Page:Encyclopædia Britannica, Ninth Edition, v. 24.djvu/452

426 AYE THEORY or dvfd, is not constant, the achromatism of the white band is less perfect than when no prism is used. If a grating were substituted for the prism, v would vary as h, and (4) would vanish, so that in all orders of spectra the white band would be seen undisplaced. The theoretical error, dependent upon the dispersive power, involved in the method of determining the refractive index of a plate by means of the displacement of a system of interference fringes (LIGHT, vol. xiv. p. 607) has been discussed by Stokes. 1 In the absence of dispersion the retardation R due to the plate would be independent of A, and therefore completely compensated at the point determined by?t = DR/6 ; but when there is dispersion it is accompanied by a fictitious displacement of the fringes on the principle explained by Airy. More recently the matter has engaged the attention of Corim, 2 who thus formulates the general principle : " Dans un systeme de /ranges d interferences produites A I aide dme lumiere heterogene ayant un spectre continu, il cxiste toujours line /range achromatique quijoue le role dc /range ccntrale et qui se trouve au point de champ oil les radiations les plus intenses prescntcnt une difference de phase maximum ou minimum. " In Fresnel s experiment, if the retardation of phase due to an interposed plate, or to any other cause, be F(A.), the whole relative retardation of the two pencils at the point u is and the situation of the central, or achromatic, band is determined, not by < = 0, but by d<p/d = 0, or = A 2 DF()/& ....... (6). Limits In the theoretical statement we have supposed the source of light to the to be limited to a mathematical point, or to be extended only in width the vertical direction (parallel to the bands). Such a vertical of the extension, while it increases illumination, has no prejudicial effect source of upon distinctness, the various systems due to different points of the light. luminous line being sensibly superposed. On the other hand, the horizontal dimension of the source must be confined within narrow limits, the condition obviously being that the displacement of the centre of the system incurred by using in succession the two edges only of the slit should be small in comparison with the width of an interference band. Diffrac- Before quitting this subject it is proper to remark that Fresnel s tion and bands are more influenced by diffraction than their discoverer Fres- supposed. On this account the fringes are often unequally broad nel s and undergo fluctuations of brightness. A more precise calcula- bands. tion has been given by H. F. Weber 3 and by H. Struve, 4 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. "When plane waves of homogeneous light (A) fall upon a parallel plate of index /j., the resultant reflected wave is made up of an infinite number of components, of which the most important are the first, reflected at the upper surface of the plate, and the second, transmitted at the upper surface, reflected at the under surface, and then transmitted at the upper surface. It is readily proved ( LIGHT, vol. xiv. p. 608) that so far as it depends upon the distances to be travelled in the plate and in air the retardation (8) of the second wave relatively to the first is given by 8 = 2 / ucos a ....... (1), where t denotes the thickness of the plate, and a the angle of refraction corresponding to the first entrance. If we represent all the vibrations by complex quantities, from which finally the imaginary parts are to be rejected, the retardation 5 may be expressed by the introduction of the factor e ~ ilcS > where i= /( - 1), and K = 2ir/A. Summa- At each reflexion or refraction the amplitude of the incident tion of wave must be supposed to be altered by a certain factor. When partial the light proceeds from the surrounding medium to the plate, the waves. factor for reflexion 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 c, f. Denoting the incident vibration by unity, we have then for the first component of the reflected wave I, for the second cc/t~ lKS , for the third cc 3 ff~ 2ilfS , and so on. Adding these together, and summing the geometric series, we find In like manner for the wave transmitted through the plate we get ,-f (3). 1 -c 2 The quantities b, c, e, f are not independent. The simplest way to find the relations between them is to trace the consequences 1 Brit. Ass. Rep.. 1850. 2 Wied. Ann., viii. p. 407. 2 Jour, de, Physique, . p. 293, 1882. 