= Lossless planar surface-plasmon dispersion
{title2=$k^2=(\omega^2/c^2)\epsilon/(\epsilon+1)$}
For a vacuum-material interface at $z=0$, write the magnetic phasor as $H_y e^{ikx-\beta_jz}$, with $\beta_1>0$ in vacuum and $\beta_2<0$ in the material. The <Ampere-Maxwell law> gives $E_x=i\beta_jH_y/(\omega\varepsilon_0\epsilon_j)$ and $E_z=-kH_y/(\omega\varepsilon_0\epsilon_j)$. Tangential field continuity gives $\beta_2=\epsilon\beta_1$. Together with $k^2-\beta_j^2=\epsilon_j\omega^2/c^2$, this gives the displayed relation and requires $\epsilon<-1$ for real propagating $k$ and decay on both sides. For $\epsilon=1-\omega_p^2/\omega^2$, the branch has $0<\omega<\omega_p/\sqrt2$ and approaches that upper frequency only as $k\to\infty$. At low frequency, $k=(\omega/c)[1+\omega^2/(2\omega_p^2)+O(\omega^4/\omega_p^4)]$.
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