Solution to the bundle-to-bundle mapping problem of geometric optics using four freeform reflectors.

Here we present a method for the coupled design of four freeform reflective surfaces that will control a bundle of rays. By this, we mean that given an input bundle of rays, we can construct an optical system that will map it to a given output bundle, where a ray-to-ray correspondence is realized as per the prescribed data. The method makes use of the Cartan-Kähler theorem of exterior differential systems. Sample imaging applications are given.