GENERAL METEOROLOGICAL OPTICS 71 
the light after refraction and reflection by the droplets 
was computed, notably by Pernter and Mébius [c. 42]. 
Bucerius [6] developed an asymptotic method (in anal- 
ogy to Debye’s treatment of cylindric functions) by 
means of which Mie’s theory of scattering of electromag- 
netic waves by dielectric spheres [c. 29] can be extended 
to large waterdrops. Applying this general theory to the 
rainbow phenomenon, he showed that it contains the 
older rainbow theories as approximations, and that the 
rainbow is an areal phenomenon that actually covers 
the entire region between the antisolar point and the 
first rmg. Meyer [36] considers the classical diffraction 
theory still adequate, particularly for the intense rain- 
bows that origmate from comparatively large droplets. 
He developed this theory further to permit the deter- 
mination of the luminous density of rainbows. He takes 
into consideration the total optical effect of all droplets 
contained in a surface element of the cloud deck, the 
thickness of the cloud, and the attenuation of the rays 
to and from the cloud. He finds the luminous density 
of the primary rainbow to be twelve times that of the 
secondary bow. 
The theoretical advancement of our knowledge of 
rainbows appears to have surpassed our fund of observa- 
tional data. Aside from older visual measurements, there 
are, to my knowledge, no results of up-to-date colori- 
metric photometry of rainbows available for application 
to the theoretical findings. In this connection an almost 
forgotten problem may be recalled, the fluctuations of 
colors during lightning and thunder [42], an observation 
requiring objective verification and explanation. 
Corona and Related Phenomena. The sun shining 
through relatively thin clouds often produces one or 
more sets of colored rings, the corona, having diameters 
of a few degrees. When poorly developed, only an aureole, 
a bluish-white disk with brownish rim, may be visible. 
After violent volcanic eruptions, a broad reddish-brown 
-ring of large radius (20° and more), Bishop’s ring, has 
been observed in dust clouds [11]. We speak of iridescent 
clouds, when the colors are not arranged concentrically 
around the sun, but are irregularly distributed over, or 
follow the contours of, the cloud. This group of coronal 
phenomena around the sun is paralleled around the 
antisolar point by a similar group: An observer, seeing 
his slightly enlarged shadow, the Brocken-specter, on 
a fog bank or cloud, often finds the shadow of his head 
surrounded by one or more sets of colored rings, the 
anticorona or glory, well-known to pilots. If the shadow 
falls on a bedewed surface on the ground at some dis- 
tance from the observer, the shadow of his head may be 
rimmed by a narrow white sheen, the heiligenschein, 
which also can be observed around one’s head-shadow 
on a beaded projection screen. 
The classical diffraction theory applied to the corona, 
under the assumption that the droplets are opaque, has 
been found to be in fair agreement with observations 
[42]. The well-known approximation formula by K. 
Exner [c. 42] for the angular radius @ of the circular 
intensity minima produced by particles of the diameter 
d in light of wave length X, is 
sin 6 = (N + a)d/d, (6) 
where JN is the order of the minimum counting from the 
center, a = 0.22 for spherical, a = 0 for nonspherical 
particles. Ramachandran [45] based his new corona 
theory on wave optics and included the wave-front 
portion that is transmitted through the droplets. In 
Fig. 10 the results of his calculation of intensities 
(1 = 0.5 ») at various diffraction angles (@) for small 
droplets (radii in » ascribed to the curves) are repro- 
duced. These curves indicate that the ring systems 
oscillate as the small droplets increase in size. Only 
relatively large droplets diffract like opaque disks of 
the same size, which explains some of the discrepancies 
formerly noted between the classical theory and 
observations. The position of the ring systems appears 
40 
INTENSITY 
40 
2.25 
20 
(0) 
o° 10° 202 30° 40° 
— > §@ 
Fra. 10.—Intensity of diffracted light as function of angular 
distance from light source. (After Ramachandran. Ordinate 
scale presumably in relative units; numbers on curves are 
drop radii in microns.) 
unaffected by the thickness and density of the clouds. 
Bucerius’ work [6], mentioned above, also includes the 
application of the rigorous diffraction theory by Mie 
[c. 29] to both corona and anticorona. The anticorona 
was similarly treated by van de Hulst [20]. This theory 
yields the intensity and polarization of the diffracted 
light and the position of intensity maxima and minima. 
Table VIII gives a comparison of the values for the 
argument of the Bessel function at which corona 
maxima occur according to the old and the newer 
theories. It is noteworthy that in Ramachandran’s 
theory the location and intensity of the maximum for 
small drops also depends on the value of (sin £)/£, where 
£ is a function of d/\. For this reason the maxima 
oscillate as shown in Fig. 10. According to Bucerius 
(6, equation (47)], the argument contains twice the sine 
of half the angle between primary and diffracted rays, 
