How Your Right Eye and Left Eye See Different Rainbows: The Most Complex Optical Illusion Ever

Written by Kerem Muldur

When we were all little children, rainbows were one of the most miraculous things in our lives. Although we didn’t care much about the scientific principles behind them, the typical answer to the question “How do rainbows appear?” was always the same boring line: “water droplets break sunlight into the seven colors of the spectrum”. Even though this explanation has nothing wrong with the true answer, it’s far from complete because it is oversimplified. In this article, we will attempt to ask questions to uncover the exact reasons and discover the physics behind this natural marvel.

Let’s revisit the common explanation of “Sunlight splits when it hits a raindrop…”; we can first delve into what leads the sun rays to make different colors from one single drop, and how it exactly occurs.

Sunlight hits the raindrops at different heights. If one array hits the surface at the middle of the raindrop, some of the light is reflected straight back to its source, some of the light enters to drop, and some of that light is reflected again as it exits the drop, and the remaining light escapes the raindrop body.

However, if the light hits the raindrop from a point away from the midpoint, this distance from the midpoint influences the reflection-refraction spectrum, called the impact parameter. We could trace how the exiting light changes position if we place this drop on a horizontal table, and the bent light exiting the drop depicted in the image hits the table surface at a specific point.

Here’s where it gets interesting: As the impact parameter increases, the angle between the incoming sunlight and the exiting refracted light also increases, causing the final light spot to move closer to the raindrop. Now, we will keep increasing the impact parameter and see what happens: When the impact parameter has not reached maximum, the spot stops moving in at a maximum angle. In the case that we use a red light laser as our light source and a glass sphere to simulate a raindrop, the maximum angle is 42 degrees. This maximum scattering angle depends on the color of light.

(Image Credit: https://education.nationalgeographic.org/resource/rainbow/)

Now, the question is, why does light bend and change its speed, and therefore bend because it changes the medium; to answer that question, we should analyze how light interacts with the electrons inside the sphere.

You might be confused by “electrons inside the sphere”; however, you can think of it like little mass particles attached to springs on 4 sides, allowing the particle to swing at a specific frequency, which is higher than all visible light’s frequencies, called the natural frequency. The light, basically an electromagnetic wave, has its frequency. As it encounters charges inside the sphere at each layer, the natural frequencies of electrons and electromagnetic waves are summed up for the resulting wave. According to the below-mentioned classical electromagnetism model equation, as the light’s frequency gets closer to the natural frequency, the resulting wave’s amplitude becomes larger, just as in the case if your pushing frequency matches the swinging frequency, there is a bigger phase kick. Therefore, high frequency color lights such as blue color light wiggle the electrons more, triggering them to release higher amplitude electromagnetic waves, producing a bigger phase kick, phase kick is the difference in phase resulting in the difference of phases between the light’s beginning phase and the one summed up with the effect of electron’s electromagnetic wave, therefore less wavelength and speed, meaning bending more.

x: how far the charge moves (its displacement)

q: charge of the particle (like electron charge)

E: strength of the electric field (from the light)

m: mass of the particle

zero squared​: frequency of the incoming light wave

one squared​: the particle’s natural frequency (how it wants to wiggle on its own)

cos⁡(ω1t) the back-and-forth motion over time

Those colors that bend more have a smaller maximum angle than high-energy color lights, and this difference provides the prism function. When the white light hits the surface of a raindrop, different colored lights that the white color contains refract at different angles, in other words, hitting different spots on the table.

With our refreshed ideas about the optics of raindrops, we have a remaining question: if each drop is a rounded shape and refracts the light like a circle, how does that rounded appearance make a complete curved structure?

To answer that question, we can analyze a final case where we pick a certain spot in the sky. Now to see this light, the caustic from that point must go straight to your eye, and in case the red light source is a raindrop, the angle should be exactly 42 degrees, explaining why rainbows take the shape of an arch with a 42-degree angle. As one drop falls and reaches the point where it is 42 degrees between itself and the observer, the observer can see the red light, but not the others because they refract at different angles. As the drop keeps falling and makes smaller degrees, like 41 degrees, the drop refracts a different color. The reason you see a multicolored rainbow is that drops at different points can refract different colors.

Moreover, if the arch should make 42-40 degrees with our eyes, light coming out of the sun will pass through the back of your head, and create a shadow at the center of this arch. Since the perspective you are observing the rainbow changes from person to person, and even from your left eye to your right eye, the arch that makes 42-40 degrees to this new observing point slightly moves; basically, the rainbow appears if refracted light makes 42-40 degrees with the observer’s point of view, and if the point of view changes, the drops that refract the light and make rainbow appear changes.

A rainbow we consider as a simple arc of colors in the sky, is, in truth, a very sensible optical illusion resulting from the balance between particle physics, optics, and mathematics. In this article, we found out that what we were told as kids, “sunlight hits a raindrop and splits into colors,” wasn’t wrong, but definitely too simple. The truth is, light bends differently depending on how it hits the raindrop, and different colors bend by different amounts because they have different electromagnetic wave frequencies. That’s all because of how light interacts with electrons inside the drop, causing phase kicks and bending based on frequency. Those raindrops make the rainbow curve as they enter into the range of 42-40 degrees, the maximum angle of caustic, and since the drops that make 42 degrees with your eye change from point of view to view, every person sees their version of it. It’s way more than just light splitting; it’s tiny physics working together to make something huge and beautiful.

References:

1- National Aeronautics and Space Administration, Science Mission Directorate. (2010). Anatomy of an Electromagnetic Wave. Retrieved [insert date - e.g., August 10, 2016], from NASA Science website: http://science.nasa.gov/ems/02_anatomy

2- How are rainbows formed? The science behind the colors. (1970, January 1). HowStuffWorks. https://science.howstuffworks.com/nature/climate-weather/atmospheric/question41.htm

3-HEWITT, P. G. (2019). FOCUS ON PHYSICS: The Physics of Rainbows. The Science Teacher, 87(1), 14–17. https://www.jstor.org/stable/26899181

4- Universita Della Santa Croce. (n.d.). On the physics of rainbows, by Federica Volpi. INTERS.org | Inters.org. https://inters.org/physics-of-rainbow

5- Electron (and other light particles) interactions with matter. (n.d.). Ossila. https://www.ossila.com/pages/electron-interactions-with-matter