Page 454 - Engineering Electromagnetics, 8th Edition

436 ENGINEERING ELECTROMAGNETICS

Figure 12.9 A dielectric slab waveguide
(symmetric case), showing light confinement
to the center material by total reflection.

Wenextconsiderthepossibilityoftotaltransmission.Inthiscase,therequirement
is simply that = 0. We investigate this possibility for the two polarizations. First,
we consider s-polarization. If s = 0, then from (71) we require that η 2s = η 1s ,or

η 2 sec θ 2 = η 1 sec θ 1
Using Snell’s law to write θ 2 in terms of θ 1 , the preceding equation becomes

2
−1/2

n 1 2 2 −1/2
η 2 1 − sin θ 1 = η 1 1 − sin θ 1
n 2
There is no value of θ 1 that will satisfy this, so we turn instead to p-polarization. Using
(67), (68), and (69), with Snell’s law, we find that the condition for p = 0is

2
1/2

n 1 2 2 1/2
η 2 1 − sin θ 1 = η 1 1 − sin θ 1
n 2
This equation does have a solution, which is
n 2
(79)
sin θ 1 = sin θ B =
2
n + n 2 2
1
where we have used η 1 = η 0 /n 1 and η 2 = η 0 /n 2 .We call this special angle θ B , where
total transmission occurs, the Brewster angle or polarization angle. The latter name
comes from the fact that if light having both s- and p-polarization components is
incident at θ 1 = θ B , the p component will be totally transmitted, leaving the partially
reflected light entirely s-polarized. At angles that are slightly off the Brewster angle,
the reflected light is still predominantly s-polarized. Most reflected light that we see
originates from horizontal surfaces (such as the surface of the ocean), and so the light
has mostly horizontal polarization. Polaroid sunglasses take advantage of this fact to
reduce glare, for they are made to block the transmission of horizontally polarized
light while passing light that is vertically polarized.