Rainbow over Hugh Owen

Click images for full-size version.

Rainbows are a common sight in a coastal town with changeable weather like Aberystwyth. However, it's rare to encounter one with all the trimmings - a full arc from horizon to horizon, a secondary rainbow on the outside of the primary with a dark band in between, and supernumerary fringes on the inside.

One afternoon shortly before Christmas 2015, the sky turned to a very unusual yellow colour from the Sun beginning to set over Cardigan Bay, and we were treated to a particularly stunning example. The pictures were taken from the Physical Sciences Building at Aberystwyth University on my mobile phone, which unfortunately doesn't have a sufficiently wide-angled lens to capture the whole arc in one piece, but they still show enough detail to appreciate the feature.

Rainbows are visible when the observer is located between the Sun and an area of cloud or mist. Rays of sunlight entering a water droplet are refracted towards the centre of the drop (because the refractive index of water is larger than that of air). If the refracted ray hits the opposite surface of the drop at or below the critical angle of total reflection, the ray is reflected back towards the Sun (and the observer). The critical angle depends on the wavelength; as a result, the bright zone created by the reflected rays has a coloured fringe - the rainbow. All wavelengths contribute to the area inside the arc, so the inside of the rainbow is brighter than the outside.

Fig.: Exaggerated-colour photo of Alexander's
dark band framed by primary and secondary bow.

If a ray hits the surface of a raindrop at a large angle, it is possible for the total-reflected ray to hit the inner surface of the raindrop below the critical angle for a second time. The result is a secondary rainbow. Because the paths of rays of different wavelengths inside the drop diverge, the secondary rainbow is more spread out than the primary one, and of course it is less bright since only a minority of rays fulfil the total reflection criterion twice. The double total reflection reverses the order in which the spectrum appears, so the colours in the secondary bow are reversed, and the outside of the secondary arc is brighter than the inside. The dark area between the two bows in which no total reflection occurs is called Alexander's dark band. In principle, it is possible to have three (or more) total reflections before the ray leaves the drop, but the higher orders require larger and larger angles at the drop's surface at the initial refraction event. For the third order, the three total reflections add up to an angle reflecting the ray in the forward direction, so the observer would have to look towards the Sun through mist to observe it.

Fig.: Exaggerated-colour photo of 
a supernumerary fringe inside the primary bow.

When two parallel rays entering the same raindrop at different angles (because of the curvature of the drop's surface) are refracted onto the same spot of the opposing surface, they will be total-reflected and leave the drop as parallel rays. However, their travel distance within the drop is different. Considering the wave nature of light, the different path length can lead to constructive or destructive interference between the two rays depending on whether two rays whose waves are in phase when they enter the drop remain so or not when they leave the drop. As a result, certain wavelengths are suppressed due to destructive interference at certain angles, and additional faint rainbows appear just inside the primary bow, known as supernumerary fringes. These can only be observed if the size distribution of the raindrops is very narrow because the path length inside the drop varies with drop size; therefore a wide size distribution averages out the interference pattern.

I recommend Les Cowley's excellent Atmospheric Optics website with much more details about the physics of rainbows and related phenomena, including photos and very instructive graphics and animations.