Somewhere Under the Rainbow

I would like to thank E. Blaise Saccocio for the beautiful picture of the double rainbow in the November 2003 issue of Physics Today (page 10). However, he is incorrect when he states that the dark zone (known as Alexander's dark belt) between the rainbows is due to interference.

The dark belt can be understood from geometrical optics. The primary (lower) rainbow represents an extreme value of the angle at which light rays are scattered after being internally reflected once by a raindrop. If a viewer on the ground looks at the top of the primary rainbow, and then lifts her head a little higher, she sees a dark sky because no light rays emerge from water droplets at angles steeper than the rainbow angle. The sky is not completely dark in the belt because of scattering due to more than one internal reflection and to light from the sky and the landscape.¹

One rainbow phenomenon that must be explained by interference is the presence of supernumerary bows beneath the primary rainbow. A partial explanation of that phenomenon was given in 1838 by George Airy.² If you look carefully, you can see one supernumerary bow (the narrow white band) underneath the primary one in Saccocio's photo. As with many topics related to rainbows, the supernumerary bow is still actively researched.³

The text accompanying E. Blaise Saccocio's double rainbow picture should have pointed out the existence of supernumerary rainbows clearly visible inside the primary rainbow. Although the primary and secondary rainbows are explainable in terms of geometric optics, the supernumerary rainbows are not, because they are a manifestation of light interference within a raindrop. In fact, it was an observation of supernumerary rainbows that prompted Thomas Young to perform the famous double-slit experiment in 1801, which confirmed the wave nature of light and led to his explanation of these rainbows in 1803. For more information and pictures, see references 1 and 2.

1. 1. Raymond L. Lee Jr, Alistair B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science, Pennsylvania State U. Press, University Park, PA (2001); see especially chap. 8, available at http://www.usna.edu/Users/oceano/raylee/RainbowBridge/Chapter_8_1.shtml.
2. 2. M. Sawicki, P. Sawicki, Phys. Teach. 38, 19 (2000). Available at http://www.jal.cc.il.us/~mikolajsawicki/rainbows.htm.

Saccocio replies: For 20-odd years, having read Jearl Walker's paper and a number of its references,¹ I have attributed the rainbow's dark zone to optical interference. A closer reading more clearly reveals that the mechanism is refractive, just as Chuck Adler and Mikolaj Sawicki point out. Both their understandings are supported by Walker;² my earlier reading likely did not focus on that part of his discussion. Walker's paper is highly detailed and describes what is and is not observable both in nature and in laboratory rainbow-simulation conditions. My thanks to Adler and Sawicki.

1. 1. J. D. Walker, Am. J. Phys. 44, 421 (1976) [INSPEC].
2. 2. Reference 1, p. 424.