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Note well the expression for the reflected wave amplitude for inplane
polarization:

(9.86) 
This amplitude will be zero for certain angles, namely those such
that:

(9.87) 
Squaring both sides and restoring cosine term to its original form:

(9.88) 
We therefore expect the reflected wave to vanish when

(9.89) 
For optical frequencies
(to simplify the algebra
somewhat) and this is equivalent to:

(9.90) 
From Snell's law this in turn is:

(9.91) 
This trancendental equation can be solved by observation from its
symmetry. It is true if and only if:

(9.92) 
The angle of incidence

(9.93) 
is called Brewster's angle. At this reflected and refracted
wave travel at right angles with respect to one another according
to Snell's law. This means that the dipoles in the second medium that
are responsible for the reflected wave are parallel to the
direction of propagation and (as we shall see) oscillating dipoles to
not radiate in the direction of their dipole moment! However, the
result above was obtained without any appeal to the microscopic
properties of the dielectric moments that actually coherently scatter
the incident wave at the surface  it follows strictly as the result of
solving a boundary value problem for electromagnetic plane waves.
Students interested in optical fibers are encouraged to read further in
Jackson, 7.4 and learn how the cancellation and reradiation of the waves
to produce a reflected wave at angles where total internal reflection
happens does not occur instantaneously at the refracting surface but in
fact involves the penetration of the second medium some small distance
by nonpropagating fields. This in turn is related to polarization,
dispersion, and skin depth, which we will now treat in some detail.
Next: Dispersion
Up: Dynamics and Reflection/Refraction
Previous: Intensity
Contents
Robert G. Brown
20071228