Polarization - PHYS143

(Updated April 2024)



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

$\mathbf{d} = A_x \sin\left(2\pi f\left(t-\dfrac{z}{c}\right)\right)\hat{\mathbf{x}}$, (1)

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

$\mathbf{d} = A_y \sin\left(2\pi f\left(t-\dfrac{z}{c}\right)\right)\hat{\mathbf{y}}$, (2)

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.

Electromagnetic waves

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.

Beauty in Physics

As an aside, physics is more than math and data. There is incredible beauty in nature that arises directly from the physics principles which you are learning about. The image at the top of this wiki page is an example. It is a photograph of crystals of a simple protein, taken with a technique known as Cross Polarization Microscopy. As the name implies, the technique makes use of a pair of polarizers in a crossed configuration. You will be studying aspects of polarization in this weeks lab. The colors and shapes in the image, and the ones which follow, are completely natural, they are not computer generated.

There is tremendous beauty present in physics. As a scientist you should strive to see this beauty in the things which we study.

Polarization Experiments

Click here to get a copy of the lab notebook. Don't forget you need to be logged into your UChicago google account to access the document.


A polarizer is an optical filter which either passes or blocks light based on its polarization. A linear polarizer has a pass axis which transmits light that is polarized parallel to the pass axis, and blocks light which is polarized along the orthogonal axis. For example a linear polarizer whose pass axis is oriented vertically will pass vertically polarized light while blocking horizontally polarized light. For the case of light which is polarized at an an angle which is neither parallel to nor orthogonal to the polarizers pass axis, only that portion of the light which projects onto the pass axis will be passed.

Polarization of light

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.

Polarization Experiments

Often in physics we create models (i.e. theories), such as the model for the polarization of light. Models are only useful if they can be used to make predictions which can be tested experimentally. Use your understanding of polarization to make predictions for each of the following cases, then use the apparatus to test your prediction.

The apparatus consists of the same optical rail, light source and diffusion screen that you used in last weeks lab. In addition for this week you also have a pair of linear polarizers mounted in holders with angle scales that indicate the orientation of the pass axis, and a light sensor which is connected to the computer. Double clicking the Light Sensor icon on the desktop will open an app that displays the light intensity readings from the sensor.

We expect your predictions of the behavior of the light for each of these cases to be based on your understanding of the physical model of polarization. Your prediction should detailed and include testable quantitative values where possible, which you can either confirm or rule out through measurement. The Goal of the lab is not for you to get the predictions correct on the first attempt. If your prediction turns out to be inaccurate you should indicate how your measurements show this, then you should modify your predictions and try again.

Some things to consider for this part of the lab.

  • Models typically assume ideal conditions. I.e. that a linear polarizer will block all light whose polarization is orthogonal to its pass axis. Reality however is that no polarizing materials are 100% efficient. How do you account for this when comparing your data to the model?
  • Your light sensor is not perfectly shielded from stray light. How do you account for this?

Case 1 : The light from the source is unpolarized.

  • If light from this source passes through a linear polarizer, how do you expect the intensity of the light to change?
  • If the polarizer's pass axis is rotated how should the intensity of the light change?

Case 2 : A linear polarizer can be used to create polarized light from an unpolarized source. If this polarized light is now viewed through a second linear polarizer how would you expect the intensity of the polarized light to change if:

  • The second polarizer's pass axis is set parallel to the pass axis of the first polarizer.
  • The second polarizer's pass axis is set orthogonal to the pass axis of the first polarizer.

Case 3 : For the setup where the two polarizers pass axes are set to be orthogonal, what would be the effect of inserting a third polarizer in between the first two, with the third polarizer's pass axis set to an angle at 45º with respect to the pass axes of the first two polarizers?

Case 4 : For light passing through two polarizers, how would the intensity vary as a function of the angle between the pass axes of the two polarizers. Plot your data for this part using the Jupyter Notebook linked below, but please try to determine the functional form you expect before opening the notebook.

Retardation of phase

Note that for this part of the lab we are not asking you to do any quantitative assessment of what you are seeing. These exercises are mostly to let you see some interesting optical effects related to the concept of retardation of phase.

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.

Note on terminology

The usage of “retarded” here comes from the latin “re-” for 'back' and “-tardus” for 'slow'. It was originally a technical term for an event that was delayed in time compared to another, and predates the medical (and then derogatory) usages.

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

$n_xL - n_y L = (n_x-n_y)L$, (4)

and the phase difference is

$\varphi = \dfrac{2\pi}{\lambda}(n_x - n_y)L$. (5)

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.

Strain birefringence

Set the polarizers to the crossed orientation. Place a piece of the yellow rubbery material between crossed-polarizers and squeeze and twist the free ends of the slotted plastic. Compression of the material induces strain which changes the orientation of polymer chains in the material producing a birefringent effect. Cross polarization is used to detect stresses in many materials, including the window glass on airplanes.

Calcite crystal

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. Look at the crystal through a polarizer and notice how rotating the polarizer affects the light passing through the crystal.

Polarization by reflection: Brewster's angle

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.

Theory shows that

$\tan\theta_B = n$. (3)

Use the spectrometer illustrated in figure 3, along with a linear polarizer, to determine whether or not the polarizations of the reflected and refracted beams are as predicted when set to Brewsters angle. Assume that the index of refraction of the microscope glass is 1.51, which is typical for a wide range of glasses. Calculate the angles necessary to configure the spectrometer so that the Brewsters condition is met. Use the telescope with a linear polarizer to check the polarization state of the reflected and refracted beams.

Note that the polarizer you are working with does not behave like an ideal polarizer, and it will not completely block orthogonally polarized light. You should however be able to convincingly demonstrate whether or not the reflected light is predominantly vertically polarized and whether or not the refracted light contains both polarizations.

Figure 3: Brewster’s angle apparatus

Report: Summary and conclusions

After the lab, you will need to write up your conclusions. This should be a separate document, and it should be done individually (though you may talk your group members or ask questions). Include any data tables, plots, etc. from the your lab notebook as necessary in order to show how your data support your conclusions.

The conclusion is your interpretation and discussion of your data. Specifically we want you to:

  • Present your data and conclusion for each of the four cases of the polarization part of the lab.
  • Present your data, calculations and summary of the polarization states of the reflected and refracted beams for the Brewster's angle part of the lab.

REMINDER: Your report is due 48 hours after the lab. Submit a single PDF on Canvas.