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



Comparing (15) with (4), we find 



S, = l{B, + C,) + i{B,-C,)coB2rf;„ (16) 



W, = l{B,-C,)sm2rlj„ (17) 



T, = i{B, + C,)-^{B,~C,)cos2^, (18) 



If the equation to the emergent front, referred to its own 

 principal axes, is 



i, = ^B,rj,^ + lC,U', (19) 



then B2, C2 are the principal curvatures. If T^aOsj = !/»2> then B2, 

 C2 , ipz ^^^ related to S2 , W2 , T2 by 



^2 = i(^2 + C2) + |(5,-C2)cos2^„ (20) 



Tf2 = i(52-C2)sin2^2, (21) 



T^ = 1 (5, + C,)- 1(5,-0,) cos 2</r2 (22) 



Solving for 5,, C, and using (11), we have 



B2 = X+Y, C2 = X-Y, (23) 



where 



Z = 1 (;S, + T,) = 1 [F^ (1 + sec2 d) + B^ + (7J, (24) 



r = i[(>S,-T,)^ + 4F,T 



-= -I [F^^ tan^ d + 2F^ tan^ d [B^ - C^) cos 2i/j^ + {B^ - C^y^f. 



(25) 



These equations, with (23), determine 5, and C,. But, as either 

 sign can be given to the square root, we are left in doubt as to 

 which of the two focal lines corresponds to 5, and which to C,. 

 To avoid confusion we must write 



7° = 1 {F^ tan^ d + B^- CJ, Y""'^ = 1 {F^ tan^ d-B. + C,), 



(26) 



where Y", Y"''^ are the values to be assigned to Y for i/fi = and 

 for i/fj = ^77. For intermediate values of ^i , we take Y intermediate 

 between Y^ and 7"^^. Since iY^ is a sum of squares, Y cannot 

 vanish unless If, = 0, or, what is the same thing, unless W^ = 0. 

 When B^ — C-^ is not zero, Wi vanishes only when ipi = or 

 ^1 = jTT. Hence Y does not change from positive to negative or 

 vice versa as ifj^ goes from to ^. 



In practical work it is convenient to determine the quantities 

 appearing in X and Y by observing 5,°, C,", 5,"^^ Cg^^^ the values 

 of B2 and C, for i/ji^ and ip^ = ^-n. We have, in accordance with 

 (26), 



B2' = F,secH + B„ C2'=F,+ C,A 



5//' = i?^^sec2^ + Ci, C2^''" = F^+B^.\ ■••^^^ 



