CHAPTER II. INTERFERENCE. 17. WE stated in the preceding Chapter, that we shall assume that the sensation of light is produced by the vibrations of the ether, without enquiring for the present into the physical constitution of the latter. We simply suppose that a medium exists which is capable of propagating waves, and that when the waves are plane, the direction of vibration is parallel to the wave front. We shall now proceed to examine how far this hypothesis is capable of explaining the interference of light. We have shown in § 12, that the superposition of two waves of light whose directions, wave-lengths, velocities of propagation and planes of polarization are the same, but whose amplitudes and phases are different, may either intensify or diminish the resultant light. Let us now suppose, that natural light is proceeding from two sources very close to one another. At a point whose distance from the two sources is large in comparison with the distance between them, the waves may be regarded as approximately plane and parallel to one another, and the displacements may be resolved into two components at right angles to one another in the front of the wave. Hence if v, v' be any two components whose directions are parallel, we may take 2π 2π v = A cos (x-Vt), v'A' cos (x- Vt-e), λ the origin being suitably chosen ; accordingly the resultant of these two vibrations is B. O. 2 If the amplitudes of the two waves are equal, then (2) and (3) become accordingly the intensity of the resultant light will be zero when e = (n+1), and a maximum when enλ. It therefore follows that when the phases of the two waves differ by an odd multiple of half a wave-length, the superposition of the two waves produces darkness, and the waves are said to interfere. In order to produce interference, it is essential that the two sources of light should arise from a common origin, otherwise it would be impossible to insure that the amplitudes of the two waves should be equal, or that the difference of phase should remain invariable; accordingly if the light from two different candles were made to pass through two pin-holes in a card, which are very close together, interference would not take place1; but if light from a single candle were passed through a pin-hole, and the resulting light were then passed through the two pin-holes, interference would take place if the two pin-holes were sufficiently close together. The phenomenon of interference was regarded as a crucial test of the truth of the Undulatory Theory, before that theory was so firmly established as it is at present; inasmuch as it is impossible to satisfactorily explain on the Corpuscular Theory, how two lights can produce darkness. 18. We shall now explain several methods, due to Fresnel, by means of which interference fringes can be produced. a, Let AB, AC be two mirrors inclined at an angle - α, where a is very small; let O be a luminous point, and let S, H be the images of O formed by the mirrors AB, AC; also let P be any point on a screen PN, which is parallel to the line of intersection of the mirrors, and to SH. 1 The effect of diffraction is not considered. 2 Euvres Complètes, vol. 1. pp. 159, 186, 268. 3 In practice the source usually consists of light admitted through a narrow slit. FRESNEL'S MIRRORS. 19 Since AO AS = AH, the difference of phase of the two = E B N streams of light which come from S and H is SP-HP. Let A0=a, NP = x, d the distance of the screen from SH. Then and since the distance of the screen is large compared with a and a sin a, we obtain SP - HP = 2αx sin a. /d = SH. x/d. Since the two images are produced by the same source, their amplitudes will be equal, whence the intensity is equal to Since a is a very small angle, we may write a for sin a, and from (5) we see that the intensity is a maximum when x= = {nλd/aa, and a minimum when x = (2n+1)λd/aa, where n is an integer. If homogeneous light is employed, a bright band will be observed at the centre N of the screen, and on either side of this bright band and at a distance λd/aa, there will be two dark bands; accordingly the screen will be covered by a series of bright and dark bands succeeding one another in regular order; the distance between two bright bands or two dark bands being equal to λd/aa. From this result we see the necessity of a being very small, for otherwise on account of the smallness of A, the bands would be so close together as to be incapable of being observed. If perfectly homogeneous light could be obtained, the number of bands would be theoretically unlimited, and with light from a sodium flame, which possesses a high degree of homogeneity, Michelson has observed as many as 200000 bands, but in practice it is not possible to obtain absolutely homogeneous light, consequently the number of fringes is necessarily limited. Since the distance between two bands is equal to λd/aa, it follows that the breadths of the bands depend upon the wavelength, and therefore upon the colour; hence if sunlight be employed, a certain number of brilliantly coloured bands will be observed. At the centre of the system, where x = 0, the difference of phase of waves of all lengths is zero, and the central band is therefore white, but its edges are red. The inner edge of the next bright band will be violet, and its edge of a reddish colour; but as we proceed from the centre, the maximum intensity of one colour will coincide with the minimum of another, and the dark bands will altogether disappear, and will be replaced by coloured bands. At a still further distance, the colours will become mixed to such an extent, that no bands will be distinguishable. 19. In Fresnel's second experiment', the light was refracted by means of a prism having a very obtuse angle, which is called a biprism. Let O be the luminous point, and let L be the focus of the rays refracted at the first face; let S and S' be the foci of the rays refracted at the second faces, and draw LG, LH perpendicular to 1 Euvres Complètes, vol. 1. p. 330. these faces. Let OA = a, and let u be the index of refraction. Then LA μa, LG = μSG. Hence if N be the point where SS' cuts AO, SN = LS sin a = LG (1 — μ ̄1) sin a = (T+ μa)(1 − μ−1) sin a cos a, where T is the thickness of the prism at A. The light may now be conceived to diverge from the two points S and S', and therefore if d be the distance of the screen from SS, the retardation will be equal to 1 ud (T+μa) (n − 1) x sin 2a, which is of the same form as in the last experiment. The breadths of the bands are accordingly equal to which involves μ as well as λ. Now the index of refraction depends not only upon the material of which the prism is made, but also on the colour of the light; hence the breadths of the bands are affected by the dispersive power of the prism, but are otherwise the same as in the last experiment. 20. A third mode of producing interference fringes is by means of bi-plates. Let CQ, Cq be two thin plates of thickness T inclined at a very obtuse angle - 2a. If O be a luminous origin situated on the axis CA of the bi-plates, a small pencil of light after passing through the plates CQ, Cq will appear to diverge from two foci S, H, such that SH is perpendicular to OC. |