Introduction

Daniel FitzGreen

Faraday Rotation

Introduction

2.1 Polarized Electromagnetic Radiation

Electromagnetic radiation (e.g. visible light) is a transverse wave, meaning that the wave oscillates perpendicularly to the direction of propagation. Radiation that travels as a transverse wave can be polarized. One can consider electromagnetic radiation (we’ll just say ‘light’ from here on out, even though these concepts apply to transverse waves, generally) in the linear framework as a combination of horizontally and vertically polarized waves:

(1) $\begin{equation*} \vec{E} = E_x \hat{x} + E_y \hat{y}. \end{equation*}$

Alternatively, one can define polarized light in terms of right-handed

(2) $\begin{equation*} \vec{E}_{rh} = E_0(\hat{x}-i\hat{y})e^{i(-kz-\omega t)} \end{equation*}$

and left-handed

(3) $\begin{equation*} \vec{E}_{lh} = E_0(\hat{x}+i\hat{y})e^{i(kz-\omega t)} \end{equation*}$

circular polarization. Hopefully it’s obvious that a linear polarization can be made from a superposition of left- and right-circularly polarized light, and vice versa.

2.2 Malus’ Law

Malus’ Law states that when linearly polarized light passes through a polarizing filter, then intensity of the output is proportional to the square of the cosine of the angle between the polarization of the incoming light and the polarizing filter:

(4) $\begin{equation*} I = I_0 \cos^2{\theta} \end{equation*}$

The squared cosine term comes from the fact that the change in amplitude of the wave is proportional to $\cos{\theta}$ , and the intensity is proportional to the square of the amplitude.

2.3 Faraday Rotation

WARNING: Understanding the dynamics between the light, the sample material, and the externally applied magnetic field requires some high-level E&M that you may not have learned, yet.

As light passes through matter, it accelerates the charged particles that make it up (mostly the electrons due to their lower mass). The force on the electron due to the EM waves is

(5) $\begin{equation*} F = m\dot{\vec{v}} = -e(\vec{E}+\cfrac{\vec{v}}{c}\times\vec{B}). \end{equation*}$

In the absence of a significant magnetic field, one need only concern ourselves with the force due to the electric field of the light. The oscillating EM wave causes the electrons to oscillate and re-radiate that EM wave. It is this constant radiating and re-radiating of the EM wave that leads to different speeds of light in different materials – though the frequency and polarization remains unchanged.

An interesting effect occurs when one applies an external magnetic field in the direction the wave is propagating, as in Figure 1 (which means we are no longer ignoring the magnetic field component in Equation (5)). The electrons in the material undergo cyclotron motion according to the direction of the magnetic field. The incoming linearly polarized light is considered as a superposition of right- and left-circularly polarized light. As the light travels through matter, it will force the electrons to rotate in right- and left-handed circles, which will be faster or slower depending on the direction of the cyclotron motion due to the magnetic field. Therefore, the light is transmitted through the material at different speeds depending on the handedness of the light. When the light is then considered in a linear framework again, one finds that the direction of polarization has rotated by some angle $\theta$ .

Another way to consider Faraday rotation is that the material the light passes through actually has two indices of refraction – $n_{lh}$ and $n_{rh}$ . Such a material is said to be circularly birefringent. These materials are typically made up of molecules that are chiral stereoisomers, meaning that molecules exist as mirror images of each other. The two different indices of refraction cause the light to pass through the material at two different speeds, resulting in a phase difference between the two circular polarizations. When the two differently-phased circular polarizations are combined to retrieve the linearly-polarized beam, one finds that the direction of polarization has rotated by some angle $\theta$ .

When the two differently-phased circular polarizations are combined to retrieve the linearly-polarized beam, one finds that the direction of polarization has changed by some angle $\theta$ . Math enthusiasts can see further details in the derivations here and here. This phenomenon was the first evidence that light and electromagnetism are related.

Math enthusiasts can see further details in the derivations here and here. Mathematically, the change in polarization is expressed as

(6) $\begin{equation*} \theta = VBd,\end{equation*}$

where $B$ is the magnetic flux density (in units of Tesla), $d$ is the length of the material in the magnetic field, and $V$ is the Verdet constant which is an inherent property of the material through which the light travels. The Verdet constant is usually quotes in units of radians-milliTesla per meter (rad $\cdot$ mT/m).

2.4 Lock-In Detection

Small changes in polarization can lead to changes in intensity that are on the order of magnitude of the noise in the signal, making even an imprecise direct measurement impossible. One option is to use a ‘lock-in’ technique, where the source of the change in a signal is modulated, then the signal at only that modulation frequency is measured. For this experiment, that means modulating the current through the solenoid at some known frequency, and looking for changes in the detector output at that frequency. A lock-in amplifier

Figure 1: A depiction of the rotation of the polarization of light as it passes through matter in a magnetic field. This manual uses $\theta$ instead of $\beta$ for the angle of rotation. Figure from Wikipedia.

achieves that goal by multiplying (or ‘mixing’) the two signals (the modulation input and experiment output). Since sine waves are orthogonal, integrating the dot product of two sine waves will always result in zero as long as the integration time is much longer than the wavelength of either sine wave (See Figure 2). The only time the output is not zero is when the frequencies of the sine waves are almost identical. In that way, an experimenter can probe the signal at exactly the frequency they need while ignoring noise at higher AND lower frequencies. Some lock-in amplifiers can reach a bandwidth of about 0.5 Hz!

Figure 2: Two sin waves of different frequencies are multiplied, then integrated over some amount of time that is longer than their wavelengths. The result is zero unless the two sine waves are approximately the same frequency.

2.5 V_rms vs V_pp

There are several definitions for the amplitude of a wave, and the instruments in this experiment aren’t always using the same ones. The peak-to-peak amplitude of a wave is the difference between the highest and lowest points of the wave, as in Figure 3(a). The peak to peak voltage from the function generator, for example, is $V_{pp}$ . Another common definition of amplitude is the root-mean-square of the amplitude ( $V_{rms}$ ), as in Figure 3(b). The best definition of amplitude to use often depends on the context of the experiment. It will be handy to know that $V_{pp} = 2\sqrt{2}V_{rms}$