The Angle of Deflection:

The applet below allows you to "experiment" with the interaction of a light ray and a rain drop. The white light ray enters the raindrop on the left. The red and blue portions of this ray are traced through the raindrop. You can move the light ray by clicking and/or dragging your mouse on the left side of the applet.

The table below the rain drop shows the angles of incidence and refraction for each color, as well as the angle of deflection for the ray. (The angle of deflection is the angle between the original direction of the light ray and its direction after striking the rain drop.)

As you move the light ray, observe the effect of the incident angle on the dispersion of the red and blue rays as well as the effect of the angle of incidence on the angle of deflection. (You can find another virtual experiment similar to this one at Circles of Light - The Mathematics of Rainbows.)

The angle through which the rain drop turns the light ray is called the angle of deflection for the ray. As the diagram below shows, the angle of deflection depends on the angle at which the light ray strikes the drop.

Deflection angle animation

The Rainbow Angle

If you graph angle of deflection vs. angle of incidence (from data collected in a virtual (or real!) experiment, or graph the function derived in the last section, you can clearly see that there is a minimum angle of deflection of approximately 138o (depending on color). This may ring a bell, since 138o is the supplement of 42o, the "rainbow angle" (sometimes called the "Descartes angle"). Why does this minimum value correspond to the angle at which you view a rainbow?

Graph animationSo, rays that strike the rain drop at an angle of incidence near the angle producing the minimum angle of deflection will tend to form a concentrated, strong beam in which the colors will be widely separated. Rays that strike the rain drop at a small angle of incidence will tend to pass through the drop, and the part of the rays that are reflected inside the drop are spread out relative to one another, while the colors within the rays are not noticeably separated.


What Next?

Well, surely this explains everything there is to know about rainbows! Not so fast, bud. The rainbow is an extremely rich phenomenon, and we have just scratched the surface. There are several couple of features of the rainbow that can be explained by simple ray optics - here are some hints...

Have fun!


last update February 10, 1999 by JL Stanbrough