What, then, is a ``convenient coordinate system''? One where
is perpendicular to the surface is good for
starters^{11.6}. 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
*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.