Relationship between the Kramers-Kronig relations and negative index of refraction

TL;DR

This paper establishes that the Kramers-Kronig relations, rooted in causality and analyticity, serve as a general criterion for the physical realization of negative index of refraction in materials.

Contribution

It links the negative index condition to the analyticity and Kramers-Kronig relations without assuming a negative solution branch for the index.

Findings

Kramers-Kronig relations imply negative index when analyticity is satisfied.

The negative index region aligns with the Depine-Lakhtakia phase velocity condition.

Causality and analyticity are sufficient for negative index without a negative solution branch.

Abstract

The condition for a negative index of refraction with respect to the vacuum index is established in terms of permittivity and permeability susceptibilities. It is found that the imposition of analyticity to satisfy the Kramers-Kronig relations is a sufficiently general criterion for a physical negative index. The satisfaction of the Kramers-Kronig relations is a manifestation of the principle of causality and the predicted frequency region of negative index agrees with the Depine-Lakhtakia condition for the phase velocity being anti-directed to the Poynting vector, although the conditions presented here do not assume {\it a priori} a negative solution branch for n.