In this lab, we will do the following:

• test Malus’ law;
• study polarization caused by reflection (Brewster’s angle); and
• explore polarizing filters and retardation plates (circular polarization).

In the discussion of wave propagation on stretched strings, it was pointed out that the vibration or displacement of the wave was transverse, i.e., perpendicular to the direction of wave travel. Longitudinal waves, such as sound waves or compression waves on springs, have displacement parallel to the direction of wave travel.

If the vibration of a transverse wave is entirely in one plane, the wave is said to be plane (or linearly) polarized. The equation of plane polarized transverse wave, traveling in the $z$-direction and vibrating in the $xz$-plane can be written as

where $\mathbf{d}$  is the instantaneous displacement in the $x$-direction, $A_x$ is the amplitude, and $\hat{\mathbf{x}}$ is a unit vector in the $x$-direction to remind us that the vibration is a vector in the $xz$-plane. Similarly, a plane-polarized wave, traveling in the $z$-direction and vibrating in the $yz$-plane, can be written as

where $\hat{\mathbf{y}}$ is a unit vector in the $y$-direction. This points out that there are two independent orthogonal polarizations for transverse waves. (Longitudinal waves cannot be described in terms of polarization.)

Transverse waves may be unpolarized, however, if the vibration is random in direction. If this is the case, the vibrations can be resolved into two independent orthogonal vibrations parallel to $x$ and $y$. Hence, an unpolarized wave can be said to be vibrating in both states of polarization simultaneously and independently.

Any plane polarized wave can also be resolved into two orthogonal directions (and it is often convenient to do so), but the components have a definite phase relationship between one another (i.e., they are not random). For this case, the two polarization components can be recombined into a plane polarized wave.

The transverse nature of electromagnetic radiation is confirmed by the observation of polarized waves. For most purposes it is sufficient to consider the vibrating electric vector component of electromagnetic radiation. The magnetic vector, however, always accompanies the electric vector and is at right angles to it.

One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.

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

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 that 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 fixed.

Figure 1: Malus’ Law configuration

Start the software (called “Interface”) on your computer's desktop. Start by turning on the lamp. Next, turn the polarizer on the apparatus to 0 degrees and press the Calibrate button to normalize your readings. (This 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.

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?

Recall that the speed of propagation of light through a transparent material is given by $v = c/n$ where $c$ is the speed of light in vacuum and $n$ is the index of refraction of the material. For some materials, this relation is more complex. These materials, called birefringent, have two different indices of refraction and exhibit interesting properties. Fig. 5 shows a block of birefringent material, characterized by the two different indices of refraction, $n_x$ and $n_y$.

In Fig. 5, a plane-polarized beam of light is sent into the material from the left. The polarization direction of the light is 45 degrees into the page. This polarization can be resolved into two components: one parallel to the $x$-axis and the other parallel to the $y$-axis. The light with the electric field component parallel to the $x$-axis ($E_x = E\cos\theta$) will have a propagation speed of $v_x = c/n_x$. The light with the electric field component parallel to the $y$-axis ($E_y = E\sin\theta$) will have a propagation speed of $v_y = c/n_y$.

Figure 5: Phase retardation of plane polarized light passing through a birefringent material.

Note that if $n_x > n_y$, then $v_x < v_y$. In this case, we may call the $x$-axis the slow axis and the $y$-axis the fast axis. Note that light with its electric field in the $x$-direction has been retarded relative to the light whose electric field is in the $y$-direction. Note that these two components arrive at the medium in phase with each other, but emerge from the material out of phase.

Let us examine how these components travel through the birefringent material. We define the optical path as $nL$ where $L$ is the geometric path length through a material of index $n$. The difference in optical paths in Fig. 5 is then

and the phase difference is

If the phase difference is $\varphi = \frac{\pi}{2},\frac{3\pi}{2},\frac{5\pi}{2},\dots,$  then as the light approaches from the left, we would see its electric field first pointing to the right, then up, then to the left, then down, and so forth. The electric field vector would appear to rotate in a circular path and the light is said to be circularly polarized.

