Retarded edge modes of a parabolic wedge

Abstract

The dispersion of the retarded edge modes of a free-electron --- metal parabolic wedge is determined. It is shown that, although the modes are decoupled within the electrostatic approximation, decoupling does not occur when the full set of Maxwell's equations is used. This feature is considered in detail and a novel solution is given, together with some numerical examples. It is shown that the dispersion equation is in the form of an infinite determinant that can be arranged in block diagonal form. The method depends upon certain expansion coefficients that, through a completely new mathematical result concerning Hermite functions, are shown to have a closed form. The validity of labeling the eigen-modes is discussed, and the convergence of the solution scheme is carefully examined. It is found that the retarded modes are quite close to the electrostatic modes, even as the light line is approached, with maximum disparity arising for the odd mode numbers.