 Lossless planar surface-plasmon dispersion

For a vacuum-material interface at $z=0$, write the magnetic phasor as $H_y{e}^{ikx-{\beta }_{j}z}$, with ${\beta }_1>0$ in vacuum and ${\beta }_2<0$ in the material. The Ampere-Maxwell law gives $E_x=i{\beta }_j{H}_y/(\omega {\epsilon }_0{ϵ}_j)$ and $E_z=-k{H}_y/(\omega {\epsilon }_0{ϵ}_j)$. Tangential field continuity gives ${\beta }_2=ϵ{\beta }_1$. Together with $k^2-{\beta }_j^2={ϵ}_j{\omega }^2/c^2$, this gives the displayed relation and requires $ϵ<-1$ for real propagating $k$ and decay on both sides. For $ϵ=1-{\omega }_p^2/{\omega }^2$, the branch has $0<\omega <{\omega }_p/\sqrt{2}$ and approaches that upper frequency only as $k\to \infty$. At low frequency, $k=(\omega /c)[1+{\omega }^2/(2{\omega }_p^2)+O({\omega }^4/{\omega }_p^4)]$.