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Primary and secondary bow supernumerary fringes imaged by Douglas Booth.

He saw these near sunset rainbows on the island of Berneray in the Outer Hebrides off the west coast of Scotland.

To the right and inside the primary bow there are 4-5 fringes.

At left is a much more rare sight, 2-3 secondary supernumerary fringes or bows.

Image ©Douglas Booth

Supernumeraries probably became so named because they were 'extras' and not supposed to exist. Classical ray theory as used by Descartes to explain the rainbow cannot account for them. The problem was further compounded by long insistence on Newton's corpuscular theory of light.

We need waves. They were used by Thomas Young in 1803 to explain supernumerary existence.

Astronomer Royal, George Biddell Airy produced a quantitative treatment in 1836 that still provides fully workable predictions.

Mie-Lorentz calculations (after 1908) are wholly accurate and were used to generate the page background simulation.

The Mie scattering prediction reproduces well the brightness inside the primary bow and the darkness of Alexanders dark band between the primary and secondary rainbows.

This simulation, adapted from Philip Laven's MiePlot, shows why secondary supernumeraries are seen so rarely (although they are probably unknowingly photographed more often).

The sun illuminated drops are all 0.5mm diameter to avoid blurring by variations in droplet size. Good supernumeraries need droplets of very similar sizes.

The primary supernumeraries are closely spaced with pronounced angular oscillations in brightness - the vertical intensity scale is linear.

In contrast, secondary supernumeraries are widely spaced and there is little variation in brightness between a fringe maximum and minimum.

A colour subtraction enhancement showing the many fringes.

Small rain droplets of quite uniform size were needed to produce this rare sight.

At left the formation of primary supernumeraries is shown. Secondary ones form in the same way.

Two classical ray paths contribute to the light at any point on a rainbow.

But light is a wave phenomenon and the wave crests of the two emerging waves can coincide or be out of phase, depending on the viewing angle and wavelength.  When the crests coincide there is a bright arc, a supernumerary bow.