The Faraday Effect

The Faraday Effect

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. You should derive this result theoretically for the simple model of a dielectric, as outlined below. Assume that μ is a constant.

Start by considering how the polarization of atoms in the dielectric, induced by the oscillating electric field of the EM wave, is influenced by the presence of a uniform static magnetic field (note, this is not the magnetic field of the EM wave, this is an additional, spatially uniform and constant in time, magnetic field). We will take the wave to be traveling in the z-direction, and the static magnetic field to be also in the z-direction.

An electron in a polarizable atom feels a force from both the static B and an oscillating electric field due to the EM wave,

E(t) = E_ωe^-iωt,

where E_ω is in the xy plane. The electron also feels the restoring force binding it to the nucleus, F_rest = -mω _o²r, but we will assume the damping force is negligible. The solution for the electron's position will then have the form,

r(t) = r_ωe^-iωt.

Write down the equations of motion for the x and y components of the electron's position, substitute in the above form of the solution, and solve for r_ω in terms of E_ω. You should find that r_ω is not in general parallel to E_ω. However r_ω will be parallel to E_ω when E_ω is circularly polarized, i.e. when E_ω = E_ω(e_x±ie_y)/√2, where (+) and (-) refer to right and left handed polarizations respectively, and e_x,y are unit vectors in the x and y directions. For these two circular polarizations, you should find that the atomic polarizations can be written as,

p_ω = -e r_ω = α_±(ω)E_ω, where α₊ ≠ α_-.

Using this result, one has that the different (±) circularly polarized waves travel through the dielectric 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.

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 with its electric field 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? You may assume that |B| is small to simplify your expression. Has the polarization rotated clockwise or counterclockwise? You may ignore reflections at the interfaces.

It will help to recall that any linearly polarized wave can always be written as a superposition of counter rotating circularly polarized waves.