Playing with Light

Learning outcomes

In this teaching unit, students learn to:

• describe polarisation as a physical property of light and explain how polarisation filters work; 
• apply Malus’ law in terms of light intensity;
• replace the concept of intensity in Malus’ law with a concept of probabilities for a small number of photons or even single photons; 
• understand polarisation states as mutually exclusive basis states of a two-level system; 
• recognise superposition as a key concept emerging from polarisation experiments.

Experiment 1 (Warming up)

Material

• Old LCD screen
• Polarised glasses
• Polarisation filters
• Calcite crystal
• Simple colour filters (frequency filters) – to compare/differentiate their effect with the one of polarisation filters). Non-polarised sunglasses will work perfectly

An experiment to break the ice could be the following: a computer is connected to an “old” LCD screen (that is not used anymore). The screen has to be set up for the purpose by peeling off the top layer. Beforehand, the teacher typed some words that appear on the screen.

Some of the students are given polarised glasses or polarisation filters, while others are given simple colour filters.

Outcome

Only the students who look through the polarisation filters or polarised glasses will be able to read the words on the screen. Those with the colour filters (or sunglasses) will see only a uniform background: without a polariser, the image on the LCD screen is not visible.

Polarisation

Light is an electromagnetic wave travelling in space and carrying energy. The electric and the magnetic field make up the electromagnetic wave. These fields oscillate in phase, orthogonally to one another.

The electromagnetic wave can oscillate in any plane containing the black line in fig. 2, which depicts the propagation direction of the wave. The light we receive from the sun or the light we switch on at night are examples of unpolarised light.

Fig. 3: Unpolarised light with its electric field $\overrightarrow{E}$ in all directions perpendicular to the direction of propagation.

One can polarise unpolarised light: the electric field then oscillates in only one plane, whereas the magnetic field oscillates in a plane that is orthogonal to that plane. 
The polarisation of light can be described with the help of a cardboard model. 

The model can be used for two different applications:

• Application 1: The electric and the magnetic fields which oscillate perpendicularly to each other (fig. 4)
• Application 2: The model shows the two components of the electric (or the two components of the magnetic field) before passing through a polarisation filter. (fig. 6)

The yellow and the white parts show the oscillations of the two perpendicular components of one of the two fields e.g. the electric field. Both are perpendicular to the direction of propagation (one component is yellow and the other one is white). The cardboard profile can be rotated along its axis to demonstrate how unpolarised light behaves. The teacher needs to clarify that the magnetic field is not visible in this model.

Polarised light can sometimes be observed naturally when light is reflected at a surface. In the lab, we can create polarised light using thin films called polarisation filters (or polarisers). A polarisation filter allows light in a specific polarisation place to pass through and absorbs all other light. In other words, only the waves with an oscillating nonzero component parallel to the direction allowed by the polarisation filter can get through. As a result, the intensity of light passing through the filter is reduced.

Polarisation filters that employed herapathite crystals were invented by Edwin Land in 1936. These crystals are characterised by long chains with free electrons that can vibrate. When light strikes these thin sheets, the electric field vibrating in the direction of the chains transfers its energy to the crystals and is thereby being absorbed. Only light whose electric field oscillates perpendicularly to the chains can pass through the crystals. The so-called transmission axis of the polarisation filter corresponds to the direction perpendicular to the crystal grids. Polarisation filters are, for example, used in sunglasses, since they reduce the light intensity of unpolarised light by half.

Here again, the students may handcraft a model made of wood or cardboard (see fig. 6). This model is powerful as it compares the polarisation effect to a gate which lets just one oscillating component of the electric field through.

Another way to polarise light is by using a particular kind of crystals, called birefringent. They split up unpolarised light passing through them into two rays whose polarisation directions are perpendicular to each other. This has an effect on what we see: the objects seen through the crystal appear twice. 

See “Splitting and reuniting laser beams - with birefringent crystals”.

Using Iceland spar, also called optical calcite (CaCO₃), students can investigate the phenomenon of birefringence by observing drawings they made on white sheets of paper. You can easily order Iceland spar, also known as Iceland crystals, online.

By putting a polarisation filter on the crystal, one can demonstrate that the polarisation directions of the two emerging rays are perpendicular to each other.

The students do the activities on the worksheet 'Birefringent Crystal and Light Polarisation'. Download it as PDF or docx.

