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Colorado Rainbows & Supernumeraries ~ An evocative scene by Haitong Yu at NCAR, National Center for Atmospheric Research, near Boulder, Colorado, USA. The secondary bow at right has reversed colours to the primary and is much broader. Inside the primary are bands of greens and purples, more widely separated higher up. These are supernumerary rainbows.         Image ©Haitong Yu, shown with permission

For any deflection angle, there are two rays passing through the rain drop. At the rainbow's edge - the minimum deflection angle - these two rays merge. At larger deflection angles, which show up inside the rainbow, the rays separate. These are the rays that form supernumeraries.

When raindrops are small the behaviours of wave crests along the ray paths becomes important. The schematic at upper right shows the crests out of phase. This would produce a dark band. Other directions produce a bright band - a supernumerary.

Supernumerary spacing depends on the raindrop size and so they show best when drops are more or less similarly sized. Small drops give a wider spacing. In the photo at top the drops must be smaller towards the upper parts of the rainbow.

Supernumerary bows

Supernumeraries occur inside the primary and outside (but rarely) the secondary. The main bows are predominantly refraction effects. Supernumeraries arise from diffraction.

Supernumerary spacing is also wavelength dependent as shown in this Airy function simulation for 0.7mm diameter drops. The considerable colour overlaps give supernumeraries of green and purple/pink hues.

Simultaneous (or near simultaneaous) pictures of rainbows like that above shot through blue and orange filters would be interesting - and maybe a first!