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



When n diminishes, D^Gn becomes more and more nearly the 

 normal to the wave front, and, for a small aperture, may be treated 

 as the normal in the estimation of distances. Thus, when n is not 

 great, DnGn may be taken as the ray distance from the wave front 

 to 6r„. 



Let OP2 be the forward direction of the chief ray of the emergent 

 beam, which, by symmetry, must lie in the plane XOY. Let 

 P2OX = 02 . When referred to axes Or2 , OS2 , 0^2 > chosen similarly 

 to Ofi, Osi, Oti, let the equation to the emergent wave front, as 

 it passes through 0, be 



^2 = IS2S2' + W2S2t2 + in^2'- (5) 



If a line through G^ parallel to OP2 cuts the emergent wave front 

 OE1E2 ... in E„, the distance E„G„ is ultimately the ray distance 

 from En to Gn . We then have 



E^Gn = Pn cos CO siu ^3 



— pn^ (1^2 cos^ CD cos^ $2 + Tf 2 COS (x) siu o) COS 62 + ^T2 sin^ co). 



The optical condition is that the time of passage of light over 

 the distance E^Gn — E^G^ in the second medium is less than the time 

 of passage over the distance DnG„ — D^G^ in the first medium by 

 (n— l)pr. Hence 



1x2 {E^Gn - E^Gi)/vo - P-i {D„G„ - D^G^)/vo = - (w - 1) pr. 



Thus, since V(,t = Xq, we have 



{P7i ~ Pi) cos CO (ju,2 sin 62 — fJii sin di) 



— {Pn^ — Pi") [2 ip-l^i COS^ $2 — p-iSi COS^ 6^) COS^ CO 

 + (j"'2^2 COS 02 — P^iWi COS dj) COS CO siu CO 

 + 2 (/^2^2 - /^l^l) sill" to] = - (W - 1) ^Aq. 



Nowp„^ — p^^ = {n — 1) Tc^, butp„ — p^ is not proportional to w — 1. 

 Hence, if the equation is to hold for all integral values of n greater 

 than 1, we must have 



jM.2 sin ^2 = i"-! sin ^1 . (6) 



Hence, the chief ray obeys the ordinary law of refraction, and 62 is 

 known when 9^ is given. Since p„^ — Pi' = {"n — 1) k^, we have 

 I" {1^282 cos^ 62 — j(Xi*S'i cos^ 0-^ cos^ CO 



+ (/t-ta^^cos^a" /XiT^iCos^i) cos CO sinco 



+ l(^2^2-/^i^i)sin2co = ^Ao/^2 (7) 



Equation (7) holds for all values of co. Putting co = 0, we have 



IX2S2 cos^ 62 — p^iSy cos^ ^1 = 2pXo/k^. (8) 



Putting CO = Itt, we have 



Pi,T2-p.,T, = 2p\lk\ (9) 



