Coordinate choice and Brewster's Law

What, then, is a ``convenient coordinate system''? One where $\hat{\mbox{\boldmath$n$}} = \hat{\mbox{\boldmath$z$}}$ is perpendicular to the surface is good for starters. The remaining two coordinates are selected to define the plane of reflection and refraction and its perpendicular. This is particularly useful because (as we shall see) the reflected and refracted intensities depend on their polarization relative to the plane of scattering.

Again, to motivate this before messing with the algebra, you hopefully are all familiar with the result taught at the kiddy-physics level known as Brewster's Law. The argument works like this: because the refracted ray consists of (basically) dipole re-radiation of the incident field at the surface and because dipoles do not radiate along the direction of the dipole moment, the polarization component with $\mbox{\boldmath$E$}$ in the scattering plane has a component in this direction.

This leads to the insight that at certain angles the refracted ray will be completely polarized perpendicular to the scattering plane (Brewster's Law)! Our algebra needs to have this decomposition built in from the beginning or we'll have to work very hard indeed to obtain this as a result!

Let us therefore treat rays polarized in or perpendicular to the plane of incidence/reflection/refraction separately.