Page:Encyclopædia Britannica, Ninth Edition, v. 24.djvu/470

Rh 444 W A V E THEORY Cormi s When the functions C and S have once been calculated, the method, discussion of various diffraction problems is much facilitated by the idea, due to Cornu, 1 of exhibiting as a curve the relationship between C and S, considered as the rectangular coordinates (x, y) of a point. Such a curve is shown in fig. 23, where, according to the definition (5) of C, S, /*&quot; i &quot; /&quot; i &quot; Jo -/0 The origin of coordinates corresponds to v= ; and the asymptotic points J, J, round which the curve revolves in an ever-closing spiral, correspond to v= K&amp;gt;. The intrinsic equation, expressing the relation between the arc a- (measured from 0) and the inclination cp of the tangent at any point to the axis of x, assumes a very simple form. For so that &amp;lt; Accordingly, and for the curvature, dd&amp;gt; (31). (32); Cornu remarks that this equation suffices to determine the general character of the curve. For the osculating circle at any point includes the whole of the curve which lies beyond ; and the successive con volutions envelop one another with out intersection. The utility of the curve depends upon the fact that the elements of arc represent, in ampli tude and phase, tho component vibra tions due to the corresponding por tions of the pri mary wave -front. and by (2) dv is proportional to ds. Moreover by (2) and (31) the retardation of phase of the elementary vibration from PQ (fig. 21) is 2?rS/A., or &amp;lt;p. Hence, in accordance with the rule for compounding vector quantities, the resultant vibration at B, due to any finite part of the primary wave, is represented in amplitude and phase by the chord joining the extremities of the corresponding arc (ff z - a-^. In applying the curve in special cases of diffraction to exhibit the effect at any point P (fig. 22) the centre of the curve is to be considered to correspond to that point C of the primary wave-front which lies nearest to P. The operative part, or parts, of the curve are of course those which represent the unobstructed portions of the primary wave. Let us reconsider, following Cornu, the diffraction of a screen unlimited on one side, and on the other terminated by a straight edge. On the illuminated side, at a distance from the shadow, the vibration is represented by JJ . The coordinates of J, J being (^ i) ) (-, -), I 2 is 2 ; and the phase is ^ period in arrear of that of the element at 0. As the point under contemplation is sup posed to approach the shadow, the vibration is represented by the chord drawn from J to a point on the other half of the curve, which travels inwards from J towards 0. The amplitude is thus subject ^fluctuations, which increase as the shadow is approached. At th*e point the intensity is one-quarter of that of the entire wave, and after this point is passed, that is, when we have entered the geometrical shadow, the intensity falls off gradually to zero, without fluctuations. The whole progress of the phenomenon is thus ex hibited to the eye in a very instructive manner. Diffrac- We will next suppose that the light is transmitted by a slit, and tion inquire what is the effect of varying the width of the slit upon the through illumination at the projection of its centre. Under these circum- a slit. stances the arc to be considered is bisected at 0, and its length is proportional to the width of the slit. It is easy to see that the length of the chord (which passes in all cases through 0) increases to a maximum near the place where the phase-retardation is f of a period, then diminishes to a minimum when the retardation is about g of a period, and so on. If the slit is of constant width and we require the illumination at various points on the screen behind it, we must regard the arc 1 Journal de Physique, iii. p. 1,1874. A similar suggestion has recently been made independently by Fitzgerald. of the curve as of constant length. The intensity is then, as always, represented by the square of the length of the chord. If the slit be narrow, so that the arc is short, the intensity is constant over a wide range, and does not fall off to an important extent until the discrepancy of the extreme phases reaches about a quarter of a period. AVe have hitherto supposed that the shadow of a diffracting Silvery obstacle is received upon a diffusing screen, or, which comes to lining nearly the same thing, is observed with an eye-piece. If the eye, of an provided if necessary with a perforated plate in order to reduce the obstacle, aperture, be situated inside the shadow at a place where the illumina tion is still sensible, and be focused upon the diffracting edge, the light which it receives will appear to come from the neighbourhood of the edge, and will present the effect of a silver lining. This is doubtless the explanation of a &quot; pretty optical phenomenon, seen in Switzerland, when the sun rises from behind distant trees stand ing on the summit of a mountain.&quot; 2 18. Diffraction Symmetrical about an Axis. The general problem of the diffraction pattern due to a source of light concentrated in a point, when the system is symmetrical about an axis, has been ably investigated by Lommel. 3 We must content ourselves here with a very slight sketch of some of his results. Spherical waves, centred upon the axis, of radius a fall upon the diffracting screen ; and the illumination is required on a second screen, like the first perpendicular to the axis, at a distance (a + b) from the source. We have first to express the distance (d) between an element dS of the wave-front and a point M in the plane of the second screen. Let denote the distance of M from the axis of symmetry ; then, if we take an axis of x to pass through M, the coordinates of M are ( 0, 0). On the same system the coordinates of dS are a sin 6 cos (p, sin 6 sin (p , (1 - cos 6) + b ; and the distance is given by d? = b&quot; + C 2 ~ 2aCsin cos + 4a(a + 1) sin ~%9. In this expression f and are to be treated as small quantities. Writing p for a sin 6, we get approximately The vibration at the wave-front of resolution being denoted by the integral expressive of the resultant of the secondary waves is -~US (2). T A Substituting pdp dtp for rfS, and for d its value from (1), we obtain as the expression for the intensity at the point , , .... (4), ... (5), where and the following abbreviations have been introduced -*, -~ C =Z (6). /v z.(c-c/ Ac&amp;gt; The range of integration is for &amp;lt;f&amp;gt; from to 2-rr. The limits for p depend upon the particular problem in hand; but for the sake of definiteness we will suppose that in the analytical definitions of C and S the limits are and r, so as to apply immediately to the problem of a circular aperture of radius r. If we introduce the notation of Bessel s functions, we have By integration by parts of these expressions Lommel develops Lommel s series suitable for calculation. Settin series. he finds in the first place o .u 1+ Tr. .... oo), &amp;lt;&quot;&amp;gt;- 2 Necker, Phil. Mag., Nov. 1832 ; Fox Talbot, Phil. Mag., June 1833. &quot;When the sun is about to emerge. . . . every branch and leaf is lighted up with a silvery lustre of indescribable beauty ..... The birds, as Mr Necker very truly de scribes, appear Uke flying brilliant sparks.&quot; Talbot ascribes the appearance to diffraction; and he recommends the use of a telescope. 3 Abh. der layer. Akad. der Wiss., ii. Cl., xv. Bd., ii. Abth. 4 Used now in an altered sense.