Why do faint extra rainbows sometimes appear just inside the main rainbow?

Settled in 1838 by G. B. Airy (1838) diffraction theory; refined with phase-shift corrections to T. Young's earlier interference picture (reviewed by P. Laven, 2017).

Thomas Young first proposed in the early 1800s that the extra arcs arise from interference between two light rays that exit a raindrop at the same angle, an idea that was historically important as early evidence for the wave theory of light, but which predicted the arcs at somewhat the wrong angles. George Airy's 1838 diffraction theory treated the rainbow caustic properly as a diffraction pattern (analogous to the shadow of a straight edge) and correctly predicted the positions and spacing of the supernumerary arcs. Philip Laven's 2017 Applied Optics paper showed that Young's simpler interference picture actually becomes accurate once a subtle 90-degree phase shift at the ray's internal focal line is included, reconciling the two historical approaches.

Those extra pastel bands inside a rainbow were one of the first pieces of direct evidence, visible in the sky, that light really is a wave.

Explanations that circulated

• Supernumerary bows are simply a second, weaker set of primary/secondary rainbows from different raindrops
• They are a lens artifact of the observer's eye, not a real atmospheric optical effect

References

• Supernumerary arcs of rainbows: Young's theory of interference, P. Laven, Applied Optics, 2017