You will use the same apparatus you used for Malus' law, however without the sensor. (All measurements in this section will be qualitative.)

At your station, you should have the Malus' law apparatus, a linear polarizer (in a metal ring), two solutions of sugar (one labeled fructose and one labeled sucrose), and an orange piece of bendy rubber.

In addition to what is at your station, the following items are also available in the common area. From left to right: $\lambda/4$ waveplate, “squeeze plastic” (squeeze at the end close to the metal holder to strain the area near where the plastic splits), calcite crystal, glucose, and corn syrup.

Look through the Malus' law apparatus at the lamp (or at an overhead light in the room). Rotate the second polarizer (closer to your eye) to get maximum extinction of the light. You may still see some blueish light passing through; this is normal for our setup.

Extinguish the light as best you can by setting the two polarizers perpendicular.

Next, hold the loose linear polarizer (in the metal ring) in between the two polarizers of the Malus' law apparatus. (Your partner may need to hold something so you have enough hands!)

Rotate the polarizer in between and describe what you observe. Why is it possible to see light through this system? Is it possible to rotate the final polarizer to extinguish the light again?

Now, instead hold the $\lambda/4$ plate (available from the common area) in the light path so that it is between the first polarizer and the second polarizer.

Rotate the $\lambda/4$  plate and describe what you observe. Why is it possible to see light through this system?

Find the position of $\lambda/4$ the plate which gives maximum light through the system. (This step ensures that the plane of the polarized light from the first polarizer is mid-way between the fast and slow axes of $\lambda/4$ the plate.)

Rotate the $\lambda/4$ waveplate (while keeping the polarizer fixed).

Keeping the $\lambda/4$ waveplate in its current position, rotate the second polarizer. (Your partner may need to hold something so you have enough hands!)

Rotate the second polarizer (while keeping the $\lambda/4$ waveplate fixed.

What do you observe? What can you conclude about the state of polarization of the light leaving the $\lambda/4$ plate? How is this different than when you had a linear polarizer in between instead?

Calcite crystal is a naturally occurring mineral which exhibits birefringence.

Place the crystal on a surface with some writing (or on the computer screen) and comment on what you observe as you look at the print through the crystal.

Now hold the single linear polarizer between the crystal and your eye and observe the print as you rotate the polarizer.

Describe what you see.

Place the calcite crystal between the crossed-polarizers in the Malus' law apparatus and rotate it.

Describe what you see.

Make sure the polarizers of the Malus' law apparatus are still crossed (to fully extinguish light). Hold the orange bendy rubber piece between the crossed-polarizers and (GENTLY!) squeeze or twist it.

GENTLY squeeze or twist the orange bendy rubber piece.

Describe what you observe.

As you distort the rubber, you are creating strain in the material. Note that light is transmitted in places where the strain is high.

OPTIONAL: There is also a hard plastic clip (in the common area) that can be squeezed. You should see strain line develop when you press the clip together.)

Make sure the polarizers of the Malus' law apparatus are still crossed (to fully extinguish light). Insert a bottle of one of the sugar solutions (either fructose or sucrose) between them, as shown below.

What do you see? Rotate the polarizer closest to your eye to test the nature of the light which passes through the syrup. Describe your findings.

Now try the other sugar solution. What is the same? What is different?

OPTIONAL: Additionally, there are samples of glucose and high-fructose corn syrup available in the common area to look at.

Some molecules are chiral, meaning that the molecule cannot be superimposed onto its mirror image. Chiral molecules are said to have a handedness (just like your right and left hands are mirror images of each other, but you cannot rotate or move your left hand to exactly overlap with your right hand).

