Negative refraction in active and passive media: a discussion of various criteria

There are three widely-used conditions for characterizing negative refraction in isotropic dielectric-magnetic materials. Here we demonstrate that whilst all the different conditions are equivalent for purely passive media, they are distinct if active media are considered. Further, these criteria can also be applied to negative refraction in acoustic materials, where we might replace the dielectric permittivity $epsilon$ with a bulk modulus $kappa$, and the magnetic permeability $mu$ with the mass density $rho$.

Causality-based criteria for a negative refractive index must be used with care

Using the principle of causality as expressed in the Kramers-Kronig relations, we derive a generalized criterion for a negative refractive index that admits imperfect transparency at an observation frequency $omega$. It also allows us to relate the global properties of the loss (i.e. its frequency response) to its local behaviour at $omega$. However, causality-based criteria rely the on the group velocity, not the Poynting vector. Since the two are not equivalent, we provide some simple examples to compare the two criteria.