Descartes ovaloides for negative refractive indices and their aplanatic cases

The possibility of material design technology to obtain negative refractive indices poses the question on the optical properties of aspherical surfaces made from such media. Here, the special aspherical surfaces connected with Cartesian ovaloides are discussed in the case of a negative refractive index. A subset of the Cartesian ovaloides contains spheres which allow aplanatic imaging in the well-known kind. A sign change of the refractive index shows changes in the virtual character of the aplanatic images.