input value, \(r_{inc}\), until convergence. The algorithm also checks for the total internal reflection (TIR) from the SOE's flat bottom and prevents it by incrementing the SOE size;
2. The intersection point, if it exists, for each of the four rays refracted through the SOE with the vertical symmetry axis is found. The vertical distance, \(SRS_{i,side}\), of each intersection point to the base of the SOE is found as \[SRS_{i,side}=CD_{i,SOE,side}\tan\left(|90-\phi_{i,side}|\right)\ :\ (i,side)=[(1,R),(g,R),(1,L),(g,L)]\] (29)  
Figure 3: (**a**) Optimization parameters of the SOE and (**b**) ray intersection geometry explained (only the innermost and outermost rays are displayed). For each intersection point, the corresponding horizontal distance, \(HD_{i,side}\), from each of the other three rays to that intersection point is found. For example, if the intersection is for ray \((1,R)\), then the corresponding horizontal distances of the other three rays are \[HD_{(g/1/g),\ (R/L/L)}=CD_{(g/1/g),\text{SOE}_{n}(R/L/L)}-\frac{SRS_{1,R}}{ \tan\left(\left|90-\phi_{(g/1/g),\ (R/L/L)}\right|\right)}\] (30)
4. The maximum value of the three horizontal distances, \(HD_{max}\), for each intersection point is found. This represents the horizontal distance from each intersection point to the farthest ray. For example, if the intersection is for ray \((1,R)\), then \[HD_{max,\ intersection\ of\ (1,R)}\ =\max\left\{HD_{i,side}\ :\ (i,side)=[(g,R),(1,L),(g,L)]\right\}\] (31)
5. The focus size, \(FS\), is taken as the minimum value of all the four \(HD_{max}\) values or \[FS=\min\left\{HD_{max,\ intersection\ of\ (i,side)}\ :\ (i,side)=[(1,R),(g,R),(1,L),(g,L)]\right\}\] (32)
6. The corresponding \(SRS_{i,side}\) is taken as the optimum SOE-receiver spacing, \(SRS_{opt}\). As a reminder, the optimum spacing here refers to the receiver position that balances the irradiance homogeneity and the concentration power.