Wilder

In the mid-1800s, James Clerk Maxwell formalized four fundamental equations about electricity and magnetism. With his equations, Maxwell theorized the existence of electromagnetic (EM) waves. Eight years after his death, Heinrich Hertz experimentally demonstrated the existence of EM waves by producing radio waves in a laboratory (1).

Electromagnetic waves are all around us. Visible light constitutes the spectrum of electromagnetic radiation that our eyes can perceive. One hundred years before Maxwell and Hertz, a French physicist named Etienne Malus studied images made with light passing through calcite crystals. He discovered that when holding crystals at some angles, the images would disappear. While he did not understand the science at the time, it was later discovered that the calcite crystals polarized light in certain directions (2). In the following section, I discuss what it means for light to be polarized. As shown in the abstract video, and as discovered by Malus with his crystals, light polarization has many surprising effects. In the experiments in this DYO, I attempt to understand how the angle between two linearly polarizing filters impacts the intensity of the light that passes through both filters. Additionally, I look at introducing a third filter in between two perpendicular filters. I attempt to learn how the angle of this middle filter impacts the intensity of light passing through all three filters, and also why the additional filter lets more light through.

What is electromagnetic radiation?

Electromagnetic waves are everywhere in our lives. For example, your car stereo receives radio waves to play music, your eye picks up visible light waves to see things, and devices at airport security use x-rays to search for hidden objects. It turns out that all of these waves are examples of the same phenomena: oscillations in the electric and magnetic fields.

To understand electromagnetic (EM) radiation, let's start by looking at a simple setup of a charge in a wire. By making a current oscillate up and down in the wire, we create an electric field that oscillates from a positive value to a negative value (3). We can graph the oscillations of the electric field with the following diagram:

As Maxwell discovered, changes to an electric field induce changes to a magnetic field, and vice versa. As the current flows in a wire, a magnetic field is formed in addition to the electric field. The forces of the magnetic field surround the wire (they are pointing in and out of the screen in rings around the wire). As the current oscillates in the wire, the magnetic field oscillates as well. The vectors of the magnetic field are perpendicular to those of the electric field (3).

By putting these two fields together on the same graph, we see that electromagnetic waves can be created by oscillating charges in a wire (3). These electromagnetic waves propagate through a vacuum at the speed of light, c, in a direction perpendicular to both the electric field and the magnetic field vectors.

The electromagnetic wave can be described by its frequency, f, and wavelength, λ, which relate to each other by c = f * λ (4). Frequency and wavelength are important—they are the reason why long-wavelength, low-frequency radio waves are different from short-wavelength, high-frequency x-rays. They are also the reason why visible light appears in different colors (the red end of the visible spectrum is longer wavelength light, while the blue end is shorter wavelength light).

What is light polarization?

Most sources of electromagnetic radiation, such as light bulbs, microwaves, and the sun, produce EM waves oriented in all directions. Consider just the propagating electric field in the EM wave above. That field will always be perpendicular to its direction of movement in space, but it could be tilted at any angle. This type of light which has photons oscillating in all planes is called unpolarized light (5).

We can pass unpolarized light through what is called a polarizing filter. All light that passes through the filter will be polarized in the filter’s direction. Classically, we can think about the action of these filters as absorbing all of the energy in the horizontal direction. While the human eye cannot tell the difference between different polarizations (only different intensities of light), light polarization is used in many optical applications such as glare-reducing sunglasses, 3D movies, and even quantum cryptography (6).

I took data at night and shut off the lights in the room before each trial. I took the layered polarizers and inserted them into the slit in the opaque cardboard tube. I switched on the light and took data for 5 seconds using a Vernier light sensor and Logger Pro. I recorded the average light intensity over the 5-second interval in an Excel spreadsheet. For my experiment with two filters, I did 19 runs of different angles between the filters evenly spaced between the range 0° to 180°. For my experiment with three filters, I did 10 runs of different angles between the first and middle filters evenly spaced between the range of 0° to 90°. For each run in both experiments, I did three trials.

