TABLE OF CONTENTS In the analysis in most of the previous chapters it was sufficient to solve the Laplace equation for the potential. Only in the chapter on reflection and transmission the wave equation was given for the bulk regions and its solution in terms of plane waves was used. In order to describe the properties of rough surfaces a more systematic analysis of the solution of the Maxwell equations including the effects on the interface is needed. In this chapter such an analysis will be given. As a first step it will be discussed how to obtain a wave equation valid not only in the bulk, but also at the surface. This extension to the surface poses a special problem, due to the singular nature of the normal components of the electric and magnetic fields at the surface. If one would try to construct the appropriate Green function to write the general solution in integral form, as one usually does, the Green function would have similar singularities. This would eliminate much of its usefulness. The way out of this dilemma is to use the solution of the wave equations for the normal components of the electric displacement field and the magnetic induction, which do not have such singular behavior at the interface, rather than for the normal components of the electric and magnetic fields. This results in a modification of one of the integral operators, which give the non-singular fields in terms of the polarization and magnetization densities, such that...
