## 2 Design of the POE  
Modeling the flat-spot Fresnel lens POE is based on a two-dimensional ray-tracing method, where one ray at the center of each prismatic groove is traced through the lens with the purpose of refracting that ray onto the center of the focal plane by optimizing the prism angle, \(\theta\).  
To simplify the design, the following assumptions were made:  
1. Solar light penetrates the Fresnel lens with no absorption, and the solar radius angle is included within the design incidence angle, \(\pm\alpha\);
2. The lens has \(g\) number of grooves at an equal width of \(w\), the groove order of \(1\leq i\leq g\) extends away from the lens' center;
3. The optimization algorithm only considers the two extreme incidence angles, \(\pm\alpha\), and only refracts the ray through the grooves' midpoints for the calculation of the prism angles.  
The full polychromatic span of the solar spectrum was incorporated in the algorithm using the segmentation method of [PERSON] [23], which was used in an earlier publication of the authors [24]. The spectrum was divided into nine wavelength intervals, each represented by a center value and a weight factor of the total irradiant energy. Ray tracing through the POE was performed at the each of those center values, and the optimized prism dimensions were averaged out to incorporate the weight factors of all segments.  
Referring to Figure 1, if a ray is incident onto the upper-surface center of the \(i\)th groove at an incidence half angle of \(+\)\(\alpha\) with respect to the surface normal, the first refraction will happen according to [PERSON]'s law as  
\[\beta=\sin^{-1}\left[\frac{\sin\alpha}{n_{lens}}\right] \tag{1}\]  
where \(\beta\) is the lens' top surface refraction angle and \(n_{lens}\) is its refractive index. This index was calculated for each material at the center values of the segmented solar spectrum and then iterated in the optimization process. For instance, the Sellmeier equation [25] for the refractive index of poly-methyl-methacrylate (PMMA), a common material in Fresnel lens fabrication, can be used. The refracted ray hits the lower surface of the \(i\)th groove at an angle of \(\gamma_{i}\) with the surface normal:  
\[\gamma_{i}=\theta_{i}+\beta \tag{2}\]  
Then, the second refraction takes place with an angle of \(\delta_{i}\) found by  
\[\delta_{i}=\sin^{-1}(\sin\gamma_{i}n_{lens}) \tag{3}\]  
For a base thickness of \(L\) and an equal groove width of \(w\), the refracted ray's path length inside the \(i\)th groove can be computed as  
\[h^{\prime}{}_{i}=\left(L+\frac{w\tan\theta_{i}}{2}\right)\frac{\sin(90+\gamma _{i}-\beta)}{\cos\gamma_{i}} \tag{4}\]  
where \(F\) is the lens' focal length. The linear vertical and horizontal distances, \(A_{i}\) and \(B_{i}\) respectively, that the ray travels from the \(i\)th groove to the focal plane are calculated by  
\[A_{i}=F-h^{\prime}{}_{i}\cos\beta+L \tag{5}\]  
\[B_{i}=A_{i}\tan(\delta_{i}-\gamma_{i}+\beta) \tag{6}\]  
The reference horizontal distance to the focal midpoint, \(B_{i,0}\), can be found as  
\[B_{i,0}=(i-1)w+\frac{w}{2}-h^{\prime}{}_{i}\sin\beta \tag{7}\]  
Through an iterative optimization using MATLAB(r), the \(i\)th prism angle, \(\theta_{i,+\alpha}\), can be chosen, in reference to the incidence half angle, \(+\alpha\), so that the difference between \(B_{i}\) and \(B_{i,0}\) is at a minimum.  
The calculations can then be repeated for the other incidence half angle of \(-\alpha\), and the optimum prism angle, \(\theta_{i,optimum}\), can be chosen such that  
\[\theta_{i,optimum}=\frac{\theta_{i,+\alpha}+\theta_{i,-\alpha}}{2} \tag{8}\]  
Each prism thickness, \(h_{i}\), can then be calculated at the optimum, \(\theta_{i,optimum}\), as  
\[h_{i}=w\tan\theta_{i,optimum} \tag{9}\]