Analysis of rotationally symmetric reflectors for illuminating systems

The design of an illuminating system consisting of a reflector with a small source, which illuminates its aperture with a preassigned continuous distribution everywhere greater than the direct illuminance distribution, is, mathematically, a singular boundary-value problem. If the rays of light striking an area en the edge of the reflector next to the aperture are to be reflected to an adjacent area on the aperture, then there exist no, one, or two solutions to the problem, depending on the size of the aperture and the reflectance of the surface. If the rays striking an area on the aperture edge of the reflector are to be reflected to the diametrically opposite edge of the aperture, then we obtain a unique solution. We derive the results only for rotationally symmetric systems.