15.1: Introduction

Rainbow over the Ala Wai Canal (Photo by Shintaro Russell).

The color of our world is defined by the interaction of objects with light. In fact, the reason we can see anything at all is due to the interactions of objects with light. Light can be thought of as both a particle and a wave. In this chapter, we’ll consider light only as a wave defined by some wavelength on the electromagnetic spectrum. Recall the electromagnetic energy spectrum from Chapter 2 with the visible portion near the center between ultraviolet and infrared:

The electromagnetic energy spectrum showing the wavelength of colors (CC BY-NC-ND 2.0).

Notice that blue and purple colors have a relatively short wavelength (~400 nm) and orange and red have a relatively long wavelength (~700 nm). While slight, this difference makes a large difference in the interaction of light with particles in our atmosphere.

Before we begin, let’s define a few important facts that will help us to understand the following material.

1. White light contains all colors. Light from the sun is considered white light.
2. The color an object appears to be is due to reflection of that color toward your eye. For example, a green leaf looks green because it absorbs red and blue light, but reflects the wavelength of light that is green.

Based on this simple description, we can say that the sky appears to be blue because the molecules in the atmosphere reflect blue light. But why do they reflect blue light more than red light? This has to do with the relative sizes of the molecules and the wavelengths of light interacting with them.

In general, there are three different laws regarding wavelength and interaction with particles. We’ll define λ as the wavelength of light, and d as the diameter of the molecule or particle.

When the wavelength of light is much much larger than the diameter of the molecule or particle, Rayleigh scattering is dominant.

\[\text { Rayleigh Scattering } \lambda \gg d \nonumber \]

When the wavelength of light is within an order or two of magnitude as the diameter of the molecule or particle, Mie scattering is dominant.

\[\text { Mie Scattering } \lambda \sim d \nonumber \]

Finally, when the wavelength of light is much much smaller than the diameter of the molecule or particle, Geometric Optics is dominant.

\[\text { Geometric Optics } \lambda \ll d \nonumber \]

Rayleigh Scattering

Rayleigh scattering is the reason the sky is blue. Rayleigh scattering occurs when the wavelength of visible light is much much larger than the particles it is interacting with. Such is the case with the tiny molecules in Earth’s atmosphere. The amount of scattering, “\(S\)”, is proportional to:

\[S \sim \frac{1}{\lambda^4} \nonumber \]

Because this relationship is dependent on wavelength (λ), shorter wavelengths will be scattered more. Purple and blue light have a shorter wavelength than orange and red light and are therefore scattered more, accounting for the blue color of Earth’s atmosphere.

Mie Scattering

Mie scattering is the reason clouds are white. Mie scattering occurs when the wavelength of visible light is approximately the same as the particle or droplet diameter (within a factor of 10 or 100). When this is true, all wavelengths of visible light are scattered roughly equally. In this case, the color illuminating is the same color reflecting. In the case of clouds, they appear white because white light is coming from the sun and illuminating them.

Geometric Optics

Finally, geometric optics is responsible for rainbows. A beam of light is called a ray. The tracing of the ray path through a raindrop is called geometric optics and helps us to understand how and why rainbows form. When light comes in contact with a density interface, for example air and water, it can reflect or refract. Refraction is responsible for everyday conundrums like the following image.

Refraction of light at an air-water interface due to the difference in density between the two media (CC BY-NC 2.0).

Refraction occurs when rays bend at a density interface. Reflection occurs when rays bounce back from an object. These two phenomenon are responsible for why we can see atmospheric optical phenomena such as rainbows, halos, and sundogs. Here we will focus only on rainbows, the geometric optics occurring in liquid rain drops.

Diagram showing angle of incidence, reflection, and refraction of a ray of light (Public Domain).

As a light ray makes contact with an interface between two media with different densities like air and water, some of the incoming (incident) light will either refract from the air into the water, reflect back into the air, or absorb into heat. In the case of light interaction with a rain drop, we have a combination of refraction and reflection. A single rainbow has two refractions and one reflection, and the second rainbow seen in a double rainbow has two refractions and two reflections.

The top right image of the following figure shows the interaction of light with a rain drop responsible for a single rainbow. Light comes in contact with a rain drop, refracts as it enters the rain drop, reflects off the back side of the rain drop, and refracts again as it exits the rain drop. The two refractions result in a splitting of the colors of light, like a prism. This is because refraction of light is dependent on wavelength; shorter wavelengths refract more than longer wavelengths.

The formation of a rainbow with light propagation. Legend: 1.) Spherical droplet 2.) Places where internal reflection of the light occurs 3.) Primary rainbow 4.) Places where refraction of the light occurs 5.) Secondary rainbow 6.) Incoming beams of white light 7.) Path of light contributing to primary rainbow 8.) Path of light contributing to secondary rainbow 9.) Observer 10.) Region forming the primary rainbow 11.) Region forming the secondary rainbow 12.) Zone in the atmosphere holding countless tiny spherical droplets (CC BY-SA 3.0).

Put concisely, a rainbow is an atmospheric phenomenon displaying a spectrum of light as a result of refraction of light in water droplets. This is possible because the wavelength of light is much much smaller than the size of the rain drop.

Primary rainbows have red on the outside of the circle and purple on the inside. There is always a 42° angle between the incident light, the primary rainbow, and the observer. This constant angle can be seen in the above figure. At a given time, rainbows appear different to different observers because of different interaction between incoming light rays and water droplets. For instance, one observer may see a well-defined rainbow in an area of large rain drops and bright sun. Another observer in a half kilometer away with smaller rain drops may see a partial rainbow.

Secondary rainbows are outer rainbows with red on the inside of the circle and purple on the outside because of the double reflection. Secondary rainbows occur at a 52° angle, hence secondary rainbows are higher than primary rainbows. They’re not as bright as primary rainbows, also because of the double reflection. Triple rainbows are also possible and have been observed, but are much less frequent.

Double rainbow in a stormy sky. Notice how the colors are flipped in the two rainbows, and how the secondary rainbow is not as bright as the primary rainbow (Photo by Shintaro Russell).

How to see Rainbows

The best chance of seeing a rainbow is when there is rain and sun low in the sky at the same time. An observer may see a rainbow if the sun is at the observer’s back and they are facing towards a region that is raining. As the sun lowers in the sky, the height of the rainbow will increase because of the constant 42° angle. Rainbows cannot be seen on the ground at noon because a 42° angle is not possible. Rainbows appear as half arcs only because the ground typically blocks the other half. Rainbows are frequent in Hawai’i because of the abundant sunlight and frequent rain.