Refractive index and wave vector in passive or active media

Materials that exhibit loss or gain have a complex-valued refractive index $n$. Nevertheless, when considering the propagation of optical pulses, using a complex $n$ is generally inconvenient—hence the standard choice of real-valued refractive index, i.e., $n_s=\text{Re}\left(\sqrt{n^2}\right)$. However, an analysis of pulse propagation based on the second-order wave equation shows that use of $n_s$ results in a wave vector different to that actually exhibited by the propagating pulse. In contrast, an alternative definition $n_c=\sqrt{\text{Re}\left(n^2\right)}$, always correctly provides the wave vector of the pulse. Although for small loss the difference between the two is negligible, in other cases it is significant; it follows that phase and group velocities are also altered. This result has implications for the description of pulse propagation in near resonant situations, such as those typical of metamaterials with negative (or otherwise exotic) refractive indices.

Figure 1

Comparison of $k$ values, as a function of $\varphi =\text{Arg}\left(ϵ\mu \right)$. The alternative choice $k_c$ is shown using a solid line when it is real valued, and dotted when imaginary (“$ık_c$”); the standard choice $\left(k_s\right)$ is given by the dashed line, with the approximate corrected form $\left(k_s^{\prime }\right)$ from Eq. (20) as shown by the dot-dashed line.