Transformation optics for surface phenomena: Engineering the Goos-Hänchen effect

Transformation optics, a geometrical recipe for metamaterial design, was originally conceived as a tool to smoothly modify the trajectory of light using continuous coordinate transformations. Here, we show how discontinuous transformations can be used as a geometric framework to understand and manipulate phenomena at the surface of nanophotonic structures. In particular, we show how the Goos-H\"anchen shift---a lateral shift exhibited by totally reflected beams---can be tailored and we provide a classification and complete analytical description of this effect in existing complex media.