The refractive index of reciprocal electromagnetic media - McCall,MW; Kinsler,P; Topf,RDM (Journal of Optics, 2016)
We study the electromagnetics of media described by identical inhomogeneous relative dielectric and magnetic tensors, ${\boldsymbol{ε }}={\boldsymbol{μ }}.$ Such media occur generically as spatial transformation media, i.e. electromagnetic media that are defined by a deformation of space. We show that such media are completely described by a refractive index $n({\bf{r}},\hat{{\bf{s}}})$ that depends on position ${\bf{r}}$ and direction $\hat{{\bf{s}}},$ but is independent of polarization. The phase surface is always ellipsoidal, and $n({\bf{r}},\hat{{\bf{s}}})$ is therefore represented by the radius vector to the surface of the ellipsoid. We apply our method to calculate the angular dependence of the refractive index in the well-studied cylindrical cloak and to a new kind of structurally chiral medium induced by a twist deformation. By way of a simple example we also show that media for which ${\boldsymbol{ε }}={\boldsymbol{μ }}$ do not in general preserve the impedance properties of vacuum. The implications of this somewhat surprising conclusion for the field of transformation optics are discussed.