Wave propagation in complex coordinates

We give an interpretation for the use of complex spatial coordinates in electromagnetism, in terms of a family of closely related inhomogeneous media. Using this understanding we find that the phenomenon of reflection can be related to branch cuts in the wave that originate from poles of at complex positions. Demanding that these branch cuts disappear, we derive a new large family of inhomogeneous media that are reflectionless for a single angle of incidence. Extending this property to all angles of incidence leads us to a generalized form of the Pöschl Teller potentials that in general include regions of loss and gain. We conclude by analyzing our findings within the phase integral (WKB) method, and find another very large family of isotropic planar media that from one side have a transmission of unity and reflection of zero, for all angles of incidence.