A word (“Quantum” in figure 8) is written on a sheet of paper, and the calcite crystal is placed on top of it. When looking through the crystal, the word is visible twice due to the birefringent nature of the calcite – which sends polarised light into two different directions that are perpendicular to each other. Then, one places a polarisation filter between the word “Quantum” and the crystal (see fig. 9). The polarisation filter should be marked at one corner for its orientation to be identifiable. The filter is then rotated relatively to the crystal until one of the two words (i.e. one of the two polarised beams) becomes invisible. Students could take a picture of this orientation. They can also observe that this arrangement of a birefringent crystal and a polarisation filter may also be obtained using two polarisation filters (also called polariser and analyser) being placed perpendicularly to each other: the polariser polarises light in a specific direction which is then totally blocked by an analyser which is placed perpendicularly to the first polariser (no components of the electric field remain).

Secondly, the polarisation filter is rotated further, until the light beam which resulted in the word to become visible is now totally blocked, whereas the second light beam (that was blocked in the first step) now makes the word become visible. The students will notice that this second rotation of the crystal corresponds to a rotation of 90 degrees with respect to the end position of the first rotation.

Experiment 2: How light passes the polarisation filters

Material

• A big laser with supports, polarisation filters (for demonstration by the teacher)
• Small laser, polarisation filters for student groups
• The Phyphox app on the students’ tablets/smart phones (to measure light intensity)

Caution: When using lasers, you should take some precautions and tell your students that they must avoid pointing the laser beam directly towards their or their peers’ faces, or even worse towards their eyes, because powerful lasers can damage the eyesight. Please also follow any safety regulations applicable in your country or institution.

Switch on the laser and show the path of the light beam without any polarisation filter. Use a screen to “stop” the light beam – a light spot is visible when the light reaches the screen.

Invite the students to observe the straight trajectory of the light beam. Insert the first polarisation filter in the light’s trajectory by fixing it on the first optical mount (see fig. 11).

Ask the students if they observe any change in light intensity. The light intensity should have considerably decreased unless the polarisation filter happens to be aligned with the polarisation direction of the laser light. Tell the students that the polarisation filter polarises the laser light in one single direction, even though we do not know its exact polarisation direction (yet). For the teacher: usually, the polarisation direction corresponds to the short side of the filter frame (see fig. 12).

Fig. 12: Blocking and transmission direction of a polarisation filter: the locking direction is the one where light with a component of the electric field in this direction is absorbed by the polarisation filter, while the transmission direction is the one where light with a component of the electric field in this direction passes through the filter.

The students can verify Malus’ law by measuring the light’s intensity – using an app on their smartphone (for example Phyphox) – as a function of the angle $\theta$.

Polarisation is a phenomenon that can be explained with the wave nature of light. 

Now insert the second polarisation filter, so that its polarisation direction is perpendicular to the polarisation direction of the first polarisation filter.

Ask the students to describe what they observe. Their answer will be: no light can be seen on the screen. Ask them how they can explain their observation.

After a while, their explanation will probably be: only light with a nonzero component in the direction of the filter’s polarisation direction (e.g. the vertical component) passes the first polarisation filter, whereas the second polarisation allows only light polarised horizontally to pass. However, since the first polarisation filter absorbs horizontally polarised light, no light passes through the second filter. Well, this is consistent with our wave view of light.

At this point, the students can measure the light intensity passing the second polarisation filter with their smartphones – depending on the angle between the polarisation directions of the two polarisation filters (e.g. 0°, 30°, 45°, 90°). By doing this, they can check whether Malus’ law also works with several polarisation filters in a row. It should be noted, however, that Malus’ law is only exactly valid for ideal polarisation filters, so you should be happy with approximations.

Now, turn the two polarisation filters to their previous positions, so that their polarisation directions are perpendicular to each other. Insert a third polarisation filter between the other two with its polarisation direction tilted by 45° with respect to the horizontal direction.

Ask the students what they expect to see on the screen. Before switching on the laser beam, let them think about it for some minutes, so that they can come up with their own ideas and “theories”. Give them the time to debate on possible outcomes and compare their theories.

Switch on the laser beam. The big surprise now is that some light passes through: a light spot can be observed on the screen.

Ask your students to try to explain theoretically what is happening. Allow some time so that they can reformulate what they said before. Would they have predicted this before having seen it?

The first polarisation filter polarises the light vertically, i.e. in a plane that is tilted by 90° with respect to a horizontal axis. Only light whose component oscillates vertically passes this filter. Subsequently, part of the light passes through the 45° polarisation filter (see fig. 16 to 18), and the third polarisation filter, which is tilted by 90° with respect to the first polarisation filter (0° with respect to the horizontal axis). Only the horizontal component of the light that was polarised in the 45° direction is now transmitted (see fig. 16).

