Chromatic dispersion, rainbows and Alexander's dark band

Primary and secondary rainbows are due to chromatic dispersion in water droplets. Here we explain their formation with ray diagrams and experiments. Both primary and secondary bows are clearly visible in the photo at right, as is Alexander's dark band, the dark sky between the two. In the primary bow, red has the largest radius; in the secondary bow, the colour order is reversed. This page supports the multimedia tutorial Geometrical Optics.
The primary (right) and secondary rainbows. Photo: Mike Gal

A model experiment to show the formation of the rainbow

An experiment using a beam of white light refracted twice and reflected once in a plexiglass cylinder

The photo at left shows a collimated beam of white light entering from the left and entering a cylindrical disc of plexiglass. It is refracted (less dense to more dense medium), and then totally internally reflacted, at the right of the disc. When it leaves the disc, it is refracted again (more dense to less dense). The beam then strikes a screen at the bottom left of the photo forming a spectrum that is seen in the second photo.

Let's imagine that this happens, not in a cylinder of plexiglass, but in a spherical drop of water, as illustrated in the schematic at right. Its direction has been changed by 140°. Using the values for water, the beam is deflected by 140°. So the viewer would see the twice-refracted-once reflected beam at 40° to the shadow of her head. This (almost) explains why the primary rainbow is an arc, with angular radius 40°, centred on the viewer's shadow.

Why a spectrum? The reason is the same as for Newton's prisms. In Snell's law and refaction, we saw that the refractive index for blue light was slightly higher than for red—in glass. The same is true for many media, including water. So blue is deflected through a bigger angle (140°) than red (138°), as shown in the small photo above and in the schematic above right.

 

The primary (40°) bow: schematic

I hope that you are now puzzled, as I was when I first thought about refraction and rainbows. After all, in any one droplet, like the one in my schematic, it looks as though the red ray lies inside the blue ray! That is, of course, true. However, if you think about how big a rainbow is, it's obvious that the blue rays we see are not coming from the same droplets of water as the red rays. Rather, if blue is deflected by 140°, we see blue coming from droplets that are at 180 − 140 = 40, while red is refracted by 138°, we see blue coming from droplets that are at 180 − 138° = 42°. So we see the blue arc with an angular radius of 40° and the red with a radius of 42°. Red outside blue, as in the photo, and as in the schematic below. No matter where you see a primary rainbow, its angular radius will be between 40 and 42°, and it is always centred about the shadow of your head. Sometimes it seems to be many kilometers in radius. If you take an open air shower at midday, you will sometimes see yourself surrounded by a circular rainbow with a radius of a metre or so.

Schematic of the primary bow: blue light is deflected 140° to mark an arc with angular radius 40°, centred on the shadow of your head. Red is deflected 138° to make an arc with radius 42°.

There is more to it than that, however. Blue and red light can both be deflected over a range of several degrees. In the experiment with the plexiglass disc, displacing the disc vertically a tiny amount y produced a spectrum with a slightly different set of angles θ. However, for any colour, there was a minimum in dθ/dy; as I increased y, θ first became smaller then became bigger. The angles 138° and 140° (for water) are the angles for this extremum. So, at the extremum for blue with the bow angle of 40°, many more droplets refract blue than at 39 or 41°, so blue is bright at that angle.

 

The secondary (50°) bow

And so to the secondary bow, the rainbow with the larger radius and the colours reversed, as shown in the photo at the top of this page. For this rainbow, the offset y of the beam has the opposite sign, and the once-refracted ray reaches the water-air interface at a greater than critical angle, and so undergoes total internal reflection.

Schematic: the secondary bow involves two refractions and a reflection.

Again, blue is deflected more than red but, because the beam has a reflection as well, it crosses over itself. This gives the blue secondary bow a radius of 53° and the red a radius of 50°: the spectrum is reversed in the secondary bow.

Alexander's dark band

Looking back at the photo at the top of the page, we see that the area inside the primary bow is brigher than the area outside the secondary bow, which is in turn darker than the area between the two bands: this dark region is called Alexander's dark band. So, what causes the dark band of sky between the primary and seconary bows? When you look at a rainbow, you have the sun at your back. The reason why the sky is not black is that light is back scattered by air molecules, water droplets and whatever else is there. Some of the light in Alexander's dark band is being refracted by water droplets at angles other than 40 or 50°: we could say that that light is making rainbows for people at other positions

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