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

TL;DR

This paper derives a causality-based criterion for negative refractive index that accounts for imperfect transparency and relates global loss properties to local behavior, highlighting differences between group velocity and Poynting vector criteria.

Contribution

It introduces a generalized causality-based criterion for negative refractive index that considers imperfect transparency and clarifies the distinction between group velocity and Poynting vector.

Findings

The criterion relates global loss properties to local behavior at a specific frequency.

Causality-based criteria depend on group velocity, not Poynting vector.

Examples show differences between group velocity and Poynting vector criteria.

Abstract

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.