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rainbow with secondary rainbow (photo by Capt. P.W. Jackson)
These are the best-known optical phenomena in the sky. A rainbow is formed when the sun shines directly on drops of rain, and appears as a coloured band at about 138° from the sun, hence at 42° from the anti-solar point. At solar elevations higher than 42° the bow is entirely below the horizon and therefore invisible in the sky. The (primary) rainbow is red outside and blue inside; the innermost colours are usually somewhat paler than the red. On its inside, the colours sometimes recur once or more often: these are the so-called 'supernumerary bows'. Within the rainbow, the sky is plainly brighter than outside it. At about 8° outside the rainbow, a second bow can be seen to appear - the secondary rainbow. The latter is considerably fainter than the primary one. Its sequence of colours is reversed; outside this bow the sky is again somewhat brighter than within. Usually, only sections of these rainbows are seen, because the entire sky is not filled with drops or not all the drops are lit by the sun. On the other hand, the rainbow can sometimes be seen as a complete circle from a high tower!
Rainbows occur mostly in showery weather, hence under cumulonimbus clouds. But also after passage of a weather front they are sometimes seen appearing beautifully while the rain passes, at least if the front is immediately followed by a sharp bright period. Moreover, they occur in waterfalls, geysers, sprinklers, fountains, flower- syringes, in sprays of sea waves, in bathroom showers, in drops of water in grass (the 'dew-bow') or in dew-drops on cobwebs and in all other places where drops of water are directly lit up by the sun. There is of course no fundamental difference between all these bows: the colour sequence is always the same and they always appear at 42° from the anti-solar point; only in the case of drops of salt water does this distance prove to be about 1° less (see photo 2).
photo 2: above the horizon, the rainbow is visible in freswater drops, below the horizon in drops of
salt seawater spray. The latter rainbow is one degree closer to the antisolar point (by J. Dijkema)
So, we must always take care to look in the right direction with the sun at our back. The colours of the rainbow are of course the same as those of which the source of light is composed; in the case of a red sunset, only the 'red rainbow' can be seen.
We can also see rainbows generated by artificial light or by moonlight; certainly in the latter case the bow seems white as a result of the poor colour sensitivity of our eyes when the light intensity is low. Once one was even seen during a total eclipse, in which the solar corona functioned as source of light! At the time the colours were pink and green. These are the colours of the brightest emission-lines of the solar corona.
However the rainbow may come about, its light always has a very high Tangential polarization. The degree of polarization of both the primary and the secondary rainbows is so great that they can be completely 'rotated away' with a polarizing filter. But with the filter in the tangential position, the bows appear very clearly. Indeed we can then often see the secondary rainbow, when it is not yet visible to the naked eye. Also it is much easier to find faint rainbows, like the red one or the dew-bow, with a polarizing filter.
The strong polarization is the result of the path that the beams of light generating the rainbow must follow through the drops. In the case of the primary rainbow, the beams are always subjected to one internal reflection against the back of the drop; this reflection occurs twice in the case of the secondary rainbow. The reflections happen to occur very close to the Brewster angle so that the light is very strongly Polarized. In contrast to what happens in a single external reflection, this light suffers a change in its direction much greater than 90° because of additional refractions. As a result, the strongly polarized phenomenon appears much further than 90° from the sun. The degree of polarization of the secondary rainbow turns out to be slightly less strong than that of the primary rainbow (90 % as against 96 %). In practice, however, it only means that both rainbows disappear entirely when a Radial ly directed polarizing filter is used.
So, the very high polarization of the light of the rainbow leads to a striking contrast: with a tangentially directed filter both rainbows are brilliantly clear, but with a radially directed filter they disappear completely.
source: Polarized light in Nature