Surface plasmons on a large-amplitude doubly periodically corrugated surface

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 $x_3>\zeta (x_1, x_2)$. The surface-profile function $\zeta (x_1, x_2)$ is periodic in both $x_1$ and $x_2$. In our numerical calculations it was chosen to have the form $\zeta (x_1, x_2)={\zeta }_0\left(cos\right(\frac{2\pi x_1}{a})$+$cos\left(\frac{2\pi x_2}{a}\right))$, while $\epsilon \left(\omega \right)$ was assumed to be given by $1-\left(\frac{{{\omega }_p}^2}{{\omega }^2}\right)$, where ${\omega }_p$ is the bulk-plasma frequency of the metal. The frequencies of the surface plasmons in this system were calculated for wave vectors ${\overrightarrow{\mathrm{k}}}_{\parallel }$ along symmetry directions in the irreducible part of the two-dimensional first Brillouin zone defined by the periodicity of $\zeta (x_1, x_2)$. For each value of ${\overrightarrow{\mathrm{k}}}_{\parallel }$ 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, ${\omega }_{\mathrm{sp}}\left({\overrightarrow{\mathrm{k}}}_{\parallel }\right)=\frac{{\omega }_p}{\sqrt{2}}$. We have considered corrugation strengths $\frac{{\zeta }_0}{a}$ up to a value of 0.25, for which the largest frequency shift with respect to ${\omega }_{\mathrm{sp}}$, at $\overrightarrow{\mathrm{k}}=(\frac{\pi }{a}, \frac{\pi }{a})$, is over 45%.