Optimize the shape of curved-Fresnel lens to maximize its transmittance

This study is to determine the range of incidence angles for a required transmittance of elementary prisms in a curved Fresnel lens and hence to determine the curvature of the lens. The p-polarized light and s-polarized light is considered separately. Calculations indicated that the transmittance of s-polarized light is always lower than p-polarized light and the total transmittance generally decreases with the increase of incidence angle monotonously according to the Fresnel equations, and therefore the transmittance of s-polarized light was recommended as the target for a lens' design. Stokes' reversible relation was derived by Fresnel formulas, which illustrates the reflectivity on both sides of the interface is equivalent when light goes through two media along with a certain path no matter which direction it is transmitted. It is found there is the optimal transmittance condition of prism (OTCP) when the incidence angle changes for a given refractive index bases on Stokes' reversible relation. According to OTCP, the transmittance of a curved Fresnel lens can be calculated for a given curvature and focus and therefore the shape of lens can be optimized. The incidence angles for the required transmittance values of 0.95, 0.90, 0.85, 0.80 and 0.75, respectively, are given for different refractive indices. For example, for the refractive index n₂₁ = 1.49 (value for the common PMMA), the range of incidence angle of s-polarized light is 0-22° if the required transmittance is 0.90; it becomes 0-48° when the required transmittance is reduced to 0.80.