432 WAVE T H E R Y Now and thus dtdn = Limit of Cacos II a 2 + 7i 2 Let ( " " -, = aw, J K The limits for u are ultimately - co and + co , and we have 2adx 2f /+*> du 2f T = .M t / I +u- In like manner the integration for y may be performed; and we find We saw that If, (the intensity at the focal point) was equal to A 2 /A 2 / 2 . If A be the area over which the intensity must be Ij; in order to give the actual total intensity in accordance with the relation between A and A is AA = A 2 / 2 . Since A is in some sense the area of the diffraction pattern, it may be considered to be a rough criterion of the definition, and we infer that the definition of a point depends principally upon the area of the aperture, and only in a very secondary degree upon the shape when the area is maintained constant. 12. Theory of Circular Aperture. We will now consider the important case where the form of the aperture is circular. Writing for brevity Kt/f=P, K1/f=q ........ (1), we have for the general expression (11) of the intensity where S = ffsin (px + qy) dx dy , ..... (: ), C"j[/"cos(px+qy )dxdy t ..... (4). When, as in the application to rectangular or circular apertures, the form is symmetrical with respect to the axes both of x and y, S = 0, and C reduces to C = ffcospxcosqydxdy, ..... (5). In the case of the circular aperture the distribution of light is of course symmetrical with respect to the focal point p = 0, q = ; and C is a function of p and q only through J(p- + <f). It is thus sufficient to determine the intensity along the axis of p. Putting (7 = 0, we get C = Jfcospx dx dy-*2-f*cospx /(R 2 - x") dx , R being the radius of the aperture. This integral is the BessiTs function of order unity, defined bv z {~T Ji(2) = - / cos (z cos 0) sin -<p d<f> .... (C). "yo Thus, if = Rcos0, pii and the illumination at distance r from the focal point is 24 l . . (8). 7 The ascending series for J](z), used by Airy 2 in his original inves tigation of the diffraction of a circular object-glass, and readily obtained from (6), is v ~3 ~5 J
- + .
When 2 is great, we may employ the semi-convergent series JiW-y(s)rfn(-{l+- 8 - 3.5.7.9.1.3.5/1 (9). 8. 16 8.16.21. 32 cos^- 1 IT) { - .I- 3 5 7>1 3//1 ( 8 z 3. 5. 7. 9. 11. 1.3. 5. 7 8 . 16 . 24 . 32. 40 . 16 . 24
- )
1 It is easy to show that this conclusion is not disturbed by the introduction at every point of an arbitrary retardation p, a function of x, y. The terms (p p) are then to be added under the cosine in (9) ; but they are ultimately without effect, since the only elements which contribute are those for which in the limit x = x, y = y, and therefore p =p. 2 "On the Diffraction of an Object-Glass with Circular Aperture." Camb. Trans., 1834. A table of the values of 2^- 1 J,(:) has been given by Lomtnel, 3 to whom is due the first systematic application of Vessel s functions to the diffraction integrals. The illumination vanishes in correspondence with the roots of the equation J 1 (s) = 0. If these be called z v z.,, z 3 , . . . the radii of the dark rings in the diffraction pattern are An A a 2irR 27rR being thus inversely proportional to R. The integrations may also be effected by means of polar co ordinates, taking first the integration with respect to $ so as to obtain the result for an infinitely thin annular aperture. Thus, if x = p cos (f> , y = p sin <j) , C = Jfcospx dx di/=f R f 2 " cos (pp cos 6) pdp dO . Now by definition 2 J o(~) (11). The value of C for an annular aperture of radius r and width dr is thus For the complete circle, c = 27r fP K J(z]zdz 2 2- 2J 1 (2R) as before. In these expressions we are to replace p by /, or rather, since the diffraction pattern is symmetrical, by kr/f, where r is the dis tance of any point in the focal plane from the centre of the system. The roots of J^s) after the first may be found from z . , -050661 -053041 -262051 ^ ^ D ~r i. - / i -i " and those of i-^z) from 151982 015399 245835 4& + 1 formulae derived by Stokes 4 from the descendin following table gives the actual values: i -forJ (e) = -forJ 1 (z) = i -forJ (z) = -forJ 1 (e) = 7T 77 1 7655 1-2197 6 57522 6-2439 2 17571 2-2330 7 67519 7-2448 O 27546 3-2383 3 7-753P 8-2454 4 37534 4-2411 9 8-7514 9-2459 5 47527 5-2428 10 9-7513 10-2463 In both cases the image of a mathematical point is thus a sym metrical ring system. The greatest brightness is at the centre, where dC =-- 2irp dp , C = irR 2 . For a certain distance outwards this remains sensibly unimpaired, and then gradually diminishes to zero, as the secondary waves become discrepant in phase. The subsequent revivals of bright ness forming the bright rings are necessarily of inferior brilliancy as compared with the central disk. The first dark ring in the diffraction pattern of the complete circular aperture occurs when ?//= 1-2197 XA/2R (15). We may compare this- with the corresponding result for a rect angular aperture of width a, I// = ^/ a and it appears that in consequence of the preponderance of the central parts, the compensation in the case of the circle does not set in at so small an obliquity as when the circle is replaced by a rectangular aperture, whose side is equal to the diameter of the circle. Again, if we compare the complete circle with a narrow annular aperture of the same radius, we see that in the latter case the first dark ring occurs at a much smaller obliquity, viz. , r/f= 7655XA/2R. It has been found by Hcrschel and others that the definition of Central a telescope is often improved by stopping off a part of the central stop, area of the object-glass ; but the advantage to be obtained in this way is in no case great, and anything like a reduction of the aper ture to a narrow annulus is attended by a development of the 3 Schlomilch, xv. p. 160, 1870. Camb. Trans., vol. ix., 1850. 5 The descending series for J (z) appears to have been first given by Sir W.
Hamilton in a memoir on " Fluctuating Functions." Roy. Irish Trans., 1840.