Dr Searle, Experiments with a plane diffraction grating 101 



from of the vertical and horizontal focal lines of the diffracted 

 beam are S2~^ and ^3"^. 



If the incident beam is stigmatic, T^ ^ S^ and W^ = 0. Then 

 S2 = k'^Sj^, Tg = S^. Hence S^ = k^T^, and so the diffracted beam 

 is astigmatic, unless k=l, i.e. unless the deviation is a minimum. 



§ 3. The principal curvatures. The principal curvatures of the 

 diffracted front can be found in terms of those of the incident front. 



Let the principal planes of the incident front intersect the 

 tangent plane at in Orj^^, 0^-^ (Fig. 2). Take these, with Of^ 

 along OPi, as axes for the front. Let the radii 

 of curvature of the sections by 0^^171 and O^^^i 

 be Bi~^ and C\~^. The equation to the incident 

 front is then 



ii = hB,rj,^ + iC\l,^ (8) 



Let Orji make an angle tpi with Osi, as in Fig. 2. 

 Then 



^1 "= ^1' Vi "" '^1 ^^^ "Ai + h ^i^ 'Ai' ^1 = ~ ^1 ^^^^ ^1 + ^1 ^°^ 'Ai' 



and hence (8) is equivalent to 



^1 = 2-^1 (^1 cos i/ji + ^1 sin 0i)2 + iCi (- Si sin tp^ + i, cos i/j^f. ... (9) 



Comparing (9) with (1), we find 



Si = i {B, + C,) + I {B, - C) cos 2<Ai, I 



If 1 = 1 (^1 - C'l) sin 2eAi, (10) 



T, = i(5i+Ci)-i(5i-Ci)cos2^,. I 



Then /Sg, Tf 2, ^2 can be found by (7). 



If the equation to the diffracted front referred to its principal 

 axes is 



^2 = ^^27)2^ + ^02^', (H) 



and if T^gOsg = «/'25 ^^^'^ ^2, O2, ^2 ^^^ related to S2, T2, W2 by equa- 

 tions similar to (10). Solving for B2, Cg, 'A2' we have 



B, 



C2 



tan 2^2 = 21f2/(.S2 - Tg) (13) 



The ambiguity in (12) has been settled so that, when i/j^ = 0, 

 B2 = k^B^, C*2 = G^. Apart from mere reversals of direction, (13) 

 gives two values of j/^a differing by hir, and corresponding to the 

 axes 0-r]2, Ot,2- The arrangement of signs in (12) implies that when 

 tjj^ = 0, ^2 = 0- Since, by (7), W^ and W2 vanish together, ijj^ 

 and 02 niust reach \tt, tt, ^tt, ... together, and it follows that, for 

 intermediate values, ^2 must lie in the same quadrant as ifj^. 



} = 1 (^2 + T2) ± [i (^2 - T2f+ W2^]\ (12) 



