318 BELL SYSTEM TECHNICAL JOURNAL 



Returning now to (93), it is evident that the problem is solved when 

 the integrals are evaluated; in particular the amplitude is given by 

 the formula, 



A = const. (1 + a) VC2 + 52. (94) 



Whatever the shape of the aperture or apertures, the values of the 

 integrals can be determined as closely as may be desired; and in two 

 instances which happily are the most frequent and useful — those of the 

 circular and the rectangular openings — the integrations lead directly 

 to familiar functions. 



Diffraction Patterns in the Focal Plane of a Lens 



If the origin is located at the centre of the circle, the integral 6" 

 vanishes — for the value of the sine-function contributed by each area- 

 element is annulled by the value contributed by the element symmetri- 

 cally placed to the other side of the centre — and the integral C for the 

 same reason becomes this: 



C = S S cos iyn^t]) cos (niy^)drjd^. (95) 



By putting 7 = and then integrating, we shall obtain the distribution 

 of amplitude along the line passing through the centre of the diffraction- 

 pattern and parallel to the axis of 3/; but this, by reason of the circular 

 symmetry of the entire system, is the same as the distribution of 

 amplitude along any radius passing through the centre of the diffrac- 

 tion-pattern, and therefore is all we need. In the expression so ob- 

 tained, replace the Cartesian coordinates heretofore used in the plane 

 of the screen by polar coordinates p and (p; then we have 



C = S S p cos {m^p cos ip)dpd(p, (96) 



the limits of integration being and R (the radius of the aperture) 



for p, and and lir for <p. 



The integral C is proportional to the Bessel function of order unity of 



the variable mR^ : 



C = 2(7ri?/m/3)/i(mi?/3). (97) 



This is a function which like the sine vanishes at intervals, though 

 not at equal intervals. The centre of the diffraction-pattern is there- 

 fore encircled by concentric rings over each of which the wave-motion 

 vanishes ; between each pair of these there is a zone where the ampli- 

 tude differs from zero and varies, attaining a maximum somewhere 

 near the middle of the zone. In the focal plane of the lens there are 

 ring-shaped zones of light, surrounded and divided by dark circles; 

 these are the "fringes." 



