(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

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.

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.

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.

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.

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.

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?

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.

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 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.

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.

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

Figure 3: Brewster’s angle apparatus

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.