Tailored Fresnel optics: I. Global shape

The key idea of Fresnel optics is to decouple the global slope from the local slope by breaking up the optical surface into small facets. The size of the facets is irrelevant as long as they are larger than the wavelength of light, so that the system behaves according to geometrical optics, and at the same time small compared the overall size of the optical surface. From the point of view of phase-space conservation, Fresnel optics suffer from a basic shortcoming. The phase-spaces of incoming and outgoing radiation beams need not automatically be equal. This results in either a dilution of radiation or losses or both. On the other hand, decoupling local from global slope allows to tailor the overall shape of the Fresnel lens independently from designing the individual facets. We show that it is possible to closely match incoming and outgoing radiation beams with a particular choice of the global shape of the Fresnel surface. This shape imultaneously minimizes dilution and blocking.