Polarization Part #2: Reflection

In my [»] previous post, I showed what light polarization was and how simple physical processes, like scattering, could affect the different electrical field directions. Here, I show one more of these processes, namely reflection. While scattering might appear connected only remotely to useful optical systems, reflections, on the other hand, are found everywhere in optics! Reflections occur every time you send a light beam to an interface that has a different refractive index than the previous one. Understanding how polarization affects reflectivity (and its counterparts, transmissivity), it therefore essential in optics.

When a light beam hits a glass interface, two processes occur: (1) refraction, (2) reflection. Refraction obeys Snell’s law, n₁×sin(θ_i)=n₂×sin(θ_t), while reflection keeps a more simple law where the angle becomes θ_r=-θ_i with θ_i the angle of incidence relative to the interface normal and θ_t, θ_r the angles of refraction and reflection respectively. All rays (incident, refracted and reflected), lie in the same plane as shown in Figure 1.

Figure 1 also depicts two electric field orientations: one perpendicular to the plane of incidence and the other one parallel to the plane of incidence. Remember that, by virtue of the [»] superposition principle, we can always decompose a random polarization direction into these two components. In this post, we develop how each of these components is affected in terms of reflection and transmission.

By developing the expression of conservation of the electrical field at the interface, it is possible to find the amplitude of the reflected and transmitted waves – as well as proving other important properties like the fact that the oscillation frequency of the resulting wave is the same as the incident wave. These ratios are known as the [∞] Fresnel equations and can be found in your favorite optical textbook such as Optics by Eugene Hect:

which stands for coefficients of reflectivity and transmission for and incident electrical field perpendicular and parallel to the interface plane respectively.

Note that these terms are the amplitude of the waves. To get the observable intensity of the reflection and transmitted waves, we use slightly different version of these terms that relates to the projected intensities,

Because refracted rays don’t have the same angle as the incident rays, a cosine correction is required for the perceived intensity.

A particular case occurs for normal incidence, θ_i=0, and we find how the reflection of an uncoated glass interface depends on the refractive index change:

which is typically around 4% for a change from air to N-BK7 but can be as large as 35% for an infrared beam on a germanium window!

Note that this is true for a single interface, for N interfaces, the transmission and reflectivity become

which leads to a typical transmission of 92% for an uncoated N-BK7 window in the visible.

If we look at the intensity coefficients for reflection, we see that the reflectivity coefficient for an electrical field parallel to the glass interface can reach zero when tan(θ_i+θ_t)~∞ which happens when θ_i+θ_t=90°. Since θ_t is linked to θ_i by Snell’s law, there is therefore a special incident angle, θ_b=atan(n₂/n₁), that cancels all reflectivity of light polarized with the electrical field parallel to the glass interface. This special angle is called the Brewster angle and is illustrated in Figure 2 and is about 57° for N-BK7 in air.

A natural consequence of the reflectivity going to zero for parallel polarization light is that all reflected light will be fully polarized perpendicular to the glass interface.

This inspired the experimental setup of Figure 3 which can measure the degree of polarization in the reflected signal for varying incidence angles. The mechanism ensures the angle configuration of Figure 1 with only one degree of freedom. You can control the angles by approaching the arms of the setup towards each other. Also, just as with my previous experiment, I recommend using a sine-wave pulsed signal and an oscilloscope configured to read the amplitude of the recorded sine wave. All CAD files can be downloaded [∞] here for reproduction.

Using the definition of the polarization ratio, P, we introduced in our previous post,

we get the experimental results of Figure 4.

The results of Figure 4 follows closely the theoretical model that we built from the Fresnel equation using n=1.5 although the polarization ratio is systematically a bit lower than the expected value. I suspect (although untested) that this comes from the fact that we have a second reflection on the rear side of microscopy slide after refraction which changes the second angle of incidence and therefore polarization ratio for the second beam. Nonetheless, the data matches the expectations surprisingly well!

We conclude that it is possible to obtain a nearly pure polarized beam by reflecting unpolarized light on a glass plate at its Brewster angle. That being said, working with reflected beams is not always the best approach if you need a signal with a low aberration count. In this case, it is better to work in transmission.

However, if we look at the coefficients of transmission, shown in Figure 5, we don’t have a clear-cut situation at Brewster’s angle. Here, we get 100% transmission for parallel polarized light but still about 90% transmission for perpendicular polarized light. The resulting transmitted wave is therefore richer in parallel polarization but is far from being fully polarized.

A common solution is to stack multiple glass plates such as to remove the perpendicular components as much as possible from the signal.

The polarization ratio after N reflections is shown in Figure 6. It is different from Figure 4 in the sense that there is no peak at Brewster’s angle and the polarization ratio keeps increasing with the angle. It should however be kept in mind that the total transmitted energy decreases quickly for larger angles, as shown in Figure 5, and that optimum transmission efficiency is still obtained at Brewster’s angle where we get 100% transmission for our desired polarization state. That being said, even with a large number of interfaces, it is much more difficult to reach a high degree of polarization efficiently when working in transmission.

From the different results we obtained here, it should now be obvious to you that refraction strongly affects the polarization of the resulting transmitted and reflected beams – something you should always keep in mind when working with optical setups that are sensitive to polarization.

Polarization has also important consequences in everyday’s life that are well known to photographers, road drivers and hikers: glazing reflection is highly polarized and can efficiently be removed by using a polarization filter with an axis perpendicular to the reflection polarization. Photographs frequently use polarization filters to remove unwanted reflection on windows, and road drivers and hikers wear sunglasses with polarization filters fixed along a predefined direction. Figure 7 illustrates the dramatic effect of polarization on windows in photography but this applies as well to reflection on water, roads etc. The image on the left shows the leaves behind the window much more clearly than on the right which shows the reflection of nature outside.

Last but not least, I also had a lot of fun playing with the experimental setup of Figure 8 where you can control the polarization axis of the laser and therefore the reflectivity/transmissivity coefficient of the beam – turning the glass window into a tunable beamsplitter! The system works at 45° and, by controlling the polarization angle using the polarizer, we can get any reflection in the range 1-10%. All CAD files can be downloaded [∞] here for reproduction with the other experimental setup.

A variation of Figure 8, shown in Figure 9, is to use two glass interfaces to produce multiple reflections of the laser beam. A part of the light is reflected at each interface (10%) but also transmitted to the next interface (90%). Using two slides you therefore get two bright peaks of intensities ~10% and ~9% of the initial laser beam. But if you pay attention, you will also see a dim spot that results from a back-and-forth reflection between the two slides! Indeed, as the light is reflected from the second slide, 10% of its intensity is reflected back on the first slide which reflects again on the second slide, leading to an overall intensity of ~0.09% of the initial laser beam intensity :) More reflections occurs, but they are even dimmer.

This concludes the first part of our polarization series on linear polarization. In the following posts, we will look into a completely different kind of polarization with even more practical implications!

Do not hesitate to share your thoughts on the [∞] community board to let me know if you enjoyed this post!

I would like to give a big thanks to Alex, Stephen, Lilith, James, Jon, Jesse, Karel, Kausban, Michael, Zach, Sivaraman, Samy, Shaun, Onur, Sunanda, Benjamin, Themulticaster, Tayyab, Marcel, Dennis, M, Natan and RottenSpinach who have supported this post through [∞] Patreon. I also take the occasion to invite you to donate through Patreon, even as little as $1. I cannot stress it more, you can really help me to post more content and make more experiments!