Without wanting to get all tedious about it, you should be able to
compute the *transmission coefficient* and *reflection
coefficient* for all of these waves from these results. These are
basically the fraction of the energy (per unit area per unit time) in
the incident wave that is transmitted vs being reflected by the surface.

This is a simple idea, but it is a bit tricky to actually compute for a
couple of reasons. One is that we only care about energy that makes it
*through* the surface. The directed intensity of the wave (energy
per unit area per unit time) is the Poynting vector
. In
equation 11.35 above, we found the time-average Poynting vector
in terms of the
-field strength and direction of propagation:
=
12&epsi#epsilon;&mu#mu;|E|^2(k)
(where we have written the direction of propagation in terms of
to avoid confusion with the normal to the *surface*
,
which we recall is
, not
).

We only care about the energy flux *through the plane surface* and
thus must form
for each wave:
I_0 = S_n & = & 12&epsi#epsilon;&mu#mu;|E_0|^2 (&thetas#theta;_i)

I_0' = S_n' & = & 12&epsi#epsilon;'&mu#mu;'|E_0'|^2
(&thetas#theta;_r)

I_0'' = S_n'' & = & 12&epsi#epsilon;&mu#mu;|E_0''|^2
(&thetas#theta;_i)

This is ``easy''^{11.8} only if
the waves are incident
to the surface, in which case one gets:
T & = & I_0'I_0 = &epsi#epsilon;'&mu#mu;&epsi#epsilon;&mu#mu;'
|E_0'|^2|E_0|^2

& = & 4nn'(n'+n)^2
R & = & I_0''I_0 = |E_0''|^2|E_0|^2

& = & (n' - n)^2(n'+n)^2
As a mini-exercise, verify that
(as it must). Seriously, it
takes only three or four lines.