The limits and extension of transformation optics

We explore an apparent limitation of transformation optics, namely that a given transformation induces polarization-dependent impedance gradients that might lead to scattering. This observation must be reconciled with the idea that the transformation optics algorithm is ‘exact’, and leads to perfect morphing of the electromagnetic field without inducing scattering. We also discuss the role of curvature in transformation optics, showing that the conventional algorithm is nothing other than an interesting re-representation of flat Cartesian (Minkowskian) space (spacetime), without any curvature. However, we also consider possible extensions to transformation optics, to schemes that embrace curvature, and include so-called anholonomic transformations. We also show that the conventional spatial transformation optics algorithm can locally be described by six numbers.