The refractive index and wave vector in passive or active media

TL;DR

This paper clarifies the correct way to define the refractive index in media with loss or gain, showing that the commonly used real-valued approximation can lead to inaccuracies in wave vector and velocity calculations, especially in exotic materials.

Contribution

It introduces an alternative definition of the refractive index that accurately reflects the wave vector in active or passive media, improving the understanding of pulse propagation.

Findings

The standard real-valued refractive index can differ significantly from the actual wave vector.

Using the alternative definition $n_c$ yields more accurate pulse propagation descriptions.

Implications for metamaterials with negative or exotic refractive indices are discussed.

Abstract

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. $n_s = \RealPart (\sqrt{n^2})$. However, an analysis of pulse propagation based on the second order wave equation shows that use of $n_s$ results in a wave vector \emph{different} to that actually exhibited by the propagating pulse. In contrast, an alternative definition $n_c = \sqrt{\RealPart (n^2)}$, 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.