Dr Searle, Experiments on focal lines formed by a zone plate 331 



On X'OX take P, Q at distances u, v from 0, and let u and v be 

 positive when P and Q are in the object and image spaces re- 

 spectively. Then, when pju, pjv are small, 



PGn + QG,, = (w^ + p.„2)^ + {V^' + p,2)* 



= u + v + ^ {pi2 + (n - 1) A;2} (l/w + 1/v}. 

 Hence, for all values of the positive integer n greater than 1, 



PG,, + QG, = PG,,_, + QG„^, + p2 (1/^ + i/„). 

 Thus, the paths PG^Q, PG^Q, ... increase by equal steps of h, where 



Let a train of spherical waves with centre P and wave length 

 X fall upon the zone plate from the object space, and let D be one 

 of the wave fronts. Let PG^, PG^, ... meet D in D^, D^, .... 

 Waves will travel out from the slits into the image space. Let 

 j& be a sphere about Q as centre, and let G^Q, G^Q, ... meet E in 

 E^, E2, .... Then the disturbances Q,t E^, E2, ... will have the same 

 phase, if the distances D^GyE^, DM.yE^, ... increase by steps of 

 ^A, where p is a positive or negative integer. When the distance 

 of E from is some thousands of wave lengths, the separate 

 wavelets due to the ring slits will merge into a single wave in- 

 distinguishable from E. We may thus speak of £' as the emergent 

 wave front. The wave D will thus give rise to the wave E, if 

 PGn + QGn exceeds PG^ + QG^ by {n — 1) pX. An image of P will 

 then be formed at Q. 



Hence, Q will be an image of P, if 11 = p\. If /p be the corre- 

 sponding focal length, and F^ the corresponding "power," we have 



1/m + 1/1; = J^ = 1//^ = 2^A/P (1) 



Thus, the power is proportional to jt) and is positive or negative 

 with j9. We here follow the custom of practical opticians, who treat 

 the power of a thin converging lens as positive. 



The zone plate thus acts as a lens with a number of positive 

 and negative focal lengths, and, for a given position of a real or 

 virtual luminous point P on X'OX, there will be a number of 

 images, some real and some virtual. 



If k"^ is found by measuring the rings with a travelling 

 microscope, A can be found, when/j, and p are known. 



§ 3. Oblique incidence. Let the luminous point now lie ofi the 

 axis X'OX at P (Fig. 2) in the plane of the figure. Let PO = u, 

 and let u be positive when P is in the object space. Let the acute 

 angle between PO and X'OX be 6. Let be a point in the plane 

 of the figure, let OQ = h, and let 6 be positive when Q is in the 



