Secondary and Higher Order Rainbows
Secondary Rainbows
Secondary rainbows are what is commonly refered to as a "double rainbow." The secondary rainbow is fainter and wider than the primary, and the order of its colors is reversed. It also has the same antisolar point as the primary rainbow.
For secondary rainbows, the light is reflected twice (rather than just once) in the water droplets. Since most of the light passes through the waterdrop, what is reflected is fainter, thus secondaries with two reflections are fainter than primaries with only one reflection. Due to the second reflection, the colors are also reversed in a secondary compared to a primary. Secondary: red on inner edge and violet on outer edge, primary: red on outer edge and violet on inner. Secondaries occur between 50.4 and 53.5 degrees from the antisolar point, with red being the smaller degree and violet being the larger.
Every primary rainbow has a secondary rainbow, however lighting conditions often make them difficult to see.
Higher Order Rainbows
Not only are there secondary rainbows, but there are also higher order rainbows. In these terms a primary rainbow is the first order rainbow and a secondary rainbow is the second order rainbow. The order refers to how many reflections within the droplet occur before the light exits the droplet to create the rainbow. Primary (first order) rainbows have only one reflection, secondary (second order) rainbows have two reflections, and third order rainbows would have three reflections within the droplet. Considering with each reflection, light passes through the droplet and is lost, the higher order the rainbow, the fainter it will be. Hence, it is rare to see more than the secondary rainbow.
Interestingly, third and fourth order rainbows are centered around the sun rather than the antisolar point. The angle the exit ray makes with the entrance ray allows the observer to see third and fourth order rainbows only from droplets around the sun, since they are directed towards the viewer that way. All droplets within the suns path reflect the light multiple times, but most do not exit towards our eyes. The third order rainbow is at 42.5 degrees around the sun, and its colors are in standard order (primary order). The fourth order rainbow is just ouside of the third, its colors in reverse order.
Fifth and sixth order rainbows do center around the antisolar point. Because they are such high orders, they are very faint and wide. The increased width is due to the additional distance traveled within the drop (from multiple reflections), allowing the wavelengths to seperate further. The fifth order rainbow is between the second and first order rainbows, and the sixth order rainbow is right below the first order rainbow.
Zeroth Order Glow
There is the zeroth order, featuring no reflection, just refraction, however it is not a rainbow, rather just a glow of light that makes the sun appear like it is brighter than usual. This occurance along with the sun's own brightness make it difficult to see third and fourth order rainbows.
Why don't zero order rays create rainbows?
This diagram illustrates rays of light that pass through a raindrop (right) and rays that reflect once off the back of a drop before exiting (bottom left). Notice how the zeroth order rays flare out, but after the first reflection the first order rays hit a hard line where they go back up. This line is the minimum angle of deviation where most of the rays cluster around (it is at 42.5 deg for red light and 40.7 for violet).
This hard line only occurs with light reflected within the drop. This dense concentration of light (called a caustic) forms a circular shape for each color (actually a cone, but we only see it head on). Without a caustic for each wavelength, all the colors overlap and wash each other out, creating white again like they do in zeroth order glow.
The zero order colors overlap to simply form a glow of white light
The first order colored discs overlap to create a rainbow. There is faint colored light in each disc that overlaps to create a white glow inside the rainbow, but the light is strongest on the edge of each disc (as it is the caustic where the rays cluster). Since each color has a different minimum angle (red at 42.5 and violet at 40.7), the "smaller" colors show through the more transparent part of the "larger" colors. Note that smaller wavelengths overlap with the faint color of each subsquent larger wavelength, making the smaller wavlength colors paler than the larger ones. Ex. the red border will be the most pure, where as purple will be the most diluted, as it is covered lightly by every other color. Yellow will be diluted by red and orange, etc
Second order operates similarly to first order, but reversed, creating a faint glow of white light on the outside of the secondary rainbow (compared to the glow of light on the inside of the primary rainbow).
Because there is white glow outside the secondary and inside the primary, there apears to be a dark band between the primary and secondary rainbows, which is called Alexander's dark band.