Supernumeraries Galore

Jürg Alean (Swisseduc.ch, Stromboli online) imaged this fogbow on Luosto mountain in northern Finland. The inner supernumerary bows are remarkable. ©Jürg Alean
IRIS Mie calculations show the effect of droplet size distribution on supernumerary visibility.

Notice how the secondary fogbow supernumeraries are more widely spaced.
Interference between waves gives supernumerary fringes. There are two wave routes for any fogbow/rainbow direction and when these waves are in phase there is a bright supernumerary fringe.

Fringe spacing depends on droplet size with smaller droplets give more widely spaced fringes. When there are drops of different size the differently spaced fringes overlap and blur each other.

The droplets forming Jürg Alean’s bow had to be of similar size to produce the many fringes. Probably the droplets had a diameter distribution (standard deviation) of <5%.

Atmospheric
Optics
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Fogbows are produced like rainbows, a single internal reflection inside water droplets directs light backwards.   The difference is that fog and mist droplets are small (1-100 micron diameter) and light waves are diffracted to form a diffuse, broad and pastel coloured bow rather than a bright rainbow.

To a lesser extent diffraction influences a rainbow too - Otherwise we would not see rainbow supernumeraries.