## 5 Conclusions  
This work explained the optimization analysis for the Fresnel lens POE and the dome-shaped SOE, which can significantly enhance the concentration acceptance product, CAP. The importance of this work lies in the deterministic and comprehensive optimization of the concentrator, utilizing polychromatic ray tracing, trigonometry, and spherical geometry in a MATLAB(r) algorithm. The optimum design outputs the POE prism geometry, the SOE radius, the receiver-SOE spacing, and the recommended receiver radius. The designed concentrator can then be simulated with COMSOL(r) to evaluate the optical efficiency, focal irradiance distribution, and ray trajectories. A proof of concept case study was selected from literature for a 625 cm\({}^{2}\) spot-lens POE with an f-number of 1.5 and a spherical dome SOE. An optimized 13.6 mm SOE radius, 84% smaller in volume than that used by the referenced work, increased the concentrator's acceptance angle from 1.0\({}^{\circ}\) to 1.15\({}^{\circ}\). A 74.9% increase in the geometrical concentration was imposed by the optimized receiver radius, which, along with the increased acceptance angle, enhanced the CAP value by 52%, compared to literature. The coherence of the optimization results was confirmed with a simulated comparative analysis, which iterated the SOE parameters up to 200% of the optimum values. The optimized parameters returned the peak efficiency in each case. The focal irradiance homogeneity was analyzed for different SOE-receiver gap sizes. A no-gap design increased the concentration power, while almost doubling the gap size significantly enhanced the homogeneity. However, the CAP value was decreased by 20% and 15% respectively, compared to the algorithmically optimized SOE-receiver spacing. Figure 10: For Case 2: (**a**) ray trajectories through the optimum SOE parameters of this work at the acceptance angle of 1.31\({}^{\circ}\) and (**b**) the receiver’s irradiance distribution in W/m\({}^{2}\) at the acceptance angle of 1.31\({}^{\circ}\). Figure 9: The concentrator’s optical efficiency (POE + SOE) at different incidence angles for Case 2. A second case study of an optimized dome-shaped SOE from literature was also compared to this work. The POE was a 314.2 cm\({}^{2}\) spot-lens with a 383\(\times\) geometrical concentration and a 27.4 cm focal length. This work's algorithm yielded an optimized dome SOE with a radius of 5.7 mm, which was about 30% smaller in volume compared to that optimized by literature. The CAP value was boosted from 0.34 to 0.45 due to the 31% increase in the acceptance angle. A comparative summary of both case studies is listed in Table 1.  
This work's method can drive further research on the algorithmic optimization of different SOE shapes and can be enhanced to include more parameters, such as optimizing the SOE material and the number of POE grooves. On the other hand, while this optimization is especially designed to maximize the CAP value, future research will work on adjusting the algorithmic approach and the analytical parameters to achieve a uniform focal irradiance for the application of concentrated photovoltaics. This will be advantageous due to the small and simple SOE design, aiding a better area utilization, easier and cheaper manufacturing, and a higher system efficiency. An ongoing research of the authors is also focused on prototyping the Fresnel lens POE with micro-machining and hot embossing and the dome SOE with glass molding. Then the prototype system will be tested with both thermal applications, such as the solar Stirling engine and solar welding, and a CPV application. Conceptualization, methodology, software, validation, formal analysis, investigation, resources, data curation, and writing--original draft preparation: H.Q. ; writing--review and editing, visualization, supervision, and funding acquisition: S.W. and W.Z. This work was supported by the University of North Texas College of Engineering, Department of Mechanical and Energy Engineering, through the faculty's start-up funding.