Seven Supernumeraries
A remarkable rainbow imaged by Denis Betsch near Strasbourg, Alsace, France. ©Denis Betsch, shown with permission

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Supernumerary arcs are narrow coloured fringes sometimes seen close to the inner purple edge of a primary rainbow. One or two fringes are a common sight but this rainbow with 7-8 arcs is rare indeed.   

Supernumerary arcs or fringes are seen when the raindrops forming the bow are small (1mm dia. or less) and of uniform size. The fringe spacing widens with decreasing drop size and therefore most fringes are seen when the drops are of uniform size and there is less blurring from overlap of different fringe spacings.   The drops must have been exceptionally uniform to produce so many fringes

Supernumeraries are diffraction effects associated with a light caustic.     

Light waves reflected once inside a rain drop form the primary rainbow. They (strictly speaking waves rather than rays) emerge in a number of directions and as they do so they fold over and intersect each other. They form a pattern of crossing rays.

The surface separating the region of space occupied by two rays intersecting and that where there are no rays is a caustic sheet.  The sheet, a fold caustic, marks the intensely bright rainbow rim.
Close to the caustic sheet, each intersecting ray/wave pair coalesces and interferes.   Coincident wave crests give brightness, out of phase crests give darkness.   

The result is a set of light and dark diffraction fringes parallel to the caustic sheet – These diffraction fringes are the supernumerary arcs.

At left, supernumerary fringes and their primary rainbow are computed using a theory developed by British Astronomer Royal George Biddell Airy (1801-1892).  His theory makes assumptions about the algebraic form of the outgoing waves from the drop. Although it is approximate it is near enough to simulate supernumeraries and far quicker then the later, fully accurate but computationally laborious Mie-Lorentz theory.

The top slice shows supernumerary fringes from a point source and monosized drops 0.7mm diameter.

The lower slice is for of the same mean size but a spread of diameters plus a 'real' sun 0.5° diameter. Both changes blur the intricate supernumerary interference structure to our eyes.