How is a rainbow formed? It is something to do with refraction, but raindrops are approximately spherical, not like tiny prisms. The diagram below shows how an incident ray from the sun is split into three components after refraction and reflection at the water-air boundaries.
Figure 1. (a) Light incident on a raindrop can take a variety of paths. Some of the incoming power is reflected at the first air-water boundary, to form ray A above. For most rays, most of the incoming power is refracted into the drop. As this ray hits the other side of the raindrop, some (most) is transmitted to form ray B above. However, at this point, some is reflected also. This reflected ray carries on until it hits the side of the drop again. Much of the power in this ray is transmitted to form ray C above. It is this type of ray that contributes to the primary rainbow.
It is the part of the beam that undergoes one reflection in the drop that contributes to the primary rainbow. Let us look at such rays more carefully. The path of any ray hitting the drop can be determined using Snell's law and some simple trigonometry. A ray that hits the drop on centre is merely reflected back upon itself. As the rays move out from the centre line the angle between the incident and reflected rays (type C above) increases, but only up to a certain point, beyond which the angle decreases again. You can see this happening by running (AFTER reading the paragraph below!) the simulation in CUPS .
Click on the highlighted words above. Read, then click on the green box to remove it. Click on the Rainbow menu item, and you will see the sperical raindrop. Move the incident ray to change which part of the drop it hits first. Note how the angle through which the ray is reflected varies with where the incoming ray hits the drop. This simulation is a MS-DOS programme, so does not run in a conventional window. To get back to this tutorial sheet while still keeping the simulation running, press the alt and tab keys together. Each time you press the tab key with alt held down will step you through the windows that are open on your PC. Release alt and tab when you reach the window you want. Likewise, you can use the same procedure to get back to the simulation. To exit the CUPS programme, click on the file button and on the exit label that comes up.
This phenomenon of an angle of minimum deviation produces a concentration of scattered rays at the angle of minimum deviation, and it is this concentration of the light that gives rise to the brighter arc of the rainbow. The reason that the rainbow appears coloured is that water has a slightly different index of refraction at different wavelengths (Blue is Bent Best) causing the rainbow angle to vary with wavelength. In water droplets the angle of minimum deviation is 137.6 degree for red light and 139.4 degree for blue light. To observe the arc of a natural rainbow we must look at raindrops at an angle of 138 degree from the sun's direction, or 42 degree from the antisolar point, as shown in figure 2. The reason for the rainbow's arc comes from this: to see the rainbow we must look in any direction that is 42 degrees from the antisolar point, and this condition describes a circle.
Figure 2. Observation of the rainbow. Note how the 138 degree angle of minimum deviation ties in with the (180 - 138 =) 42 degree angle that the bow is from the antisolar point. Anywhere that you can look at this angle to the antisolar point gives the rainbow condition. How much of the potentially circular rainbow we can see depends on the angle of the sun, and where the raindrops are.
Note that if we look inside the rainbow arc we will see lots of "white" light being reflected back to us by the drops. But outside the rainbow arc there will be no light coming back due to a single reflection. The contrast in the amount of sunlight reflected by the drops inside and outside the bow is clearly seen in the next photo.
The second rainbow that can sometimes be seen alongside the main bow is due to rays that have undergone two internal reflections. The second order rainbow can also be explored in the CUPS simulation, and is seen in the photo at the head of this page. The primary and secondary bows, and bows of higher order, can be examined in a second year lab experiment.
In the CUPS simulation the graph at the bottom right hand corner shows how much of the light is reflected at different angles. The peaks show the angles of the first and second order rainbow, as well as the lack of light reflected between the first and second order rainbow.
Books and articles on rainbows and related phenomena
This page was created by B Sinclair, last modified 9.00