Nontensorial Transformation Optics

We present an alternative version of transformation optics that allows us to mold the flow of light without rotating or scaling the electromagnetic fields. The resulting media experience unusual force densities, are nonreciprocal, and exhibit loss or gain. Because of these singular features, a variety of effects and devices unreachable by standard transformation optics can be achieved, including reflectionless light compression, optical modes with arbitrary in-plane polarization, and special isolators.