This paper has the best simplified classical explanation that I could find:

http://web.mit.edu/8.13/WWW/JLExperiments/JLExp_08.





Explanation of what the paper means:



The paper models each electron in the Terbium-glass material as a charged mass on a spring, with a velocity perpendicular to the propagation of the light (and the magnetic field). The spring represents the attractive electric force between the nucleus of the atom and the electron. This ideal spring has an equilibrium position of length equal to zero. (See diagram above)



Light propagates, or moves through space, along the x-axis, by convention. The z and y axes are perpendicular to each other and to the x axis. Light is a combination of an electric field whose magnitude varies sinusoidally with the propagation of the light, and a perpendicular (transverse) magnetic field whose intensity also varies sinusoidally with the propagation of the light. The maximum magnitude of E (the electric field) is equal to the maximum magnitude of B (the magnetic field) divided by c (the speed of light in a vacuum, 299,792,458 m/s). Visible light is one type of electromagnetic wave. (See diagram above)



Linearly polarized light is an electromagnetic wave whose electric field is confined to one plane and whose magnetic field is confined to another plane that is perpendicular to the first. Linear polarized light is thus called a plane wave. (See diagram above)



Circularly polarized light is composed of two plane waves of equal amplitude but differing in phase by 90 or 270 degrees. The sum of the two periodic electric field vectors gives a net electric field vector with constant magnitude, and that is rotating either clockwise or counter clockwise (both from the point of view of the light source), depending on whether the phase difference is 90 degrees or 270 degrees. Right-handed circularly polarized light and left-handed circularly polarized light have electric field vectors that spiral in opposite directions as the light propagates through space. (See diagram above)



When working with the actual device, the linear polarization is considered to be the direction of the Z-axis (the one with the electric field). An equal combination of right and left handed circularly polarized light waves, in phase with each other, add up exactly to a linear polarized light wave. This is because the horizontal plane waves are 180 degrees out of phase with each other and will thus cancel each other out. The vertical plane waves, which are in phase with each other, just give a net vertical plane wave that has twice the amplitude.



The electric field applies an attractive or repulsive force on the electron. Because the rotating field vector of the circularly polarized light is periodic in nature in the zy plane (perpendicular to the propagation), it will apply a periodic force on the electron, causing it to experience a periodic displacement in said plane. This periodic displacement occurs in opposite directions for right hand and left hand polarizations, because the periodic electric field is cycling in opposite directions. This results in the electron being displaced some distance from its equilibrium position at all times.



However, linearly polarized light would only cause a periodic up/down motion of the electron on the “spring.” If one averages the net displacement of the electron over time, it would be zero. Thus, the electron just periodically gains and loses energy, with a net change in its energy of zero.



The amplitude of the electron’s angular motion (or equivalently, in the case of linearly polarized light, the displacement from the electron’s equilibrium position) is directly proportional to its energy; the energy is stored in the “spring.” Thus, when the electric field of the circularly polarized light wave acts on the electron, it must either add or subtract (depending on its handedness) from the electron's energy.



This energy that the electron gains is lost from the light, causing the amplitude of the light to decrease (or the wave dissolves altogether). Normally, with no external magnetic field present, the linearly polarized light wave is re-emitted by the periodically moving electron. The periodically moving electron naturally creates a self-propagating electromagnetic wave: it creates a periodic electric field due to its charge and periodic motion, and a transverse periodic magnetic field due to its charge, velocity, and periodic motion. This causes the "spring" to return to its original position, the electron to again reach its equilibrium position, and the electron to lose its previously gained energy. This light wave is identical to the original right or left handed polarized light wave that first interacted with the electron.



If there is a magnetic field, the re-emitted wave will be different from the absorbed wave. Because the affected electron has been given a velocity perpendicular to the applied magnetic field by the periodic electric field of the circularly polarized light wave, it experiences a force up that stretches the "spring" a little bit farther up or down. (Remember that the electron is already experiencing a circular periodic motion and has an unusually high or low energy due to the periodic electric field from the circularly polarized light; the "spring" has already been stretched by the light.) The force on a moving charged particle is equal to qvB, which is the charge of the particle times the velocity of the particle times the magnitude of the magnetic field that is perpendicular to the velocity of the charged particle. This force is what causes the small increase or decrease in the electron's energy, depending on the direction of the force -- which depends on the direction of the electron's velocity -- which depends on the handedness of the circularly polarized light that interacted with the electron.



When the "spring" pulls the electron back to its original equilibrium position, the electron loses energy and emits an electromagnetic wave, just like in the case discussed earlier where no external magnetic field is present. However, depending on the handedness of the light, the electron falls a slightly different distance (the "spring" releases a slightly different amount of energy). Therefore, the circularly polarized light waves that are released have slightly different wavelengths, as the energy of an electromagnetic wave is inversely proportional to its wavelength.



The index of refraction is n, which equals c, the speed of light in a vacuum, divided by the speed of the light propagating through the medium. As do many materials, the glass rod medium inherently has different indexes of refraction at different wavelengths of light. This can be explained by the different losses of energy from the oscillating electric field to the electron, which is made to oscillate. The closer the frequency of the light (a constant times one over the wavelength) to the natural resonant frequency of the electron as a mass on a "spring" (depends only on mass and spring constant), the more energy lost to the oscillating electron and the higher the amplitude of the electron. The higher the amplitude of the electron's periodic motion, the longer it takes the electron to re-emit the absorbed electromagnetic wave, and thus the longer it takes for the light to propagate through the medium a certain distance. The slower propagation is equivalent to a higher index of refraction. The phenomenon of different indexes of refraction existing for light at different wavelengths in a medium is called dispersion.



So, when one gets the two aforementioned circularly polarized waves with slightly different wavelengths propagating in such a medium, due to the applied external magnetic field of the permanent magnets, one travels slightly faster than the other due to dispersion. Finally, this in turn causes these two waves to become slightly out of phase.



The sum of two circularly polarized waves of equal magnitude that are out of phase is a rotated linearly polarized wave and a leftover left or right handed circularly polarized component. When this mix of waves reaches the Polaroids, only the rotated linearly polarized light can pass through, and the leftover circularly polarized component is screened out. Thus, one can observe the apparent rotation.





Summary:



Ultimately, it is the spiraling electric field vectors of the two polarized light components of the linearly polarized light that give the electrons their velocities in the zy plane and thus allow the applied external magnetic field to act on the electrons, causing them to lose slightly different amounts of energy and to thus re-emit light at slightly different wavelengths. Then, due to the natural dispersion of the medium, the two circularly polarized waves become out of phase; they then will add to net a slightly rotated observable linearly polarized wave component and a small left-over right or left handed circularly polarized component that cannot pass through the second linear polarizer.

