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 behavior at $\omega$. However, causality-based criteria rely on the group velocity, not the Poynting vector. Since the two are not equivalent, we provide some simple examples to compare the two criteria.

Figure 1

A double ($ϵ$ and $\mu$) plasmon system exhibiting NRI, with both plasma frequencies being ${\omega }_p\simeq 1.4$. It compares the (${\chi }_E$) NPVE condition (thick lines) to the (${\chi }_G$) NPVG one (thin lines). The results shown are for $\gamma =0.02$ (solid lines), $\gamma =0.04$ (dashed lines), and $\gamma =0.06$ (dot-dashed lines).

Figure 2

A system exhibiting narrow band NRI. It combines a lossy resonance at ${\omega }_1=1$ (with ${\gamma }_1=0.05$, ${\sigma }_1=-5$) and an active one at ${\omega }_2=1.05$ (with ${\gamma }_2=0.01$ and ${\sigma }_2=1.02$). (a) The real parts and imaginary parts of $ϵ$ and $\mu$. (b) Comparison of the NPVE (${\chi }_E$) and NPVG (${\chi }_G$) criteria. (c) Expanded view around ${\omega }_2=1.05$, showing also the NPVG approximation from Eq. (6) (labeled ${\chi }_{GA}$), and $Q_1$ and $Q_2$.