  In the last ten years, the technology of differential geometry, ubiquitous in
gravitational physics, has found its place in the field of optics. It has been
successfully used in the design of optical metamaterials, through a technique
now known as "transformation optics". This method, however, only applies for
the particular class of metamaterials known as impedance matched, that is,
materials whose electric permittivity is equal to their magnetic permeability.
In that case, the material may be described by a spacetime metric. In the
present work we will introduce a generalization of the geometric methods of
transformation optics to situations in which the material is not impedance
matched. In such situation, the material -or more precisely, its constitutive
tensor- will not be described by a metric only. We bring in a second tensor,
with the local symmetries of the Weyl tensor, the "$W$-tensor". In the
geometric optics approximation we show how the properties of the $W$-tensor are
related to the asymmetric transmission of the material. We apply this feature
into the design of a particularly interesting set of asymmetric materials.
These materials are birefringent when light rays approach the material in a
given direction, but behave just like vacuum when the rays have the opposite
direction with the appropriate polarization (or, in some cases, independently
of the polarization).
