of the interference of light to this phenomenon. His not wholly satisfactory explanation was mathematically examined in 1835 by Richard Potter (Camb. Phil. Trans., 1838, 6, 141), who, while improving the theory, left a more complete solution to be made in 1838 by Sir George Biddell Airy (Camb. Phil. Trans., 1838, 6, 379).
The geometrical theory first requires a consideration of the path of a ray of light falling upon a transparent sphere. Of the total amount of light falling on such a sphere, part is reflected or scattered at the incident surface, so rendering the drop visible, while a part will enter the drop. Confming our Geometrical theory.attention to a ray entering in a principal plane, we will determine its deviation, i.e. the angle between its directions of incidence and emergence, after one, two, three or more internal reflections.
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| Fig. 1. |
Let EA be a ray incident at an angle i (fig. 1); let AD be the refracted ray, and r the angle of refraction. Then the deviation experienced by the ray at A is i-r. If the ray suffers one internal reflection at D, then it is readily seen that, if DB be the path of the reflected ray, the angle ADB equals 2r, i.e. the deviation of the ray at D is π−2r. At B, where the ray leaves the drop, the deviation is the same as at A, viz. i-r. The total deviation of the ray is consequently given by D=2(i-r)+π−2r.
Similarly it may be shown that each internal reflection introduces a supplementary deviation of π−2r; hence, if the ray be reflected n times, the total deviation will be D=2(i-r)+n(π−2r).
The deviation is thus seen to vary with the angle of incidence; and by considering a set of parallel rays passing through the same principal plane of the sphere and incident at al angles, it can be readily shown that more rays will pass in the neighbourhood of the position of minimum deviation than in any other position (see Refraction). The drop will consequently be more intensely illuminated when viewed along these directions of minimum deviation, and since it is these rays with which we are primarily concerned, we shall proceed to the determination of these directions.
Since the angles of incidence and refraction are connected by the relation sin i=μ sin r (Snell’s Law), μ being the index of refraction of the medium, then the problem may be stated as follows: to determine the value of the angle i which makes D=2(i-r)+n(π−2r) a maximum or minimum, in which i and r are connected by the relation sin i=μ sin r, μ being a constant. By applying the method of the differential calculus, we obtain cos i=√(μ2−1)/(n2+2n) as the required value; it may be readily shown either geometrically or analytically that this isia minimum. For the angle i to be real, cos i must be a fraction, that is n2+2n>μ2−1, or (n+1)2>μ2. Since the value of μ for water is about 43, it follows that n must be at least unity for a rainbow to be formed; there is obviously no theoretical limit to the value of n, and hence rainbows of higher orders are possible.
So far we have only considered rays of homogeneous light, and it remains to investigate how lights of varying refrangibilities will be transmitted. It can be shown, by the methods of the differential calculus or geometrically, that the deviation increases with the refractive index, the angle of incidence remaining constant. Taking the refractive index of water for the red rays as 10881, and for the violet rays as 10981, we can calculate the following values for the minimum deviations corresponding to certain assigned values of n.
| n | Red. | Violet. |
| 1 2 3 4 |
π – 42°.1 2π – 129°.2 3π – 231°.4 4π – 317°.07 |
π – 40°.22 2π – 125°.48 3π – 227°.08 4π – 310°.07 |
To this point we have only considered rays passing through a principal section of the drop; in nature, however, the rays impinge at every point of the surface facing the sun. It may be readily deduced that the directions of minimum deviation for a pencil of parallel rays lie on the surface of cones, the semi-vertical angles of which are equal to the values given in the above table. Thus, rays suffering one internal reflection will all lie within a cone of about 42°; in this direction the illumination will be most intense; within the cone the illumination will be fainter, while, without it, no light will be transmitted to the eye.
Fig. 2.
