I know—it sounds like a Vaudeville jazz group.
In fact, Alexander’s dark band is the darker region separating the primary rainbow from the secondary bow. The secondary rainbow is reversed in its color sequence. It’s weaker than the primary rainbow. The light for the second rainbow comes from sunlight that bounces twice inside the spherical raindrops that are suspended in the air after a rainstorm.
The band of sky between the bows looks darker compared to the region inside the primary bow because a certain amount of light bounces back at angles below 42 degrees. In fact, in this photo there are faint “supernumerary bows” inside the primary bow.
Alexander’s Dark Band is named after Alexander of Aphrodisias, who first described the phenomenon. While the main bow forms at 42 degrees from the antisolar point (see last week’s post), the secondary bow forms at 51 degrees.
Here is a version of the double rainbow that I painted for Dinotopia: First Flight. To get the curvature exactly even, I attached the paintbrush to an improvised beam compass (basically a long wooden bar pivoting on a nail). It’s the best way in traditional oil paint to keep the arc geometrically perfect.
John Everett Millais Blind Girl of 1856 shows a double rainbow correctly painted in terms of the sequence of colors (red at the outside of the main bow). But he failed to show the lighter region inside the inner bow, and the darker band between the two bows.
The majesty of Millet’s conception comes from the knowledge that the blind girl is unaware of the glory behind her. Millet is careful to show the light coming almost exactly from the front. The shadows are cast just a little to the right of the trees.
The sun is behind us, a bit to the left. The antisolar point (the point 180 degrees opposite the sun) is outside the frame of the picture, just to the right of the flock of crows. It forms the center of the arc of both rainbows.
Rainbow photo by Eric Rolph, courtesy Wikipedia, link.
Last week's post on the science of rainbows, link.
Another explanation of all this, with a nice diagram of the angles, link.