Optimization of the Collection Efficiency of a Hexagonal Light Collector using Quadratic and Cubic B\'ezier Curves

Reflective light collectors with hexagonal entrance and exit apertures are frequently used in front of the focal-plane camera of a very-high-energy gamma-ray telescope to increase the collection efficiency of atmospheric Cherenkov photons and reduce the night-sky background entering at large incident angles. The shape of a hexagonal light collector is usually based on Winston's design, which is optimized for only two-dimensional optical systems. However, it is not known whether a hexagonal Winston cone is optimal for the real three-dimensional optical systems of gamma-ray telescopes. For the first time we optimize the shape of a hexagonal light collector using quadratic and cubic B\'ezier curves. We demonstrate that our optimized designs simultaneously achieve a higher collection efficiency and background reduction rate than traditional designs.