Polarization Revisited: The Brewster Angle

Note well the expression for the reflected wave amplitude for in-plane polarization: E_0'' = E_0 &mu#mu;&mu#mu;' n'^2(&thetas#theta;_i) - nn'^2 - n^2^2(&thetas#theta;_i) &mu#mu;&mu#mu;' n'^2(&thetas#theta;_i) + nn'^2 - n^2^2(&thetas#theta;_i)

This amplitude will be zero for certain angles, namely those such that: &mu#mu;&mu#mu;' n'^2(&thetas#theta;_i) = nn'^2 - n^2^2(&thetas#theta;_i)

Squaring both sides and restoring the cosine term to its original form^11.9: (&mu#mu;&mu#mu;')^2 n'^2^2(&thetas#theta;_i) = n^2^2(&thetas#theta;_r) We therefore expect the reflected wave to vanish when &mu#mu;n'&mu#mu;' n = (&thetas#theta;_r)(&thetas#theta;_i) For optical frequencies $\mu \approx \mu'$ (to simplify the algebra somewhat) and this is equivalent to: n'(&thetas#theta;_i) = n(&thetas#theta;_r) From Snell's law this in turn is: nn'(&thetas#theta;_i) = n'n(&thetas#theta;_r) This trancendental equation can be solved by observation from its symmetry. It is true if and only if: (&thetas#theta;_i) = n'n = (&thetas#theta;_r)

The angle of incidence &thetas#theta;_b = ^-1(n'n) is called Brewster's angle. At this particular angle of incidence, the 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 do 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 non-propagating fields. This in turn is related to polarization, dispersion, and skin depth, which we will now treat in some detail (skipping optical fibers per se).