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.