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

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 antidirected to the Poynting vector, although the conditions presented here do not assume a priori a negative solution branch for $n$.

Figure 1

Plot of the family of solutions for $n$ for which a Drude model is used for $\epsilon$ and a Lorentz model is used for $\mu$.

Figure 2

Plot of the $n_{+}$ solution branch, together with its KK transformation for (a) the Lorentz-Lorentz model and (b) the Drude-Lorentz model. Note that the KK transformed $n$ does not match $\kappa$.

Figure 3

(a) Plot of the refractive index under the Veselago condition [choosing the $n<0$ solution branch ($n_{-}$) when $\epsilon$ and $\mu$ are simultaneously negative] for the Lorentz-Lorentz model. (b) A comparison between that Veselago refractive index solution and the solution determined by the Depine-Lakhtakia condition. We see that that Veselago criterion is not sufficiently general, because it does not sustain the validity of the KK relations.

Figure 4

(a) A comparison between the Depine-Lakhtakia condition and the analytic condition for the Lorentz-Lorentz model. (b) The analytic condition imposed on the solution branch restores the KK relations, and hence causality.