From Fermat's Principle to Invisibility

We present the details of an invisibility cloak whose implementation would not require unphysical material properties, i.e. refractive indices that are singular or less than unity. To achieve this aim, we take the Non-Euclidean Cloak developed by Ulf Leonhardt and Thomas Tyc [1] and combine its refractive index profile with that of the Invisible Sphere [2] to raise all indices above one. We eliminate the singularity of the Invisible Sphere by a transmutation [3].