A novel waveguide-optical integrator is introduced for applications to LEDs. The concept is based upon a Kohler illuminator made of Luneburg lenses. Typical Kohler illuminators are formed by pairs of thin lenses, and perform badly when the paraxial approximation is rough, i.e., when the angular span of the incoming rays is wide. In contrast, the new illuminator performs ideally for angular spans up to 90o (±45o), and has only a 3% loss for a 180o angular span. In general such an illuminator cannot be made in 3D, because adjacent Luneburg lenses overlap. It can, however, be implemented in planar optics, by using Rinehart geodesic lenses, which moreover do not use gradient index material. This waveguide device has application in illumination engineering as a light mixer, particularly for LEDs. Another light mixer using a combination of two kaleidoscopes with a geodesic lens is also presented. Irradiance at the exit of a kaleidoscope has good light mixing if the kaleidoscope is long enough, but the intensity is never well mixed, irrespective of the length. Inserting a Rinehart geodesic lens produces a 90-degree phase-space rotation of the rays, i.e., it exchanges irradiance and intensity. A further kaleidoscope assures complete mixing in both irradiance and intensity.