Following the wave theory of light explanation, James Clerk Maxwell came up with the electromagnetic theory behind rainbow formation. Nonetheless, the Fresnel-Airy theory is oftentimes used instead given its practicality.
The development of science behind rainbow formation has a long history that involves several scientists and mathematicians. We will only be touching on two of such people for the purpose of this course.
Why touch on the history though? Because the history provides us with several explanations to the details and tonalities of rainbow formation!
Given that rainbows are a result of reflection in rainbows, then how do light rays leave the raindrop at different angles?
René Descartes believed that rainbows occur because when a light ray hits and enters a raindrop, it undergoes internal reflection before it exits. As mentioned before in the section about refraction, the light ray changes it direction when it moves from one medium to another (in this case, air and water). The change in direction can be calculated using Snell's Law.
Snell's Law basically states that sin i = n sin r, given that:
i is the angle of refraction,
n is the index of refraction (n is ~1.33 for the above scenario)
Fundamentally, this is what occurs.
However, we have to remember that raindrops are in fact, spherical (take note that this is still an approximation of a raindrop's shape).
In a raindrop, the total deflection can be calculated using the following equation:
D = 180° + 2i - 4r, given that
i is the angle of incidence,
r is the angle of refraction (the first angle when the ray meets the droplet)
i is the angle of incidence,
r is the angle of refraction (the first angle when the ray meets the droplet)
Since we know that there can only be one direction for light rays to come from (given that we only have one sun which can only be at one location), Descartes took this into account. He tried to figure out the science behind rainbows by observing the interaction between light rays coming from one direction and rain drops.
Descartes took this one direction to be at the infinite left.
There is some higher-level math involved in this step. Since this is out of our context, we will leave the math out. Do drop us an email if you would like to learn more about the math! :)
Even though light rays enter a rain drop at various angles, their deflection ends up to be somewhat equal. How do the arcs of a rainbow arise then? What Descartes discovered was that light rays cross and gather along another one light ray. This gathering and crossing basically a concentration of light in one direction, which results in a caustic ray! This also indicates the maximum deflection possible - at 42°.
He also discovered that the height of incoming light rays increase in a pattern- from 0 to the radius of the rain drop. As this height increases from 0 to the radius, the height of outgoing light rays decrease as they tend towards the caustic ray. As the height passes that of the caustic ray, it subsequently increases again.
Caustic rays come together to give us the bow of a rainbow. This gives rise to the 6 arcs of the rainbows, because the outgoing light rays leave the raindrop at various angles.
The passage above acts as a reminder of the complexity and beauty behind the relationship between nature and science. Rainbows are not simply a result of reflection in raindrops.
Descartes managed to figure out the science behind rainbow formation and where rainbows were located, but he failed to find out why a rainbow was made up of 6 colours (though he figured out why there were 6 bands).
Isaac Newton discovered that white light was made up of all 6 colours of the rainbow, and that refraction was what caused white light to split into the various colours. He was the first to set up a demonstration showing that a second prism had the ability to combine the 6 colours (that remained constant) produced by a first prism from incident white light.
This theory is known as the Corpuscular Theory of Light!
He had also proved that blue light experiences more refraction than red light; this being the catalyst to the future explanation behind the colours of the rainbow.
(With reference to the section of Descartes) We now know that arcs get their colour because caustic rays of different colours are scattered upon leaving the raindrop!
How did he help Descartes?
Descartes managed to figure out the science behind rainbow formation and where rainbows were located, but he failed to find out why a rainbow was made up of 6 colours (though he figured out why there were 6 bands).
Isaac Newton discovered that white light was made up of all 6 colours of the rainbow, and that refraction was what caused white light to split into the various colours. He was the first to set up a demonstration showing that a second prism had the ability to combine the 6 colours (that remained constant) produced by a first prism from incident white light.
This theory is known as the Corpuscular Theory of Light!
He had also proved that blue light experiences more refraction than red light; this being the catalyst to the future explanation behind the colours of the rainbow.
(With reference to the section of Descartes) We now know that arcs get their colour because caustic rays of different colours are scattered upon leaving the raindrop!
It is interesting to note the Corpuscular Theory of Light could only explain primary and secondary rainbows (also known as double rainbows). Other rainbows such as supernumeracy rainbows (rainbows with smaller rainbows inside of the main one) have to explained by George Biddell Airy's theory- which talked about how the intensity of the colours of rainbows was dependent on the size of water droplets. His theory, however, was based on Thomas Young's work. Young was the one who managed to observe the conditions necessary for light to behave as waves.
