Supernumerary Rainbows

beautiful picture of the side of a rich rainbow arc with thinner fainter rainbows on its inside edge. the sun is casting golden light on the trees behind the roof of a house. John Entwistle

Supernumerary rainbows are the faint rainbows that can sometimes be seen under the primary rainbow. They occur when the water droplets are smaller than 1mm wide. Supernumeraries & Drop Size - AtOpt

At the bottom of my primary rainbows page I illustrate how each color makes a slightly different sized cone of concentrated light (which we see head on as a circle) and how they overlap to form a rainbow. Primary rainbows with supernumeraries overlap very similarly, each color is simply in banded rings.

seven rings with two rings nested in each. The largest is the red rings, and the smallest is the violet rings.
The rings overlapped so that they form a rainbow with two smaller fainter rainbows nested inside of it. [TO BE ADDED] animation of the rings superimposing over each other, starting with violet, resulting in a rainbow with smaller fainter rainbows nested inside of it.
diagram showing how the colors of light in a rainbow overlap for large water drops and small ones. Primary Colors (1), Supernumerary Colors (2) - AtOpt

These two simulations show the concentration of each color and how they overlap to form a primary rainbow made from large verus small drops. The left is a primary rainbow formed by larger water droplets, and the right is a primary rainbow with supernumary bands formed from 0.7 mm droplets. Both feature staggered colors, allowing a rainbow to form instead of just overlapping fully to make white light, but in the supernumary simulation, each wavelength of light is banded. Notably, further from the primary, the supernumerary bands are more overlapped, creating washed out rainbows further out as the wavelengths mix more.

two diagrams showing the bands of light for two different drop sizes. One is for monochromatic blue, and the other is for a full rainbow. For both the rainbow and the blue, the 0.8mm drops create thinner bands, the supernumerary ones being very faint. the 0.4mm drops create wider bands and more saturated supernumeraries. Primary Rainbow Drop Size - AtOpt

The smaller the droplets, the wider the supernumary bands are spaced and the brighter they are. Supernumerary bands are more noticable when the raindrops are uniform in size, if they are too varied, their different banding patterns will overlap and wash each other out. Supernumeraries & Drop Size - AtOpt

So why exactly do smaller water droplets form bands of light and large ones don't? Well, wavelengths of light are very small (400nm-700nm) so only drops that are small enough will affect how the waves travel through the drops.

Waves

Light travels in waves, and waves can interact with other waves in what is called interference. They can combine with each other to create larger waves, or cancel each other out to make smaller waves.

diagram of two waves each of constructive and destructive interference.

This diagram shows contructive and destructive interference. When the peaks and valleys of two waves line up, they add together, increasing the amplitude of the wave (left). When a peak lines up with another waves valley, they cancel each other out, decreasing the amplitude of the resultant wave (right).

Interference does not change wavelength, just the height of the peaks and valleys, which affects the brightness of the light. Tall waves are bright, whereas shallow waves are dim (a flat line is no light).

diagram showing waves traveling along side of each other so that their peaks and valleys line up

Consider this diagram again, showing how waves form wavefronts. The lines in the next diagram represent the peak of a wavefront.

straight wavefronts entering a waterdrop where they become curved and exit curved. The Mathematics of Rainbows - AMS

Most notable is the fact that when the wavefronts from the sun hit the curved surface of the drop, they too become curved. The smaller the drop, the more severe the curve of the surface is. Large water drops are so large compared to the wavelengths of light that its curved surface is very slight and negligible, like how the horizon of the earth appears flat to us. This is why only small drops give notable supernumerary bands.

We know that once the exiting rays hit 42 deg from the entrance rays, the light folds back in on itself and flares back upward, making a hard line that forms the primary bow. This is where we get the overlap of wavefronts in this diagram.

diagram of many rays entering a water drop. They bend slighty inside the drop, and then some exit on the other side, bending a little more and flaring out from another. Before exiting the drop, some rays reflect off of the back of it, and exit on the entrance side. These rays flare down and out from another, until hitting a line where they start to flare upwards. Rainbow Caustics - AtOpt

Consider this diagram again (first mentioned in secondary and higher order rainbows). It shows how upon exit, the reflected rays hit a hard line and flare back up, overlapping the other reflected rays. This is a ray diagram, with just straight lines, unlike the previous it doesn't show light in its wave form. With illustrated wavefronts, we will see an interference pattern.

wavefronts entering a water drop, reflecting off the back and exiting. They are straight before entering, and then curved according to the drop inside and exiting the drop. The curved lines overlap where they do in the previous diagram. Like a Bridge Over Colored Water: A Mathematical Review

Here the exit wavefronts are extended to show how they overlap. Where the curved lines overlap perfectly, showing the same amount of space between the lines as the non-overlapping waves, is constructive interference, the peaks and valleys of the wave add together to create stronger more intense light. Where the waves overlap so one line is directly in the midde of the space between two lines is destructive interference, a peak is directly overlapping the valley of the other wave and cancelling it out, making no light. The first place there is constructive interference is the caustic line, the primary rainbow. The second area of constructive interference is the first supernumerary band.

Each wavelength of light has a slightly different pattern from the same sized drop, which allows the colors to show up in each others dark bands, forming rainbows. Infact, thats why there is a white glow on the inside rim of a primary rainbow made from large drops. Since the curvature is slight, the interference pattern produces tight and narrow bands for each color, which overlap to form a glow inside of the rainbow instead of distinct supernumerary bands. Similarly if there are too many different sized drops, they would wash each other out since the interference pattern for each size drop is different.

Suggested Next Page: Fogbows