10/12/2014 -- What are double rainbows?

posted Nov 16, 2014, 6:16 PM by Patrick Poole   [ updated Nov 16, 2014, 7:13 PM ]
A common sight after storms, rainbows are an excellent everyday example of the behavior of light. Some people are familiar with the idea that there is no end to a rainbow, but have you thought about why? And what goes into making that elusive double rainbow everyone longs to see? Below is a quick introduction to the physics of rainbows and some tips on how to optimize your chances for seeing beauty across the sky.

Note: below you’ll see me use the “color” and “wavelength” of light interchangeably—this is because light has some properties like a wave, and it turns out that the color of light we see is described by the wavelength of that light wave. This isn’t an obvious idea, so I’ll return to it in a later post.

Storms and sprinklers can leave a fine mist of water droplets hanging in the air

Water droplets can hang in the air after a storm, or you can make your own using a lawn sprinkler. The droplets naturally form into spheres due to surface tension (the details of why are another post), and this spherical shape determines a lot about how light turns into a rainbow when it hits the droplet.

Light entering these droplets will refract (split into its constituent colors)

The sun emits a lot of visible light (along with a lot of other wavelengths of light which we’ll ignore for the moment), and all this light enters the tiny droplet at all different points along its surface (like shining a spotlight on a ping pong ball). Light that enters the droplet will bend its trajectory—this is called refraction, and happens when light moves from one material into another.

Red light rays traveling through a water droplet

The bent light then travels through and then hits the inside of the back surface of the droplet. Some of this light will go through this surface (bending again because of refraction), but some of it reflects off and starts traveling back toward the front of the droplet—this is reflection, just like from a mirror.

This reflected light will reach the front side of the droplet, where some of it will reflect again (we’ll come back to this twice reflected light later), but most of it will go through to the outside of the droplet, bending again from refraction as it does.

Water droplets are small, so this light that traveled through the droplet emerges not too far away from where it started, but now it’s going in a different direction. The three processes that happened inside the droplet when light reached a surface—refraction upon entering, reflection from the inside back, and refraction again from exiting—depend on the angle that the light had as it approached that surface, and on the wavelength of the light.

For a moment let’s focus on one color, say red. The refraction that occurs when light enters the droplet depends on the angle that light hits that droplet surface, so light that hits the droplet at different points are seeing different angles of that surface—in optics we would say the light has a different angle of incidence depending on where it hit the sphere. Imagine throwing a ping-pong ball at a tortoise: the direction the ball leaves as it bounces off the curved shell depends on its angle of incidence.

Because this first refraction, and the reflection and exit refraction later, all depend on the angle of incidence the light sees, light that enters the droplet at different points will exit the droplet at different angles. You can see some of the angles possible for red in Figure 1 above: light that hits right in the middle of the droplet will come straight back out as though it hit a mirror, and light that hits the upper part of the sphere will exit on the lower part, but at different angles depending on where it initially entered.

Let’s look at what angles the light can leave with. At most the light leaves goes exactly backwards compared to how it entered—in terms of angles, this is 180° different from how it came in. It turns out that the most different angle light can leave a droplet at is 138° for red light—this has to do with the nature of refraction and what surface angles the incident light can see on the droplet. So if red light is hitting our droplet everywhere on its surface, we’ll get some red light that comes back towards us at every angle between 138° and 180°, but nothing below 138° (this will be important in a bit).

Blue refraction
Now let’s go back to the other colors—everything will happen just as before, with the refraction upon entering, reflection from the back inside, and refraction upon exiting. The difference is that refraction is depends on wavelength, so red and blue light that enter at the same spot on the droplet will leave at slightly different angles. It turns out that red gets refracted the most, so the range of red light leaving the droplet is the largest: 138°-180°. Blue light gets reflected the least, and its range is only about 150°-180°. You can see some blue ray angles in Figure 2, and the full spectrum produced from one ray of white light in Figure 3.
Rainbow refraction
How much light reflects changes as the angle of incidence changes

Here comes the crucial point that makes rainbows: the reflection at the back inside isn’t perfect like it would be from a normal mirror. Only a small fraction of the light that hits the back inside of the water droplet will be reflected—a lot of it goes right through. It turns out that the reflection from the rear side also depends on the angle of incidence, such that more light is reflected if the angle is smaller. This is called Fresnel reflection, and always happens when light goes between two different materials. So for example there is a lot more red light that will eventually leave the droplet at 138° than there will be at 180°. For each color, the smallest angle it can leave the droplet with will have the most light--see Figure 4.

Angle refraction

So, when you happen to be standing with the sun behind you, so that rays can come over your head and go through this refraction/reflection/refraction process within a water droplet, and if some droplets are floating such that their 138° light will go towards you, you’ll see more red light from those droplets. Some droplets just a bit below those will be at the proper angle to send you their orange light, and some just below those will send you their yellow light, and so on until you see the full rainbow.

Single bounce makes the first rainbow, and a double bounce makes a (weaker) second rainbow

This discussion above, and the pictures there are all for light that enters the top half of the water droplet. Light that enters the bottom half will throw its rainbow light upwards, and you would see it if you were in a hot air balloon, say. However, it is possible for light entering the bottom of a droplet to undergo reflection twice before it leaves—in this case the light still comes out towards the ground, but it’s a lot weaker because of the two internal reflections. In this case most of the red light will leave at an angle of 232.5°, so this second rainbow will be at a different position that the primary one. It’s also going to be a lot weaker because of the light that was lost at the second internal reflection. If you look closely at the image at the top of this page you’ll notice that the colors are flipped on the second rainbow—that’s because of the second internal reflection.

In principle you can have any number of internal reflections and get any number of rainbows from water droplets, but so much light is lost that they’re usually impossible to see—some scientists can observe the third rainbow, but only with specialized optical equipment.

Rainbows occur at precise internal bounce angles, and there will be more light below the rainbow and above due to the rest of the light

Double reflection all the way!

Because no visible wavelengths will deflect at angles below 138°, the light above a rainbow is always darker than the light below (check this out in the picture at the top of this page). Above the rainbow you’re only seeing light that’s coming towards you from the other side, but below the rainbow you’re seeing light coming toward you from the background and from behind you because it’s being reflected by the water droplets. That's why it's always darker below a rainbow than above!

Rainbow tips—light from in front reflects back at you

For best rainbow-viewing results, make sure the sun is behind you, and if possible move so that whatever is behind the rainbow is as dark as possible (retreating thunderclouds make a good backdrop)—that way there isn’t a lot of light to wash out the rainbow light reflecting back towards you. If you're making your own water droplets with a sprinkler, try changing the direction you look at the spray from and see how it affects the rainbow you see.

Bonus physics—why isn’t there an end of the rainbow?

The reason there is no end to a rainbow is because rainbows form in arcs (or a full circle if you’re lucky enough to view one from high above the ground) Think about this one based on what we discussed above about the importance of the angle light leaves the water droplets. Hint: it has to do with the light source, (probably the sun) and where you are standing, and which water droplets in the air will make the proper angle between the sun, the droplet, and your eyes as it reflects a certain color down at you.

Thanks to Serge Melkl for the double rainbow picture and Lance Robotson for the sprinkler rainbow picture.