Surface-plasmon dispersion relation in the presence of surface roughness

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Abstract
The dispersion relation for surface plasmons on a statistically slightly rough surface is obtained from our earlier results for the effect of surface roughness on the electrostatic image potential. It is found that this curve consists of two branches, in contrast with the single branch that is obtained in the case of a flat surface. On the basis of a Gaussian model for the correlation function of the surface roughness, explicit results are obtained for the dispersion relation and for a response function that enters the theory of the scattering of electrons by surface plasmons. Our results are similar to those obtained recently by Kretschmann et al. [Phys. Rev. Lett. 42, 1312 (1979)] by a different approach and for a different model of the correlation function of the surface roughness, and are consistent with recent experiments that demonstrate the splitting of the surface-plasmon dispersion relation.