Problem 1 [15 points]
Consider as a very simplified model for the dielectric function
(This is the model discussed in lecture in the limit that the damping force vanishes.) Assume ω0 < ωp.
a) Using the dispersion relation k2 = ω2ε(ω)/c2, make a sketch of k vs ω for electromagnetic wave propagation.
b) Re-draw this sketch as ω vs k, and show that for each value of k > 0 there are two allowed values (modes) of ω for the electromagnetic waves. Show that at both small k and at large k one of these two modes has a dispersion relation characteristic of an electromagnetic wave in the vacuum, i.e. ω = vpk where vp is only weakly dependent on k. Show that the other mode has a frequency ω that is, to lowest order, independent of k and so like some internal atomic mode of vibration. Show that these two modes exchange their characteristic behavior as one crosses from small k to large k. In this intermediate region the modes have a mixed character and are referred to as "polaritons".
c) Show that, at a given value of k, the higher frequency mode has a phase velocity that satisfies vp > c, while the lower frequency mode satisfies vp < c. Show that for both modes the group velocity always satisfies vg < c.
Problem 2 [40 points]
In 1845, Faraday made the experimental discovery that a linearly polarized light wave (i.e. EM wave) could have its direction of polarization rotated by passing it through a static magnetic field in a dielectric. This was an important step in demonstrating that light was in some way an electromagnetic phenomenon. The purpose of this problem is to derive this result theoretically. For this problem, assume that µ is a constant.
a) [20 points] Start by considering how the polarization of atoms in the dielectric is influenced by the presence of a uniform static magnetic field. Consider an electron in a uniform magnetic field
B oriented in the z-direction, and an oscillating electric field due to an EM wave,
E(t) = Eωe-iωt,
where Eω is in the xy plane.
Assume that there is a restoring force on the electron, Frest = -mω o2r, but assume the damping force is negligible. The solution for the electron's position will then have the form,
r(t) = rωe-iωt.
Write down the equations of motion for the x and y components of the electrons position, substitute in the above form of the solution, and solve for rω in terms of Eω. Show that rω is not in general parallel to Eω. Show that rω will be parallel to Eω when Eω is circularly polarized, i.e. when Eω = Eω(ex±iey), where (+) and (-) refer to right and left handed polarizations respectively, and ex,y are unit vectors in the x and y directions. For these two circular polarizations, show that the atomic polarizations can be written as,
pω = -e rω = α±(ω)Eω, where α+ ≠ α-.
b) [5 points] Using the above result, show that the two different (±) circularly polarized waves travel through the medium according to different dispersion relations,
c2k±2 = ω2µε±(ω),
where the dielectric functions, ε±(ω) = 1+4πNα±(ω), are related to the atomic polarizabilities in the usual way. N is the density of polarizable atoms.
c) [10 points] Consider now a slab of the dielectric of thickness L (the surfaces of the slab are perpendicular to the z axis). Suppose a plane wave, linearly polarized in the x direction, enters the slab at z=0. Show that when the wave exits the slab at z=L, the direction of polarization has been rotated. What is the angle of rotation? Has it rotated clockwise or counterclockwise? You may ignore reflections at the interfaces.
Hint: To do this part, recall that any linearly polarized wave can always be written as a superposition of counter rotating circularly polarized waves.