Ideal shape of Fresnel lens for visible solar light concentration

This paper presents theoretical research based on the optimal transmittance condition of a prism to find an ideal shape for Fresnel lenses to concentrate visible solar light. First, the ideal-shape equation was derived out through a simplified method that uses one refraction on the midline of a prism to replace the two refractions, respectively, on its upper and lower interfaces. It has been assumed that the Fresnel lens is thin enough to consider each prism as a point, then all the simplified points form a curve. The differential equation of this curve was built up, which has been solved and expressed by a parametric formula. The parametric formula is defined as the ideal-shape equation of Fresnel lens. Second, the optimal combination of the total refracted angle θ and refractive index n has been analyzed to determine the maximal transmittance. The quantitative analysis has indicated that only one group of (θ, n) can achieve the optimal Fresnel lens' transmittance. Finally, the maximal geometrical concentration ratio C_g of ideal Fresnel lens has been discussed. When material is defined, there is a unique θ that makes the geometric concentration maximal for visible solar light. Generally, materials with low refractive index can be used to design a Fresnel lens with larger C_g.