4 Wied. Ann., xv. p. 49. of supposing 8 = in (2) and (3). For it is evident a priori that with a plate of vanishing thickness there must be a vanishing re flexion, and a total transmission. Accordingly, the first of which embodies Arago s law of the equality of reflexions, as well as the famous " loss of half an undulation." Using these we find for the reflected vibration, and for the transmitted vibration I -c" -1 o IKO ~/" 1-C 2 6 The intensities of the reflected and transmitted lights are the Inten- squares of the moduli of these expressions. Thus sities. T , ., < n , i v i , (1 - cos /tS) 2 + sin 2 *5 Intensity of reflected light = c- (1 - e 3 cos K$)- + e 4 in 2 S 4e 2 sin 2 (i/c5) 1 - 2e- cos *5 + c 1 Intensity of transmitted light = . ; l-Se-cos/cS + c 4 (8), the sum of the two expressions being unity. According to (7) not only does the reflected light vanish com- Zero re- pletely when 8 = 0, but also whenever %k$ = mr, n being an integer, ilexionat that is, whenever 8 = 7i. When the first and third medium arc certain the same, as we have here supposed, the central spot in the system thick- of Newton s rings is black, even though the original light contain nesses. a mixture of all wave lengths. The general explanation of the colours of Newton s rings is given under LIGHT, to which reference must be made. If the light reflected from a plate of any thickness be examined with a spectroscope of sufficient resolving power ( 13), 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. The relations between the factors b, c, c, f have been proved, Prin- independcntly of the theory of thin plates, in a general manner ciple of by Stokes, 5 who called to his aid the general mechanical principle reversi- of reversibility. If the motions constituting the reflected and bility. 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 reproduc tion 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) being 1, the re flected (AR) and refracted (AF) rays are denoted by b and c. When b is reversed, it gives rise to a reflected / ray b 2 along AI, and a refracted ray be along AG (say). Fig. l. When c is reversed, it gives rise to (/along AI, and ce along AG. Hence bc + cc = Q, b~ + cf=l, which agree with (4). It is here assumed that there is no change of phase in the act of reflexion or refraction, except such as can be represented by a change of sign. Professor Stokes has, however, pushed the applica tion of his method to the case where changes of phase are admitted, and arrives at the conclusion that "the sum of the accelerations of phase at the two reflexions is equal to the sum of the accelera tions at the two refractions, and the accelerations of the two re fractions are equal to each other." The accelerations are supposed to be so measured as to give like signs to c and/, and unlike to b and e. The same relations as before obtain between, the factors b} c > e >f> expressing the ratios of amplitudes. 6 When the third medium differs from the first, the theory of thin Thin plates is more complicated, and need not here be discussed. One plate in particular case, however, may be mentioned. When a thin trans- contact 5 "On the Perfect Blackness of the Central Spot in Newton s Rings, and on the Verification of Fresnel s Formula; for the Intensities of Reflected and Kefructed perfect Kays," Cainb. and Dub. Math. Jour., vol. iv. p. 1, 1849 ; reprint vol. ii. p. 89. reflector. t> It would appear, however, that these laws cimuot be properly applied to the calculation of reflexion from a thin plate. This is sufficiently proved by the fact that the resultant expression for the intensity founded upnn them does not vanish with the thickness. The truth is that the method of deducing the aggregate reflexion from the consideration of the successive partial reflexions and refrac tions is applicable only when the disturbance in the interior of the plate is fully represented by the transverse waves considered in the argument, whereas the occurrence of a change of phase is probably connected with the existence of additional superficial waves ( 27). The existence of these superficial waves may be ignored when the reflected and refracted waves are to be considered only at distances from the surface exceeding a few wave-lengths, but in the application to thin plates this limitation is violated. If indeed the method of calculating the aggregate reflexion from a thin plate were sound when a change of phase occurs, we could still use the expressions (2) and (3), merely under standing by 6, c, e, f, factors which may be complex ; and the same formal rela tions (4) would still hold good. These "do not agree with those found by Stokes by the method of reversion ; and the discrepancy indicates that, when there are

changes of phase, the action of a thin plate cannot be calculated in the usual way.