 Complex refractive index

For a plane wave proportional to $e^{i(N\omega z/c-\omega t)}$, the complex refractive index separates phase propagation from attenuation. In a nonmagnetic medium $N^2={ϵ}_r$, so $n^2-{\kappa }^2={ϵ}_r^{\prime }$ and $2n\kappa ={ϵ}_r^{\prime \prime }$. The passive branch has $\kappa \ge 0$. If $\kappa \ll n$ and ${ϵ}_r^{\prime }>0$, then $n\simeq \sqrt{{ϵ}_r^{\prime }}$ and $\kappa \simeq {ϵ}_r^{\prime \prime }/(2\sqrt{{ϵ}_r^{\prime }})$. These signs assume the $e^{-i\omega t}$ convention for the complex permittivity.