Why does a rainbow contain a pure spread of spectral colors?

Category: Physics
Published: January 30, 2014

A rainbow does not contain a pure spread of spectral colors, although it is somewhat close. A spectral color is a color that contains only one wavelength component of its electromagnetic wave. In contrast, a non-spectral color contains many wavelengths and is therefore a mixture of spectral colors. Simple lasers produce effectively pure spectral colors. The visible "electromagnetic spectrum" is a continuous spread of all of the spectral colors, arranged according to wavelength (i.e. red, orange, yellow, green, blue, and violet). Furthermore, the complete electromagnetic spectrum is a continuous spectrum and contains an infinite number of spectral colors. Just because we do not have a common name for the spectral color between red and orange does not mean that it is not a spectral color. When we display the spectrum of a certain light beam (or the spectrum of the light from a certain object), we are really just showing the spectral colors contained in that light, as well as their intensities and locations on the wavelength scale. Natural white light, such as from the sun, contains all spectral colors and therefore displays a continuous spread when separated into a pure spectrum of spectral colors (ignoring the narrow absorption lines). For instance, when white light enters and exits a glass prism, the different spectral color components of the light bend different amounts due to the dispersive nature of the glass. The different colors exit the prism at different angles, leading to a pure spectrum that becomes visible when it reflects off a wall or screen (strictly speaking, a prism only creates a pure spectrum if the original beam of light is very thin).

These mathematical plots show the angles at which different light rays exit a raindrop depending on the light's color and where it entered the raindrop. The left plot shows how red light gets redirected and the right plot shows how violet light gets redirected. Although most of the red light comes out where the arrows are thickest at 42.1°, which corresponds to the bright red outer edge of the rainbow, we see that some red light comes out at all angles between 0° and 42.1°. Similarly, although most of the violet light comes out where the arrows are thickest at 40.6°, we see that some violet light comes out at all angles between 0° and 40.6°. Therefore, the colors in a rainbow are slightly mixed and do not form a pure spectrum. Although the image on the left and right looking nearly identical, if you look close, you see that the angles are slightly different. Public Domain Image, source: Christopher S. Baird.

Unlike the spread of colors created by a prism, the spread of colors created by a spherical raindrop is not a pure spectrum. (By the way, raindrops are round and not tear shaped.) While the brightest part of a rainbow (the colorful outer edge) is close to a pure spectrum, each point in the spread contains a mixture of spectral colors. The more you look inwards from the outer edge of a rainbow, towards the arc's center, the more spectral colors there are mixed together, until finally the entire interior region of a rainbow is faint white, indicating a complete mix of all colors. The reason that a point in a rainbow contains a mix of spectral colors is ultimately because the front surface of a raindrop is round. This means that different parts of the original light beam encounter the raindrop's curved surface at different angles and bend different amounts, even for a single color. The diagram above shows how each color gets bent into many angles. Although pure red is mostly bent by a raindrop into a 42.1° viewing angle to form the outer edge of a rainbow, some of the red is bent into all angles between 0° and 42.1° because of the curved surface of the raindrop. Similarly, pure orange is mostly bent into the 41.9° viewing angle, but some orange is bent into all lower angles as well. The color in a rainbow at 42.1° is therefore red, the color at 41.9° is orange plus a little bit of red, the color at 41.7° is yellow plus a little bit of orange and red, etc. The end result is that the colors in a rainbow tend to blur together and wash each other out. The extended shape of the sun also sends light into the raindrop at slightly different angles and further blurs the colors together.

A prism and a raindrop are in principle very similar. They both spread white light out into a span of colors through refraction. The main difference though is that a prism has flat surfaces, leading to a pure spectrum, while a raindrop has a round surface, leading to an impure spectrum. Unfortunately, in everyday language, the phrases "rainbow" and "visible spectrum" are used to mean the same thing, even though scientifically, they are not exactly the same.

The top image shows a high-quality photograph of a rainbow and the bottom image shows a mathematically pure spectrum (insofar as a computer screen is able to render colors). As this comparison makes obvious, a rainbow is not a pure spectrum. The colors of a rainbow are more blended together and washed out. This blurring in the rainbow is due to the inherent structure of the rainbow itself and is not due to the limitations of the camera. Also note that the last bright color of the rainbow is purple (purple = red + violet) while the last visible color of the pure spectrum is violet. Public Domain Images, source for top image: U.S. National Park Service, source for bottom image: Wikipedia.

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Topics: color, color spectrum, colors, light, optics, rainbow, rainbows, spectrum