In order to bound yet keep realistic optical shapes for freeform illumination optics, fractional optimization is employed.
By allowing the shape variables to freely vary, optimization leads to bizarre shapes that cannot be manufactured.
Therefore, fractional shape variables that are referenced to other system variables are set. Optimization of the pseudo-collimation
of an LED using a hybrid optic is presented. The shape of the hybrid optic is described by non-uniform
rational B-splines (NURBS). Two examples of this system are optimized: maximum transfer efficiency into an intensity
distribution of ±10° and maximum transfer and uniformity over this same angular range.
R. John Koshel, R. John Koshel,
"Fractional optimization of illumination optics", Proc. SPIE 7061, Novel Optical Systems Design and Optimization XI, 70610F (17 September 2008); doi: 10.1117/12.810198; https://doi.org/10.1117/12.810198