One-way invisibility in isotropic dielectric optical media

Optical materials with a distribution of loss and gain can be used to manipulate waves in fascinating ways, seemingly impossible with ordinary lossless materials. Some recent results have shown that (for planar media) if the spatial distributions of the real and imaginary parts of the permittivity are related to one another by the Kramers-Kronig relations, then reflection can be eliminated. Moreover, if an additional "cancellation condition" is satisfied, then a material can be made invisible for incidence from one side. Here, we give a simple demonstration of these results that should be accessible to undergraduates. In addition, we show how this simple method can be used to prove results about the reflection from permittivity profiles, without ever requiring an exact solution of the Helmholtz equation.