In a classical experiment interference-bands were employed to examine whether light moved faster or slower in glass than in air. For this purpose a very thin piece of glass may be interposed in the path of one of the interfering rays, and the resulting displacement of the bands is such as to indicate that the light passing through the glass is retarded. In a better form of the experiment two pieces of parallel glass cut from the same plate are interposed between the prism and the screen, so that the rays from O1 (fig. 1) pass through one part and those from O2 through the other. So long as these pieces are parallel, no shifting takes place, but if one be slightly turned, the bands are at once displaced. In the absence of dispersion the retardation R due to the plate would be independent of λ, and therefore completely compensated at the point determined by u = DR/b; but when there is dispersion it is accompanied by a fictitious displacement of the fringes on the principle explained by Airy, as was shown by Stokes.
Before quitting this subject it is proper to remark that Fresnel’s bands are more influenced by diffraction than their discoverer supposed. On this account the fringes are often unequally broad and undergo fluctuations of brightness. A more precise calculation has been given by H. F. Weber and by H. Struve, but the matter is too complicated to be further considered here. The observations of Struve appear to agree well with the corrected theory.
§ 8. Colours of Thin Plates.—These colours, familiarly known as those of the soap-bubble, are seen under a variety of conditions and were studied with some success by Robert Hooke under the name of “fantastical colours” (Micrographia, 1664). The inquiry was resumed by Sir Isaac Newton with his accustomed power (“Discourse on Light and Colours,” 1675, Opticks, book ii.), and by him most of the laws regulating these phenomena were discovered. Newton experimented especially with thin plates of air enclosed by slightly curved glasses, and the coloured rings so exhibited are usually called after him “Newton’s rings.”
The colours are manifested in the greatest purity when the reflecting surfaces are limited to those which bound the thin film. This is the case of the soap-bubble. When, as is in other respects more convenient, two glass plates enclosing a film of air are substituted, the light under examination is liable to be contaminated by that reflected from the outer surfaces. A remedy may be found in the use of wedge-shaped glasses so applied that the outer surfaces, though parallel to one another, are inclined to the inner operating surfaces. By suitable optical arrangements the two portions of light, desired and undesired, may then be separated.
In his first essay upon this subject Thomas Young was able to trace the formation of these colours as due to the interference of light reflected from the two surfaces of the plate; or, as it would be preferable to say, to the superposition of the two reflected vibrations giving resultants of variable magnitude according to the phase-relation. A difficulty here presents itself which might have proved insurmountable to a less acute inquirer. The luminous vibration reflected at the second surface travels a distance increased by twice the thickness of the plate, and it might naturally be supposed that the relative retardation would be measured by this quantity. If this were so, the two vibrations reflected from the surfaces of an infinitely thin plate would be in accordance, and the intensity of the resultant a maximum. The facts were notoriously the reverse. At the place of contact of Newton’s glasses, or at the thinnest part of a soap-film just before it bursts, the colour is black and not white as the explanation seems to require. Young saw that the reconciliation lies in the circumstance that the two reflections occur under different conditions, one, for example, as the light passes from air to water, and the second as it passes from water to air. According to mechanical principles the second reflection involves a change of sign, equivalent to a gain or loss of half an undulation. When a 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 (l − sin2 α′) = 2μt cos α′ |