Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals. I. Formalism and surface plasmon polariton

A fully relativistic description of the electron energy loss and the induced radiation emission in arbitrary arrays of nonoverlapping metallic cylinders is presented in terms of the multiple scattering method on the basis of vector cylindrical waves. Numerical analysis is given for dilute and dense arrays of Aluminum cylinders with a nanoscale diameter. The results of the electron energy loss spectrum are well correlated with the dispersion relation of coupled surface plasmon polaritons, and can be interpreted with an effective medium approximation when the electron runs inside the arrays. In addition, the cavity modes localized in the grooves between the cylinders can strongly affect the electron energy loss spectrum.