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



image space. Let the acute angle between OQ and X'OX be (/>. 

 Then, since OG^ = pn , 



PG^^ = w2 + 2wp„ sin e + p,\ QG,^ = b^ - 26p„ sin (f> + p,\ 



Hence, by expanding, 



PG^=u{l + l {2p^ sin e/u + p„7w,2) _ 1 (2p„ sin d/u + p„7w2)2+ ...}. 



Thus, as far as terms in 1/u, 



PG^^u + p„ sin d + ip,„2 (], _ sin2 d)lu = u+p^ sin 6+ i^/ ^oss 0/^. 



Similarly, QG^ = b- ,o„ sin (^ + ip„2 cos2 c/,/6. 



Hence 



PGn + ^6^n - (P(^i + QG^) = ip,, - p^) (sin d - sin </.) 



+ i (Pn' - Pi') (cos2 ^/w + C0S2 C/./6). 



Fig. 2. 



If this difference is (n - 1) ^A, there will be concentration of light 

 at Q. Now p,2 _ 2 ^ (^ _ 1) j,2^ but p, - p, is not proportional 

 to w - 1. Hence, if the equation is to hold for all integral values 

 of n, we must have sin </, = sin 6. Thus cos^ ,/, = cos^ d, and then, 

 l/u + 1/6 = 2pX sec2 ^/Z;2 _ gg^a ^/y^ _ ^^ g^^a ^i^ _ _ (2) 



Let H^, H^, ... lie on a straight line through perpendicular 

 to the plane of Fig^2, and let OH,, = p^. Then, if ^ is a point on 

 UF, the path PH^R is, for given values of OP = w and OR = c 

 equal to the path PH„R when P and P He on the axis. Hence if 

 c be positive when R Hes in the image space, there will by (1) 

 be a concentration of light at R, when ^ v /' 



Vu+llc=l/f^ = F,. (3) 



As in I 2, we can speak of the emergent wave front, but in this 

 case the front will not be spherical. By symmetry, the plane of 

 -big. 2 and a plane through PO and perpendicular to the latter 

 plane are the principal planes of the emergent wave front The 

 radu of curvature of the principal sections of this front are b and 

 c, as given by (2) and (3). 



