A few weeks ago during dinner there was a sudden burst of sunlight in the midst of a rain shower. Knowing that these were ideal rainbow-making conditions, we jumped up from the table and ran out to investigate. Sacrificing our meal was entirely worth it: this photo was taken just a few hundred metres down the road from our house. Once again, I am blown away by the beauty of physics in the world around me.
The bright feature in the photo labeled A is the primary bow, which is always observed with the red band on the outer edge of the arc and the violet band on the inner edge. Light coming from the Sun, behind you, scatters in all directions from the raindrops in front of you. The light doesn’t scatter equally in all directions though, as shown in the following image, modified from a 2016 paper by Alexander Haußmann. In this plot of intensity versus scatter angle, zero degrees represents light that is scattered straight back towards you, while light scattered directly away from you is at an angle of 180°.
(image from Haußmann’s 2016 paper)
We can see that the intensity is pretty steady at small angles, coming straight back towards you, then reaches a peak at 42°. This is the angle where you see the primary rainbow: 42° is the maximum angle of the cone of light that scatters straight back towards you from one bounce inside a raindrop.
At the very edge of this cone of scatter, we see the separation of the different colours of light that come from the Sun: the highest wavelength, red, scatters back at a slightly higher maximum angle (42.2°) than the lower wavelengths, like blue, which has a maximum angle of 41.2°, as we see in the sky with the red on the outside edge and violet on the inside of the primary bow.
Primary bow from one bounce inside the droplet 
Secondary bow from two bounces inside the droplet
Moving along the relative intensity graph from 42° to the next large spike at 51°, there is a region known as Alexander’s band, named after Alexander of Aphrodisias (circa 200 AD), which is marked B in my photo at the top. The band ends at 51°, which is the minimum scatter angle for light to bounce twice inside the drop and then reach our eyes as sketched here. The angle of 51° corresponds to the location of the secondary bow (C).
Alexander’s band is dark because it spans the ‘dead zone’ between the maximum angle for scattering with one bounce and the minimum angle for scattering with two bounces inside the droplet. Since light can’t bounce 1 ½ times, the intensity in this in-between range is quite low and the sky looks darker here than above the secondary bow or below the primary bow. It’s not midnight black though, because light is being scattered from clouds, other droplets, the front surface of droplets, etc.
The secondary bow, with its sharp spike at 51°, will always appear with the red band on the inside and the violet band on the outside, opposite to the order seen in the primary band. It’s not always easy to see this feature though; the intensity is less than that of the primary bow. With each bounce, some of the light continues through the droplet rather than reflecting, which means that the tertiary (and higher order) bows are rarely observed. The added challenge with seeing the tertiary bow is that it appears at about 140°, very much in the forward direction, away from the observer. To see the tertiary or quaternary bows, you have to be facing the Sun with scattering raindrops between you and the Sun, which, shining brightly, tends to wash out faint features like these weak bows. The angular spread of the colours increases with each order of bow, so this smearing out also makes them harder to see. The fifth order bow is predicted to appear just inside the secondary bow, as seen as the little spike in the intensity graph at around 50°.
We have marveled at, and tried to understand, rainbows for millennia. Way back in 1637, Descartes published a chapter in his Discours de la Méthode in which he correctly calculated the angles observed for the primary and secondary bows, although the colours were still a mystery in his day. Newton famously solved this mystery by deducing that white light is made up of all these colours. But it wasn’t until the early 1800s when Thomas Young linked the colour of light to its wavelength that physics was able to explain the observation of “supernumary arcs”, shown here in a cropped piece of my photo, marked as D.
These bands of light and dark regions just inside the primary bow are classic interference patterns caused by light scattered at the same angle but travelling different path lengths through a raindrop. Young’s theory correctly predicted the effect of drop size on this feature of the rainbow and provided further support for the wave theory of light.
Once James Clerk Maxwell figured out that light is transmitted in the form of electromagnetic (EM) waves (around 1864), a new theory emerged to understand light scattering from small objects like raindrops, known as Mie scatter after the German physicist Gustav Mie. By breaking the incoming EM wave into a series of partial waves, we then calculate how each scatters from a sphere of a particular size and electrical properties, and then add up the results. Intensive calculations with Mie theory were not possible until we had the help of computers in the latter part of the 20th century, but now millennia of wonder and delight has been explained, matching observations beautifully.
Refinements continue today, with tweaks to deal with non-spherical droplets (affectionately known as the ‘hamburger bun’ model), different sizes of droplets within the rain shower, effects of the intensity of rain in different regions of the shower, and more. Science marches on, so grab your rubber boots and let’s dash out, chasing rainbows. My little helper unicorn likes to refresh with a rainbow treat when we get back inside!
References:
Rainbows in nature: recent advances in observation and theory by Alexander Haußmann 2016 Eur. J. Phys. 37 063001
More to rainbows than meets the eye by Simon Davies 2016 IOP Publishing News
The Mysterious Physics of Rainbows by Jon Butterworth, 2017, The Atlantic
Wonder Why 