series of waves constituting any particular coloured light is reflected
from an infinitely thin plate, the two partial reflections are in
absolute discordance and, if of equal intensity, must give on superposition
complete darkness. With the aid of this principle the
sequence of colours in Newton’s rings is explained in much the
same way as that of interference fringes (above, § 5).
 |
| Fig. 2. |
The complete theory of the colours of thin plates requires us to take account not merely of the two reflections already mentioned but of an infinite series of such reflections. This was first effected by S. D. Poisson for the case of retardations which are exact multiples of the half wave-length, and afterwards more generally by Sir G. B. Airy (Camb. Phil. Trans., 1832, 4, p. 409).
In fig. 2, ABF is the ray, perpendicular to the wave-front, reflected at the upper surface, ABCDE the ray transmitted at B, reflected at C and transmitted at D; and these are accompanied by other rays reflected internally 3, 5, &c., times. The first step is to calculate the retardation δ between the first and second waves, so far as it depends on the distances travelled in the plate (of index μ) and in air.
If the angle ABF = 2α, angle BCD = 2α′ and the thickness of plate = t, we have
| δ = μ (BC + CD) − BG = 2μBC − 2BC sin α sin α′ = 2μBC (1 − sin2 α′) |
(1). |
In (1) α′ is the angle of refraction, and we see that, contrary to what might at first have been expected, the retardation is least when the obliquity is greatest, and reaches a maximum when the obliquity is zero or the incidence normal. If we represent all the vibrations by complex quantities, from which finally the imaginary parts are rejected, the retardation δ may be expressed by the introduction of the factor ε−iκδ, where i = √(−1), and κ = 2π/λ.
At each reflection or refraction the amplitude of the incident wave must be supposed to be altered by a certain factor which allows room for the reversal postulated by Young. When the light proceeds from the surrounding medium to the plate, the factor for reflection will be supposed to be b, and for refraction c; the corresponding quantities when the progress is from the plate to the surrounding medium will be denoted by e, f. Denoting the incident vibration by unity, we have then for the first component of the reflected wave b, for the second cefε−iκδ, for the third ce3fε−2iκδ, and so on. Adding these together, and summing the geometric series, we find
| b +cefε−iκδ1 − e2ε−iκδ | (2). |
In like manner for the wave transmitted through the plate we get
| cf1 − e2 ε−iκδ | (3). |
The quantities b, c, e, f are not independent. The simplest way to find the relations between them is to trace the consequences of supposing δ = 0 in (2) and (3). This may be regarded as a development from Young’s point of view. A plate of vanishing thickness is ultimately no obstacle at all. In the nature of things a surface cannot reflect. Hence with a plate of vanishing thickness there must be a vanishing reflection and a total transmission, and accordingly
| b + e = 0, cf = l − e2 | (4), |
the first of which embodies Arago’s law of the equality of reflections, as well as the famous “loss of half an undulation.” Using these we find for the reflected vibration,
| −e(1−ε−iκδ1 − e2ε−iκδ | (5), |
and for the transmitted vibration
| 1 − e21 − e2 ε−iκδ | (6). |
The intensities of the reflected and transmitted lights are the squares of the moduli of these expressions. Thus
| = 4e2 sin2 (12κδ1−2e2 cos κδ+e4 | (7); |
| Intensity of transmitted light = (1−e2)21−2e2 cos κδ+e4 | (8), |
the sum of the two expressions being unity.
According to (7) not only does the reflected light vanish completely when δ = o, but also whenever 12κδ = nπ, n being an integer, that is, whenever δ = nλ. When the first and third mediums are the same, as we have here supposed, the central spot in the system of Newton’s ring is black, even though the original light contain a mixture of all wave-lengths. If the light reflected from a plate of any thickness be examined with a spectroscope of sufficient resolving power, the spectrum will be traversed by dark bands, of which the centre corresponds to those wave-lengths which the plate is incompetent to reflect. It is obvious that there is no limit to the fineness of the bands which may be thus impressed upon a spectrum, whatever may be the character of the original mixed light.
