430 WAVE THEORY Aualyti- cal ex- pressiou. Uncer- tainty in the ap- plicatiou of Huy- geus s principle, The amplitude of the light at any point in the axis, when plane waves are incident perpendicularly upon an annular aperture, is, as above, cos K(at - r-)) - cos n(at - r.,) = 2 sin KCit . sin K(I - r.,} , r,,, r l being the distances of the outer and inner boundaries from the point in question. It is scarcely necessary to remark that in all such cases the calculation applies in the first instance to homo geneous light, and that, in accordance with Fourier s theorem, each homogeneous component of a mixture may be treated separ ately. When the original light is white, the presence of some components and the absence of others will usually give rise to coloured eii ects, variable with the precise circumstances of the case. Although what we have to say upon the subject is better post- poned until we consider the dynamical theory, it is proper to point out at once that there is an element of assumption in the appli- cation of Huygens s principle to the calculation of the effects pro duced by opaque screens of limited extent. Properly applied, the principle could not fail ; but, as may readily be proved in the case , of sonorous waves, it is not in strictness sufficient to assume the expression for a secondary wave suitable when the primary wave is undisturbed, with mere limitation of the integration to the transparent parts of the screen. But, except perhaps in the case of very fine gratings, it is probable that the error thus caused is insignificant ; for the incorrect estimation of the secondary waves will be limited to distances of a few wave-lengths only from the boundary of opaque and transparent parts. 11. Fraunliofcr s Diffraction Phenomena. A very general problem in diffraction is the investigation of the distribution of light over a screen upon which impinge divergent or convergent spherical waves after passage through various diffracting apertures. When the waves are convergent and the recipient screen is placed so as to contain the centre of convergency the image of the original radiant point, the calcu lation assumes a less complicated form. This class of phenomena was investigated by Fraunhofer (upon principles laid down by Fresnel), and are sometimes called after his name. z We may conveniently commence with them on f account of their simplicity and great importance - 7^ in respect to the theory of optical instruments. o I If / be the radius of the spherical wave at the ./ ^ place of resolution, where the vibration is repre- r< " ^x seuted by cos nat, then at any point M (fig. 4) in *~>g- 4 - the recipient screen the vibration due to an element dS of the wave-front is ( 9) dS , . N -- sin K(at - p) , Ap p being the distance between M and the element dS. Taking coordinates in the plane of the screen with the centre of the wave as origin, let us represent M by {, n, and P (where dS is situated) by x, y, z. Then so that In the applications with which we are concerned, |, ?j are very small quantities ; and we may take -/{ -* At the same time dS may be identified with dxdy, and in the de nominator p may be treated as constant and equal to /. Thus the expression for the vibration at M becomes E.xpres- and for the intensity, represented by the square of the amplitude, sion fol ia tensity. ^ = A This expression for the intensity becomes rigorously applicable when f is indefinitely great, so that ordinary optical aberration disappears. The incident waves are thus plane, and are limited to a plane aperture coincident with a wave-front. The integrals are then properly functions of the direction in which the light is to be estimated. In experiment under ordinary circumstances it makes no differ ence whether the collecting lens is in front of or behind the diffract ing aperture. It is usually most convenient to employ a telescope focused upon the radiant point, and to place the diffracting aper tures immediately in front of the object-glass. What is seen through the eye-piece in any case is the same as would be depicted upon a screen in the focal _plane. Theorems Before proceeding to special cases it maybe well to call attention of Bridge, to some general properties of the solution expressed by (2). 1 Phil. Nov. 1858. If, when the aperture is given, the wave-length (proportional to K" 1 ) varies, the composition of the integrals is unaltered, provided | and 77 are taken universely proportional to A. A diminution of A thus leads to a simple proportional shrinkage of the diffraction pattern, attended by ail augmentation of brilliancy in proportion to A- 2 . If the wave-length remains unchanged, similar effects are pro duced by an increase in the scale of the aperture. The linear dimension of the diffraction pattern is inversely as that of the aperture, and the brightness at corresponding points is as the square of the area of aperture. If the aperture and wave-length increase in the same proportion, the size and shape of the diffraction pattern undergo no change. We will now apply the integrals (2) to the case of a rectangular Rect- aperture of width a parallel to x and of width b parallel to y. angular The limits of integration for x may thus be taken to be - a and aperture + a, and for y to be - %b, + ^b. We readily find (with substitution for K of 27T/A) , sin 2 -~ sin - ~^-~~ , a-b* /A /A ,, ,- as representing the distribution of light in the image of a mathematical point when the aperture is rectangular, as is often the case in spectroscopes. The second and third factors of (3) being each of the form sin-uju^, we have to examine the character of this function. It vanishes when u = mir, m being any whole number other than zero. When u = 0, it takes the value unity. The maxima occur when = tan u , ........ (4), and then sin 2 /zt 2 = cos 2 w ....... (5). To calculate the roots of (5) we may assume -m + 7r-2/=-7/, where y is a positive quantity which is small when Substituting this, we find cot y= U - y, whence 7 u is lar Y5315 This equation is to be solved by successive approximation. It will readily be found that In the first quadrant there is no root after zero, since tan u > u, and in the second quadrant there is none because the signs of u and tan u are opposite. The first root after zero is thus in the third quadrant, corresponding to m = l. Even in this case the series converges sufficiently to give the value of the root with considerable accuracy, while for higher values ofm it is all that could be desired. The actual values of u/v (calculated in another manner by Schwerd) are 1 4303, 2 4590, 3-4709, 4 4747, 5 4818, 6 4844, &c. Since the maxima occur when u = (m + ^)ir nearly, the successive values are not very different from 4 4 4_ Q 2 9K J > 40 2 The application of these results to (3) shows that the field is Diffrac- brightest at the centre | = 0,rj = 0, viz., at the geometrical image, tioii It is traversed by dark lines whose equations are pattern. Within the rectangle formed by pairs of consecutive dark lines, and not far from its centre, the brightness rises to a maximum ; but these subsequent maxima are in all cases much inferior to the brightness at the centre of the entire pattern (1 = 0,77 = 0). By the principle of energy the illumination over the entire focal Total plane must be equal to that over the diffracting area ; and thus, in illumin- accordance with the suppositions by which (3) was obtained, its atioii. value when integrated from |=-ooto|=+oo, and from rj= - oo to 77 =+oo should be equal to ab. This integration, employed originally by Kelland 2 to determine the absolute intensity of a secondary wave, may be at once effected by means of the known formula It will be observed that, while the total intensity is proportional to ab, the intensity at the focal point is proportional to a*b 2 . If the aperture be increased, not only is the total brightness over the focal plane increased with it, but there is also a concentration of the diffraction pattern. The form of (3) shows immediately that, if a and b be altered, the coordinates of any characteristic point in the pattern vary as a 1 and b 1 . Diffusio The contraction of the diffraction pattern with increase of of imagt aperture is of fundamental importance with reference to the due to resolving power of optical instruments. According to common Unite optics, where images are absolute, the diffraction pattern is sup- wave-
2 Ed. Trans., xv. 315. length.