Surface plasmons on a large-amplitude doubly periodically corrugated surface

Abstract

On the basis of the Rayleigh hypothesis we have derived the dispersion relation for surface plasmons propagating in an arbitrary direction along a doubly periodically corrugated metal surface, and have solved it numerically. The system considered consists of a vacuum in the region ${x}_{3}>\ensuremath{\zeta}({x}_{1}, {x}_{2})$. The surface-profile function $\ensuremath{\zeta}({x}_{1}, {x}_{2})$ is periodic in both ${x}_{1}$ and ${x}_{2}$. In our numerical calculations it was chosen to have the form $\ensuremath{\zeta}({x}_{1}, {x}_{2})={\ensuremath{\zeta}}_{0}(cos(\frac{2\ensuremath{\pi}{x}_{1}}{a})$+$cos(\frac{2\ensuremath{\pi}{x}_{2}}{a}))$, while $\ensuremath{\epsilon}(\ensuremath{\omega})$ was assumed to be given by $1\ensuremath{-}(\frac{{{\ensuremath{\omega}}_{p}}^{2}}{{\ensuremath{\omega}}^{2}})$, where ${\ensuremath{\omega}}_{p}$ is the bulk-plasma frequency of the metal. The frequencies of the surface plasmons in this system were calculated for wave vectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{\ensuremath{\parallel}}$ along symmetry directions in the irreducible part of the two-dimensional first Brillouin zone defined by the periodicity of $\ensuremath{\zeta}({x}_{1}, {x}_{2})$. For each value of ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{\ensuremath{\parallel}}$ there is an infinity of branches of the dispersion curve with frequencies above and below the dispersion curve for surface plasmons on a flat surface, ${\ensuremath{\omega}}_{\mathrm{sp}}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{\ensuremath{\parallel}})=\frac{{\ensuremath{\omega}}_{p}}{\sqrt{2}}$. We have considered corrugation strengths $\frac{{\ensuremath{\zeta}}_{0}}{a}$ up to a value of 0.25, for which the largest frequency shift with respect to ${\ensuremath{\omega}}_{\mathrm{sp}}$, at $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}=(\frac{\ensuremath{\pi}}{a}, \frac{\ensuremath{\pi}}{a})$, is over 45%.