 
Summary: Geometric optics and rainbows: generalization
of a result by Huygens
John A. Adam
Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia 23529, USA
(jadam@odu.edu)
Received 21 February 2008; revised 11 April 2008; accepted 9 May 2008;
posted 9 May 2008 (Doc. ID 93030); published 4 June 2008
In 1652 Huygensderived a formula specifyingthe rainbowangle for theprimary bow ðk ¼ 1Þ in termsof the
refractive index only. A generalization of this result for any k 1 is outlined, along with an alternative
representation. The details of the derivation can be found in (Adam, Mathematics Magazine, 2008, under
review), but the results as stated may be of interest to the atmospheric optics community. © 2008 Optical
Society of America
OCIS codes: 010.1290, 080.2720, 000.3860, 000.3870.
It is well known that for k internal reflections and two
refractions in a spherical raindrop, the angle through
which an incoming ray is deviated (as a function of the
angle of incidence i) is defined by
DkðiÞ ¼ k þ 2i  2ðk þ 1Þ arcsin
sin i