 |
| Fig. 3. |
The relations between the factors b, c, e, f have been proved, independently of the theory of thin plates, in a general manner by Stokes, who called to his aid the general mechanical principle of reversibility. If the motions constituting the reflected and refracted rays to which an incident ray gives rise be supposed to be reversed, they will reconstitute a reversed incident ray. This gives one relation; and another is obtained from the consideration that there is no ray in the second medium, such as would be generated by the operation alone of either the reversed reflected or refracted rays. Space does not allow of the reproduction of the argument at length, but a few words may perhaps give the reader an idea of how the conclusions are arrived at. The incident ray (IA) (fig. 3) being 1, the reflected (AR) and refracted (AF) rays are denoted by b and c. When b is reversed, it gives rise to a reflected ray b2 along AI, and a refracted ray bc along AG (say). When c is reversed, it gives rise to cf along AI, and ce along AG. Hence bc+ce = 0, b2+cf = 1, which agree with (4). It is here assumed that there is no change of phase in the act of reflection or refraction, except such as can be represented by a change of sign.
When the third medium differs from the first, the theory of thin plates is more complicated, and need not here be discussed. One particular case, however, may be mentioned. When a thin transparent film is backed by a perfect reflector, no colours should be visible, all the light being ultimately reflected, whatever the wave-length may be. The experiment may be tried with a thin layer of gelatin on a polished silver plate. In other cases where a different result is observed, the inference is that either the metal does not reflect perfectly, or else that the material of which the film is composed is not sufficiently transparent. Some apparent exceptions to the above rule, exhibited by thin films of collodion resting upon silver surfaces, have been described by R. W. Wood (Physical Optics, p. 143), who attributes the very curious effects observed to frilling of the collodion film.
For study of the colours of thin plates there are no more interesting subjects than the soap-film. For projection the films may be stretched across vertical rings of iron wire coated with paraffin. In their undisturbed condition they thin from the top, and the colours are disposed in horizontal bands. If, as suggested by Brewster, a jet of wind issuing from a small nozzle and supplied from a well-regulated bellows be allowed to impinge obliquely, parts of the film are set in rotation, and displays of colours may be exhibited to a large audience, astonishing by their brilliance and by the rapidity with which they change. Permanent films, analogous to soap-films, are best obtained by Glew’s method. A few drops of celluloid varnish are poured upon the surface of water contained in a large dish. After evaporation of the solvent, the films may be picked up upon rings of iron wire.
As a variant upon Newton’s rings, interesting effects may be obtained by the partial etching of the surfaces of picked pieces of plate-glass. A surface is coated in parallel stripes with paraffin wax and treated with dilute hydrofluoric acid for such a time (found by preliminary trials) as is required to eat away the exposed portions to a depth of one quarter of the mean wave-length of light. Two such prepared surfaces pressed in the crossed position into suitable contact exhibit a chess-board pattern. Where two uncorroded, or where two corroded, parts overlap, the colours are nearly the same; but where a corroded and an uncorroded surface meet, a strongly contrasted colour is developed. The combination lends itself to projection and the pattern seen upon the screen is very beautiful if proper precautions are taken to eliminate the white light reflected from the first and fourth surfaces of the plates (see Nature, 1901, 64, 385).
Theory and observation alike show that the transmitted colours of a thin plate, e.g. a soap film or a layer of air, are very inferior to those reflected. Specimens of ancient glass, which have undergone superficial decomposition, on the other hand, sometimes show transmitted colours of remarkable brilliancy. The probable explanation, suggested by Brewster, is that we have here to deal not merely with one, but with a series of thin plates of not very different thicknesses. It is evident that with such a series the transmitted colours would be much purer, and the reflected much brighter, than usual. If the thicknesses are strictly equal, certain wave-lengths must still be absolutely missing in the reflected light; while on the other hand a constancy of the interval between the plates will in general lead to a special preponderance of light of some other wave-length for which all the component parts as they ultimately emerge are in agreement as to phase.
On the same principle are doubtless to be explained the colours of fiery opals, and, more remarkable still, the iridescence of certain
