In my previous post about rainbows, I described how the light of the rainbow was reflected back to our eyes by falling water droplets. For a raindrop at the top of the rainbow arc, light follows a path that enters near the top of the raindrop, bounces off the back, and then exits from the bottom:

The angle between the incoming light ray and the reflected ray ranges from 42.4º for red light to 40.5º for violet light. All other light rays, entering the drop either closer to its centre or closer to its edge, are reflected back at smaller angles—and it’s their smeared and superimposed light which accounts for the white glow visible within the arc of the rainbow, above. I used the name “Offset” for the parameter that measures how close to central the incoming light ray hits the droplet. It’s measured like this:

And by plotting the deflection of light rays at various offsets, along with light transmission along the reflected pathway, I showed how the rainbow forms at the point of maximum deflection, corresponding to an offset of about 0.86:

I called the ray that follows this maximum-deflection route the “rainbow ray”. (See my previous post for much more detail.)

I also produced a little diagram of how light is lost from the rainbow ray each time it encounters a surface at which it is either reflected or transmitted, like this:

So the rainbow ray arrives at your eye containing only about 4.5% of the light that entered the water drop.

What I want to talk about this time is the fate of the light that undergoes a second internal reflection (labelled “0.5%” in the diagram above).

It’s possible for light from this second reflection to exit the drop when it next encounters the water-air interface, like this:

The second reflection takes the light to the front of the raindrop, and it exits on an upward course, crossing the path of the incoming ray. For this ray to reach the eyes of an observer looking towards the antisolar point (the centre of the rainbow arc), we have to flip the geometry upside-down, like this:

So now the incoming light enters near the bottom of the raindrop, and the reflected light is deflected downwards, towards an observer on the ground.

This light is the source of the secondary rainbow, which is larger and fainter than the primary rainbow I’ve been describing so far. (A secondary rainbow is dimly and partially visible in the photograph at the head of this post.)

Like the singly reflected light that forms the primary bow, the doubly reflected light that forms the secondary bow has its own characteristic angle of deflection, but this time (because of the flipped light-path described above) it’s the minimum angle of deflection from the antisolar point where light is concentrated to form the secondary rainbow ray:

I’ve kept the scale of the transmission curve the same, to allow comparison with that of the primary rainbow. And the range of the angle-of-deflection axis is the same, but it spans from 45º to 90º this time, rather than the 0º to 45º of the primary plot.

The minimum deflection occurs at an offset of about 0.95. The deflection is 50.4º for red light, and 53.8º for violet.

So there are several things going on with this secondary rainbow. Firstly, because the light enters closer to the edge of the raindrop, the refraction is greater, and that causes the separation between red and violet light to be greater—so the secondary rainbow has a width of 3.4º, compared to just 1.9º for the primary rainbow. Secondly, the fact the light-path is flipped over compared to the primary reverses the colour sequence—red is on the inside of the secondary bow, but on the outside of the primary. Thirdly, the additional reflection means more light is lost from the rainbow ray as it passes through the drop. This is partially offset by the fact that the rainbow-ray offset is closer to the offset of maximum light transmission—so the light transmitted into the secondary bow works out to be about 43% of that transmitted into the primary. Finally, because the rainbow ray occurs at a minimum deflection from the antisolar point, the rest of the sunlight entering the drop is reflected to greater angles than the rainbow ray—it lights up the sky outside the secondary bow.

So if we imagine a raindrop falling vertically towards the antisolar point, it at first sends a doubly reflected mixture of light, appearing white, towards the observer’s eye. When it has fallen to 53.8º from the antisolar point, it sends a relatively pure, doubly reflected violet light to the observer—the top of the secondary bow. As it falls from 53.8º to 50.4º, it reflects all the spectral colours in sequence, ending with red light at the inner rim of the secondary bow. Then, it quite literally goes dark. It is too low to send any doubly reflected light in the observer’s direction, but too high to send any singly reflected light. The region of sky between the secondary and primary bow is therefore noticeably darker than either the region outside the secondary, or inside the primary.

This dark region is called Alexander’s Dark Band*, in honour of the Greek philosopher Alexander of Aphrodisias, who described it in about 200AD.

After passing through the Dark Band, the raindrop lights up with singly reflected red wavelengths when it reaches 42.4º, and then runs through the spectral colours until it reaches violet at 40.5º. Below that point, it reflects a mixture of wavelengths (white light again) until it hits the ground.

The whole sequence looks like this:

The obvious question now is this: if the primary rainbow forms from singly reflected light, and the secondary from doubly reflected light, what happens to triply reflected light? Is there a tertiary rainbow? And if so, where is it?

That’s the topic for next time.

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* Not to be confused with the 1911 Irving Berlin hit, Alexander’s Ragtime Band.