Figure 1: Average light intensity coming through two filters is shown for differing angles between the filters. Experimental data (green points) has been inputted into Desmos and a curve of best fit has been fitted to the function and graphed over the interval of measured angles (0° to 180°). Values for percent average absolute deviation between the three trials for each run fell between 0.700% and 1.878%, but error bars are not shown as they cannot be easily rendered on Desmos. Initial light intensity hitting the first filter remained constant because the LED was taped at a set distance from the filters. Light intensity was measured using a Vernier Light Senor which was also taped at a set distance from the filters throughout the experiment.

Figure 2: Average light intensity coming through three filters is shown for differing angles between the first and middle filter. The third filter was held at a constant 90° angle in relation to the first filter throughout the experiment. Experimental data (green points) has been inputted into Desmos and a curve of best fit has been fitted to the function and graphed over the interval of measured angles (0° to 90°). Values for percent average absolute deviation between the three trials for each run fell between 0.969% and 2.477%, but error bars are not shown as they cannot be easily rendered on Desmos. Initial light intensity hitting the first filter remained constant because the LED was taped at a set distance from the filters. Light intensity was measured using a Vernier Light Senor which was also taped at a set distance from the filters throughout the experiment.

In order to understand these experimental results, we must first understand how an electromagnetic wave of one polarization can be represented as a superposition of electromagnetic waves in two polarizations. Looking back at the diagram of an EM wave in the "Background Theory" section, if we consider only the electric field portion of the wave, and we orient the wave so that it is coming out of the screen and polarized at a 45° angle, it would look like the image below:

In this sense, the EM wave with amplitude E is a superposition of the EM waves with amplitudes E_x and E_y, where E_x= E*cos(θ) and E_y= E*sin(θ). When we think about the first experiment with two filters, the first filter polarizes the light vertically by absorbing all of the energy in the horizontal direction, and the second filter polarized this light in the direction θ° from vertical by absorbing all of the energy in the (θ+90)° direction off of vertical.

We can write the vertically polarized light (the vectors in pencil to the right of filter 1 as a superposition of its components in the θ direction and the (θ-90) direction. This is shown by the vectors in purple. When the light passes through filter 2, all of the energy in the (θ-90) direction is absorbed (the smaller purple vector), leaving only the light polarized in the direction of θ (the longer purple vector). As you can see from the diagram, the amplitude of the light passing through both filters corresponds to cos(θ) times the amplitude of the vertically polarized light. So, the light emitted through both filters will have an amplitude that varies with the cosine of the angle between the two filters.

Figure 1 shows that 658 lux was the highest value of light intensity which was measured when θ=0°. With the above explanation, we might expect the data to vary with a cosine curve—as θ increases, the amount of light passing through both filters changes given 685*cos(θ). However, the data actually corresponds much more closely to the function 658*cos²(θ). This is an intriguing result and suggests that the light meter is not actually measuring the amplitude of the waves.

In classical physics, the energy a of wave is proportional to the square of its amplitude, or E∝A². As we just saw in the diagram above, the amplitude of the light passing through both filters is A*cos(θ), where A is the amplitude of the light that has passed through the first filter. The energy of the light passing through both filters would then equal k*(A*cos(θ))² = k*A²*cos²(θ), where k is the constant of proportionality. In the first experiment, we found that k*A² = 658. This relationship that the intensity of light coming through two filters varies with the square of the cosine between the filters, which can also be written as I(θ) = I_0*cos²(θ), was first discovered by Antoine Malus in 1808 and is known as Malus's Law (7).

