Conformal optical devices based on geodesic lenses

Conformal transformation optics provides a simple scheme for manipulating light rays with inhomogeneous isotropic dielectrics. However, there is usually discontinuity for refractive index profile at branch cuts of different virtual Riemann sheets, hence compromising the functionalities. To deal with that, we present a special method for conformal transformation optics based on geodesic lenses with special closed surfaces. The requirement is a continuous refractive index profile of dielectrics, which shows the almost perfect performance of designed devices. We demonstrate such a proposal by achieving conformal transparency (invisibility without cloaking region) and reflection. We can further achieve conformal invisible cloaks by two methods with perfect conductors. The conformal transformation optics method based on geodesic lenses may also find applications in other waves that obey the Helmholtz wave equation in two dimensions.