Surface plasmon polariton

A transverse magnetic polarization electromagnetic wave bound to an interface through coupling to the material's collective charge response. Its fields propagate along the interface and decay away from it on both sides. In an ideal nonmagnetic lossless planar vacuum-material interface, a real negative relative permittivity below $-1$ supports the bound branch described by the lossless planar surface-plasmon dispersion. Material losses generally make the propagation constant complex.

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, $k=(\omega /c)[1+{\omega }^2/(2{\omega }_p^2)+O({\omega }^4/{\omega }_p^4)]$.