2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/209887Using 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 $ω$. It also allows us to relate the global properties of the loss (i.e. its frequency response) to its local behaviour at $ω$. 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.slightly longer version of published PRLOpticstext