Rainbows, Rainbows Everywhere!

A photograph showing a rainbow over Glasgow Southside in 2004. Image: NaturPhilosophieRainbows: Technicolor Symphonies in the Sky

Rainbows are one of Nature’s most gorgeous optical spectacles to behold, brightening up clouded skies with an ephemeral palette of colours when the light falls just right… 

Two physical phenomena are involved: refraction and reflection.  When a drop of water in the atmosphere meets a ray of white sunlight, the light refracts into the drop, reflects from the drop’s inner surface, then refracts out of the drop.  As with a glass prism, the first refraction separates the sunlight into its component colours, and the second refraction increases the separation.


Refraction and Reflection Combined

We focus our attention on a single water droplet in the blue (or violet) band of the arc.  Blue light is only one part of the spectrum of colours – each one shining out from the raindrop at a different angle.  Blue shines in our direction, but other colours shoot out in different directions and cannot be seen from our standpoint.  Taking an adjacent droplet, it also shines blue light at us, as well as the other nearby raindrops.  When we look at a raindrop in the red band, the red light shines out at us.

A diagram showing how rain drops suspended in the atmosphere can act as a prism to refract and reflect the sun light, and create rainbows.

Between blue and red, all the colours are refracted and reflected from a multitude of water droplets in just the right way for those light wavelengths to be shining in the observer direction.  And beyond the edge of the arc, no raindrop emits light in our direction, although they may well shine coloured wavelengths too.

Snell’s Law

When sunlight refracts, it follows Snell’s law, also called Snell-Descartes law or the law of refraction.

Snell’s Law describes the relationship between the angles of incidence and refraction, when light or any other electromagnetic wave passes through the boundary between two media, like glass or water.  It states that the ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of phase velocities in the two media, or equivalently the reciprocal ratio of the two refractive indices:

\frac {\sin\theta_1}{\sin\theta_2} = \frac {v_1}{v_2} = \frac {n_2}{n_1}

Basically, it gives a way of predicting the amount by how much a light ray bends.  This law is a consequence of Fermat‘s principle of least time, which itself follows from the propagation of light as waves.


Double Rainbows

A photograph taken in 2004 of a double rainbow over Glasgow Northside. Image: NaturPhilosophie
Double Rainbow over Glasgow Northside, 2004

Under certain conditions, light bounces off the inside of the drop more than once, coming out at a different angle.  The result is a fainter secondary arc, known as a double rainbow, where the order of the colours are reversed.

A third, or fourth, additional rainbow is theoretically possible, but the emitted light is usually too faint to be seen or captured on camera.