By understanding a vertically polarized EM wave as a superposition of two other waves, one of which is polarized in the direction of the second filter and one of which is polarized 90° off of the direction of the second filter, the unintuitive second experiment starts to make more sense. If we were to pass a photon through two filters that are perpendicular to each other, the first filter would polarize it vertically. We could write it as a superposition of components corresponding to the second filter, but this would end up having the amplitude of waves in the horizontal direction, which is also the direction of the second filter, to be zero, and therefore no waves could pass through both filters. When we slide a second filter in between these two filters, the vertically polarized photons passing through the first filter can be written as a superposition of two photons in the direction of the second filter and in a direction perpendicular to the second filter (this is the first experiment). Now, these photons which have amplitudes of A*cos(θ) are at an angle of θ compared to the first filter, and an angle of (90-θ) compared to the third filter. So long as θ does not equal zero, these angled photons can all be written as a superposition of two other waves with an amplitude in the horizontal direction that corresponds to B*cos(90-θ) where B=A*cos(θ), so the amplitudes of the photons passing through all three filters can be found with A*cos(θ)*cos(90-θ) where A equals the amplitude of the waves that pass through the first filter. My experimental results found that the light passing through all three filters had an energy of 537*cos²(θ)*cos²(90-θ). If the light meter measures the light energy, then our theoretical values would find that the energy of the light passing through all three filters is E = k_2*A²*cos²(θ)*cos²(90-θ) where k_2*A² = 537.

All of the theoretical explanations about waves and energy that I have given so far in this discussion accords with my experimental results. However, this is technically not a correct description of how light behaves. Electromagnetic waves are actually quantum objects, and therefore they only exist in discrete energy levels. More specially, all photons have the energy of n*h*f where n is a positive integer, h is Planck's constant, and f is the frequency of the photon. If a vertically polarized photon with the energy h*f, which is the lowest possible energy that a photon can have, hits a filter at 45° from vertical, we could try to split the photon into a superposition of two photons, one of which is in the 45° direction, and this photon would pass through. Except in that case, the photon would have the energy of h*f*cos²(45), or h*f*0.5, and we know that this is impossible because a photon cannot have an energy of 0.5*h*f. Scientists have discovered that what actually happens when photons pass through a filter is either the photon's entire energy passes through and the photon collapses to a new orientation in space, or none of its energy passes through. The likelihood of a photon completely passing through a filter and keeping all of its energy ends up being equal to the cosine squared of the angle between the filters. Notice that this quantum explanation gives the exact same results for my experiment. I used the classical understanding of EM waves in my explanations because it gives a much more intuitive description of these phenomena. It is important to note, however, that nature EM waves actually follow this quantum explanation, and pass through the filters probabilistically, rather than the classical explanation of being broken up components of superpositions (8).

(1) “Brief History of Maxwell's Equations .” MIT OpenCourseware, Massachusetts Institute of Technology, www.ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-013-electromagnetics-and-applications-spring-2009/lecture-notes/MIT6_013S09_res_maxwell.pdf.

(2) Murphy, Douglas B., et al. “Introduction to Polarized Light.” Nikon's MicroscopyU, www.microscopyu.com/techniques/polarized-light/introduction-to-polarized-light.

(3) OpenStax. “Physics.” Lumen, https://courses.lumenlearning.com/physics/chapter/24-2-production-of-electromagnetic-waves/#:~:text=Electromagnetic%20waves%20are%20created%20by,is%20ordinarily%20a%20transverse%20wave.

(4) Anderson, Matt, director. EM Waves. YouTube, YouTube, 5 Aug. 2014, www.youtube.com/watch?v=bwreHReBH2A.

(5) Murphy, Douglas B., et al. “Introduction to Polarized Light.” Nikon's MicroscopyU, www.microscopyu.com/techniques/polarized-light/introduction-to-polarized-light.

(6) Tatualia Teacher Follow, Edzon. “Uses of Polarized Light.” SlideShare, www.slideshare.net/edzontatualia/uses-of-polarized-light.

(7) E. Collett, Field Guide to Polarization, SPIE Press, Bellingham, WA (2005), https://www.spie.org/publications/fg05_p03_maluss_law?SSO=1#:~:text=In%201808%2C%20using%20a%20calcite,when%20viewed%20through%20the%20crystal.

(8) “Some light quantum mechanics (with minutephysics).” YouTube, uploaded by 3Blue1Brown, 13 Sep. 2017, https://www.youtube.com/watch?v=MzRCDLre1b4.