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 x₃>ζ(x₁, x₂). The surface-profile function ζ(x₁, x₂) is periodic in both x₁ and x₂. In our numerical calculations it was chosen to have the form ζ(x₁, x₂)=ζ₀(cos(2πx₁a)+cos(2πx₂a)), while ε(ω) was assumed to be given by 1-(ωₚ²ω²), where ωₚ is the bulk-plasma frequency of the metal. The frequencies of the surface plasmons in this system were calculated for wave vectors →k_∥ along symmetry directions in the irreducible part of the two-dimensional first Brillouin zone defined by the periodicity of ζ(x₁, x₂). For each value of →k_∥ 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, ωₛₚ(→k_∥)=ωₚ√2. We have considered corrugation strengths ζ₀a up to a value of 0.25, for which the largest frequency shift with respect to ωₛₚ, at →k=(πa, πa), is over 45%.