Fig. 2 represents sections of the drop and the cones containing the minimum deviation rays after 1, 2, 3 and 4 reflections; the order of the colours is shown by the letters R (red) and V (violet). It is apparent, therefore, that all drops transmitting intense light after one internal reflection to the eye will lie on the surfaces of cones having the eye for their common vertex, the line joining the eye to the sun for their axis, and their semi-vertical angles equal to about 41° for the violet rays and 43° for the red rays. The observer will, therefore, see a coloured band, about 2° in width, and coloured violet inside and red outside. Within the band, the illumination will be faint; outside the band there will be perceptible darkening until the second bow comes into view. Similarly, drops transmitting rays after two internal reflections will be situated on covertical and coaxial cones, of which the semi-vertical angles are 51° for the red rays and 54° for the violet. Outside the cone of 54° there will be faint illumination; within it, no secondary rays will be transmitted to the eye. We thus see that the order of colours in the secondary bow is the reverse of that in the primary; the secondary is half as broad again (3°), and is much fainter, owing to the longer path of the ray in the drop, and the increased dispersion.
Similarly, the third, fourth and higher orders of bows may be investigated. The third and fourth bows are situated between the observer and the sun, and hence, to be viewed, the observer must face the sun. But the illumination of the bow is so weakened by the repeated reflections, and the light of the sun is generally so bright, that these bows are rarely, if ever, observed except in artificial rainbows. The same remarks apply to the fifth bow, which differs from the third and fourth in being situated in the same part of the sky as the primary and secondary bows, being just above the secondary.
The most conspicuous colour band of the principal bows is the red; the other colours shading off into one another, generally with considerable blurring. This is due to the superposition of a great number of spectra, for the sun has an appreciable apparent diameter, and each point on its surface gives rise to an individual spectrum. This overlap ing may become so pronounced as to produce a rainbow in which colour is practically absent; this is particularly so when a thin cloud intervenes between the sun and the rain, which has the effect of increasing the apparent diameter of the sun to as much as 2° or 3°. This phenomenon is known as the “white rainbow” or “Ulloa’s Ring or Circle,” after Antonio de Ulloa.
We have now to consider the so-called spurious bows which are sometimes seen at the inner edge of the primary and at the outer edge of the secondary bow. The geometrical theory can afford no explanation of these coloured bands, and it has been shown that the complete phenomenon of the rainbow Physical theory. is to be sought for in the conceptions of the wave theory of light. This was first suggested by Thomas Young, who showed that the rays producing the bows consisted of two systems, which, although emerging in parallel directions, traversed different paths in the drop. Destructive interference between these superposed rays will therefore occur, and, instead of a continuous maximum illumination in the direction of minimum deviation, we should expect to find alternations of brightness and darkness. The later investigations of Richard Potter and especially of Sir George Biddell Airy have proved the correctness of Young’s idea. The mathematical discussion of Airy showed that the primary rainbow is not situated directly on the line of minimum deviation, but at a slightly greater value; this means that the true angular radius of the bow is a little less than that derived from the geometrical theory. In the same way, he showed that the secondary bow has a greater radius than that previously assigned to it. The spurious bows he showed to consist of a series of dark and bright bands, whose distances from the principal bows vary with the diameters of the raindrops. The smaller the drops, the greater the distance; hence it is that the spurious bows are generally only observed near the summits of the bows, where the drops are smaller than at any lower altitude. In Airy’s investigation, and in the extensions by Boitel, J. Larmor, E. Mascart and L. Lorentz, the source of light was regarded as a point. In nature, however, this is not realized, for the sun has an appreciable diameter. Calculations taking this into account have been made by J. Pernter (Neues ilber den Regenbogen, Vienna, 1888) and by K. Aichi and T. Tanakadate (Jour. College of Science, Tokyo, 1906, vol. xxi. art. 3).
Experimental confirmation of Airy’s theoretical results was afforded in 1842 by William Hallows Miller (Camb. Phil. Trans. vii. 277). A horizontal pencil of sunlight was admitted by a vertical slit, and then allowed to fall on a column of water supplied by a jet of about Blyth of an inch in diameter. Primary, secondary and spurious bows were formed, and their radii measured; a comparison of these observations exhibited agreement with Airy’s analytical values. Pulfrich (Wied. Ann., 1888, 33, 194) obtained similar results by using cylindrical glass rods in place of the column of water.
In accordance with a general consequence of reflection and refraction, it is readily seen that the light of the rainbow is partially polarized, a fact first observed in 1811 by Jean Baptiste Biot (see Polarization).
