Resolving the wave-vector in negative refractive media: The sign of $\sqrt{Z}$

TL;DR

This paper investigates how to determine the correct sign of the wave-vector in complex media, including negative refractive materials, by analyzing the analytic properties of the wave-vector in the complex plane to ensure causality.

Contribution

It provides a comprehensive analysis of the sign ambiguity of the wave-vector in complex media using complex analysis and causality constraints, identifying eight distinct physical cases.

Findings

Eight physically distinct wave-vector cases identified

Sign determination linked to causality and complex analysis

Framework applicable to negative refractive media

Abstract

We address the general issue of resolving the wave-vector in complex electromagnetic media including negative refractive media. This requires us to make a physical choice for the sign of a square-root imposed merely by conditions of causality. By considering the analytic behaviour of the wave-vector in the complex plane, it is shown that there are a total of eight physically distinct cases in the four quadrants of two Riemann sheets.