Short digression on vectors

Read more on "Short digression on vectors" below:

Polarisation in the particle picture

Up to now, we have described everything in the “wave picture” of the light, but what about the particle picture? In the particle picture, light consists of many extremely small quanta of energy, small particles, called photons. Any single one of these photons passes through the polarisation filter … or not. How can we explain the behaviour that we just observed in the particle picture? If we describe light as a beam composed of a large number of photons, the light intensity is proportional to the number of photons. In this case, students can use Malus' law with regards to $N_0$, the number of photons emitted by the laser and $N$, the number of photons that pass through the polarisation filter: $$N=N_0{\cos }^2\theta$$ But what would happen if only a single photon would hit the polarisation filter? Photons are indivisible, so we cannot say that only one part of a photon passes the filter. We need to use another mathematical approach to describe it. We need to work with probabilities. The probability for one photon to pass the polarisation filter is given by Malus' law which must be adapted for a small amount of photons. If each photon passing through the polarisation filter has a certain probability $p_{\mathrm{through}}$ of passing through the filter, this can be written as follows: $$I=I_0\cdot p_{\mathrm{through}}=I_0{\cos }^2\theta$$ The light intensity ${\displaystyle I}$ measured after the polarisation filter is proportional to the incoming light intensity ${\displaystyle I_0}$ multiplied with the probability  $p_{\mathrm{through}}$ that a photon passes through the polarisation filter. For a large number of photons, one can again use the classical version of Malus’ law (right part of the equation above).

When the polarisation direction of the photon and the polarisation direction of the polarisation filter are parallel, $\theta =0°$, i.e. $\cos \theta =1$. When both polarisation directions are orthogonal, $\theta =90°$, i.e. $\cos \theta =0$. The square of ${\cos }^2\theta$ ensures that we get a positive result between 0 and 1 – which is suitable to describe probabilities.

For the time being, we assume that polarisation is a property of the photon, a characteristic that we will not examine in detail at this point.

Table 1: Relation between the polarisation direction of the polarisation filter and the light intensity behind the polarisation filter

The probability for a photon to be absorbed is:

$$p_{\mathrm{abs}}=1-p_{\mathrm{through}}=1-{\cos }^2\theta$$

Connection to quantum physics

Now let's slowly try to establish a connection to quantum physics. When a photon is polarised horizontally, it is in a specific “ state ” that we can write as ${\displaystyle \left(\begin{array}{c}1\\ 0\end{array}\right)}$. When a photon is polarised vertically, it is in the state ${\displaystyle \left(\begin{array}{c}0\\ 1\end{array}\right)}$, and it is in the state $\left(\begin{array}{c}{\cos }^2\theta \\ {\sin }^2\theta \end{array}\right)$ for any other polarisation direction. A special feature of the states ${\displaystyle \left(\begin{array}{c}1\\ 0\end{array}\right)}$ and ${\displaystyle \left(\begin{array}{c}0\\ 1\end{array}\right)}$ is that they are independent of each other: A vertically polarised photon cannot be “horizontally polarised” at the same time, and vice versa. This means that the two states “vertically polarised” and “horizontally polarised” are mutually exclusive.

Let us go back to the polarisation filters. They interact (“do something”) with the incident light. The light that is transmitted by the polarisation filter allows conclusions to be drawn about the polarisation of the incident light. This corresponds to a measurement : the polarisation filter examines the incident light, so to speak. A polarisation filter with a vertical polarisation direction allows the photon to pass through if it is vertically polarised and absorbs it if it is horizontally polarised.

Now, let us examine the experiment with the three polarisation filters again (see fig. 15 and 16). The middle polarisation filter with a polarisation direction of 45° analyses whether the incident photon has a non-zero polarisation component in the 45° direction. Actually, the photon that made it through the polarisation filter with horizontal polarisation direction does have a component in the 45° direction. And this 45° component has, in turn, a vertical component which passes through the third polarisation filter.

Fig. 20: The photon interacts with the various polarisation filters– first with the one with horizontal polarisation direction, then with the one with a polarisation direction of 45°, and lastly with the polarisation filter with a vertical polarisation direction. The probabilities that a photon can pass the filters is given by the length of the red arrowsblue lines.

In the particle approach, the probability that a horizontally polarised photon passes the 45° polarisation filter is $\frac{1}{2}$. The probability that it subsequently passes through the polarisation filter with vertical polarisation is again $\frac{1}{2}$. The overall probability that a single photon is able to get through these two polarisation filters is: $\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4}$. So, some photons will be able to reach the screen after having passed all three polarisation filters! We can’t tell whether a specific single photon will make it through, but statistically, one out of four photons will pass through the last two polarisation filters.

Remark about lab lasers

Normal lab lasers emit polarised light. This means that you can avoid using the first polarisation filter, two would be sufficient. You should know the polarisation direction of the light emitted by your laser. You can analyse it using a polarisation filter. Just find out the direction where the least laser light is transmitted (ideally none). The polarisation direction of the laser light corresponds to a direction that is perpendicular to that one.