Table of Contents
Polarization of Light: Malus' law
An ordinary light source consists of a very large number of randomly oriented atoms which emit light. The emitted light is only polarized for a short period of about $10^{-8}$ sec. Over longer times, the polarization changes randomly in time at such a rate as to render individual polarization states indiscernible. Thus, we say that such a light source is unpolarized.
If unpolarized light is passed through two polarizers in succession, Malus' Law says that the intensity of the transmitted light has the form
| $\begin{equation} I = I_0\cos^2\varphi \end{equation} \tag{1}\label{eq:Malus} $ |
where $\varphi$ is the angle between the pass-axes of the two polarizers and where $I_0$ is the intensity of the light after passing through the first polarizing filter.
Why does this equation hold? In which direction is the light polarized after emerging from the second polarizer at angle $\varphi$?
The polarizers we shall use are plastic they are good electrical conductors (at optical frequencies) along one direction, but poor conductors in the orthogonal direction. We have no knowledge of the actual direction of the pass axis of the polarizers. However, we will still be able to measure the difference, $\varphi$, between the pass axis angles.
Use the apparatus shown in Fig. 1. One polarizer is mounted in an angle scale and the second polarizer is magnetically attached to the face of the detector housing.
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| Figure 1: Malus’ Law configuration |
Start the software on your computer's desktop. Start by turning on the LED. Next, turn the polarizer on the apparatus to 0 degrees and press the Calibrate button to normalize your readings. This basically sets the maximum intensity to 1000, making it easier to quickly judge how the light levels are changing. After this, you can use the Single button to take a reading from the light sensor.
Varying the angle of the polarizer in the degree scale, measure and plot the transmitted intensity (detector readout) on the $y$-axis as a function of polarizer angle on the $x$-axis.
IMPORTANT: Choose random increments of polarizer angle, e.g., sometimes 10 degrees, sometimes 7 degrees, sometimes 16 degrees. Spacing the angle irregularly helps insure that the fitting software will not be fooled!
To see how well your data agree with the form of Eq. $\eqref{eq:Malus}$, use the Google Colab notebook below to perform a fit. Do your data fit this functional form? Are your data consistent with Malus' law?
Phase Shifts and Circular Polarization
Many transparent materials can be characterized by a single index of refraction (e.g. glass, water, air. etc.). A few materials may have light propagate at different speeds depending on what direction it travels through it. Substances with two distinct indices of refraction are thus called birefringent. When passing polarized light through such a material, some interesting things can happen…
quarter waveplates
We'll start with a plastic disc known as a quarter waveplate. For visible light, its thickness is carefully engineered to produce a change of phase of 90 degrees (one quarter of a wavelength). This can result in converting linearly polarized light to circularly polarized light. In reality, the light will be elliptically polarized to a degree depending on the orientation of the quarter waveplate.
* experiment with turning waveplate *
* experiment with turning polarizer *
Other materials
Uniaxial Birefringence
There are other materials that can change the phase of a light wave passing through them to differing degrees.
* Calcite *
Biaxial Birefringence
* Mica *
Strain birefringence
* Plastic *
Rotary dispersion
Sugars