phy1020.DVI

57.6 Higher-Order Rainbows

Both the primary and secondary rainbows are easy to observe in Nature, but what about high-order bows,
corresponding to three or more reflections of light rays inside each raindrop?
Confirmed observations of third- and fourth-order bows in Nature have only very recently been made for
the first time, in 2011.^2 (See Figures 57.6 and 57.7.) There is at this time also evidence for observation of a
fifth-order rainbow in 2015.^3
It can be shown (Ref. [16]) that the rainbow angle of thek-th order rainbow (corresponding tokinternal
reflections in each drop) is given by

kDk.180ı/C2ik2.kC1/rk (57.1)

where thek-th angle of incidenceikis given by

ikDcos^1

s

n^2 w 1

k.kC2/

(57.2)

and thek-th angle of refractionrkis found from Snell’s law:

rkDsin^1



1

nw

sinik



(57.3)

Herenwis the index of refraction of water. Sincenwvaries depending on the color of light, these equations
can be used to find the rainbow angle for both red and violet light, and from that deduce the width of each
bow. The results of these calculations through the 20-th order rainbow are shown in Table 57-1, and illustrated
in Figure 57.5. Notice that as the rainbow orderkincreases, the rainbows get both fainter and wider.

(^2) SeeApplied Optics, 50 , 28, pp. F129-F141 (2011).
(^3) See Edens, H.E. (2015) Photographic observation of a natural fifth-order rainbow,Appl. Opt. 54 , B26-34.