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 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.

Figure 1: Malus’ Law apparatus

Turn the detector on and set the $I-V$ switch to the $I$ (current) position. In this configuration, more light to the detector gives a greater current through a resistor. Thus, voltage is our indicator of light intensity.

Figure 1.1: Photodetector configuration panel. Note that the top switch (I-V) should be to the right and the bottom switch upwards (away from OFF) when performing this experiment.

Connect the OUTPUT of the detector to the voltage (red wire) and common (black wire) jack of the DVM. Set the DVM to measure DC voltage. With the lamp turned on, position the detector so that the light is entering the detector opening and adjust the bipolar power supply voltage to about -10 V.

Varying the angle of the polarizer in the degree scale, measure and plot the transmitted intensity (detector voltage) 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. (1), use the Google Colab notebook below to perform a fit. Do your data fit this functional form? Are your data consistent with Malus' law?

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 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.

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 birefringement 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.

The apparatus is shown in Fig. 6. You can (and should) turn the white light source into a monochromatic light source by flipping up the green filter, as shown in Figure 6b.

Figure 6a: Apparatus for studying phase retardation, strain birefringence, and rotary dispersion.

Figure 6b: Green filter used to turn white light into monochromatic light.

First polarization observation

To begin, move the first polarizer and quarter wave ($\lambda/4$) plate holders out of the way and look at the lamp through just the second polarizer, as depicted above.

Is the light from the lamp naturally plane-polarized?

Second polarization observation

Rotate the first polarizer back into place. Then look through the second polarizer at the lamp, rotating the second polarizer to get maximum extinction of the light. You may still see some blueish light passing through; this is normal for out setup.

Third polarization observation

Now, turn the $\lambda/4$ plate back into the light path so that it is between the light source (with 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 polarizer in the light source is mid-way between the fast and slow axes of $\lambda/4$ the plate.

Fourth polarization observation

Now, rotate the second polarizer on the stand as shown in the above diagram.

What do you observe? What can you conclude about the state of polarization of the light leaving the $\lambda/4$ plate?

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 a linear polarizer between the crystal and you eye and observe the print as you rotate the polarizer.

Describe what you see.

Remove the $\lambda/4$ plate from between the polarizers and return the polarizers to the crossed orientation. Place a piece of plastic between crossed-polarizers and squeeze the free ends of the slotted plastic.

Describe what you observe.

Note that light is transmitted in places where the strain is high.

Cross the polarizing filters and insert a bottle of corn syrup 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.

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 equal parts fructose and glucose, but it is possible to create sugar solutions of different concentrations.

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!)

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!

Write your conclusion in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write your conclusion by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

The conclusion is your interpretation and discussion of your data. In general, you should ask yourself the following questions as you write (though not every question will be appropriate to every experiment):

• What do your data tell you?
• How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
• Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
• Do your results lead to new questions?
• Can you think of other ways to extend or improve the experiment?

In about one or two paragraphs, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.