Solving the inverse problem for the design of omnidirectional lenses with an ultimately high numerical aperture

We introduce a type of lens that focuses a plane wave into a spherical one, where light comes from all directions. Our method also suggests the design of ideal optical tweezers or, in the reverse direction, photodetection of nearly all of the light emitted from an omnidirectional source. Our design trades off simple isotropic nonmagnetic materials for requiring a high refractive index. Despite many challenges, we believe that our proposed lens can evolve, with optimization and some help from conventional lenses, to fabricable lenses that will help in boosting collection efficiency of photons in ion-photon quantum experiments.