An example of a “left-handed” and a “right-handed” version of an amino acid. Notice that even though both molecules have the same chemical formula, you cannot rotate or translate the left-handed molecule so that it has the same shape as the right-handed molecule. (Source: Wikimedia Commons)

Chiral molecules will rotate the plane of polarization of light passing through them, and molecules of opposite handedness will rotate polarizations in opposite directions.

The molecule $C_6 H_{12} O_6$ is what we call “sugar” and it comes in two forms: fructose (having one handedness) and glucose (having the opposite handedness). Because these molecules have different shapes, the body processes them differently. Common table sugar (called sucrose) is made up of a molecularly bonded pair of fructose and glucose.

The high-fructose corn syrup we are using here contains – as the name suggests – more fructose than glucose (about 55% fructose to 45% glucose).

Here is a great video from Steve Mould that goes into why the chirality of the sugar molecule leads to a change in polarization as light passes through. (You may want to wait until you get home to watch it… it's 18 minutes long!)

It is also possible to polarize light by reflecting it from a shiny, insulating material such as glass.

Consider the case shown in Fig. 2. Suppose unpolarized light is incident on the surface of glass. (Unpolarized light may be represented by light with two, orthogonal polarizations as shown on the left side of Fig. 2.) The light entering the glass (the refracted ray) forces electrons in the glass to oscillate. The oscillating electrons re-radiate, giving rise to the reflected ray. In the special case that the angle between the reflected ray and the refracted ray is 90 degrees, the angles of incidence and reflection are said to be the Brewster's angle, $\theta_B$.

At the Brewster angle, the electrons in the path of the refracted ray oscillate in two directions: one in and out of the page and the other parallel to the reflected ray. However, the electrons vibrating parallel to the reflected ray cannot radiate in the direction of the reflected ray. It follows, therefore, that at the Brewster angle, the reflected ray must be plane polarized in and out of the page.

Figure 2: Polarization by reflection at Brewster’s angle from the surface of a shiny, insulating material such as glass. Note that $n_2$ is the glass's index of refraction here.

The polarizer holder for this apparatus. Note that you can either detach the polarizer only or remove the entire fixture. It may take a little force and/or twisting to free the adapter from the telescope.

Remove the polarizer from the end of the telescope. Look through the polarizer at the light scattered off the ceiling. Rotate the polarizer and note whether the light intensity changes.

Is the light from the ceiling polarized?

Now look through the polarizer at the light reflected from the floor while rotating the polarizer.

Is the light from the floor polarized?

Figure 3: Brewster’s angle apparatus

We will now do a more quantitative measurement to find Brewster’s angle. The light source and collimator shown in Fig. 3 produce a well-defined beam direction. Place the glass slide as shown in Fig. 3 so that light will be reflected from the glass. Move the telescope to find the reflected beam of light. To test whether the reflected light is polarized, replace the polarizer onto the end of the telescope. While rotating the polarizer observe whether the light reflected from the glass changes intensity.

What does this finding say about the polarization of the reflected light?

Set the polarizer to give the minimum intensity of the reflected ray. Now slowly rotate the table holding the glass slide, while also moving the telescope to keep the reflected ray in the telescope’s field of view. Find the angle of reflection which gives the absolute minimum reflected intensity. Since the intensity will be low, cover the spectrometer with the cloth. At this angle the reflected light is most polarized and meets the condition of Fig. 2. Lock the table holding the glass slide so it will not turn!

Read and record the angular position of the telescope at this angle of maximum polarization. It is sufficient to measure to the nearest degree. Estimate your uncertainty in this angle. 

In order to find Brewster’s angle, $\theta_B$, remove the glass slide holder (leaving the slide holder table in place!) and move the telescope so you can see the light directly from the collimator on the telescope cross-hairs. Note and record this angular position of the telescope. You now have enough information to determine $\theta_B$.

Theory shows that

From your value of $\theta_B$, calculate $n_2$, the index of refraction for the glass.

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!

Answer the questions/prompts in the post-lab assignment below and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write the answers to these questions by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.