100 Dr Searle, Experiments with a plane diffraction grating 



1 



When R and F^ approach 0, F^R becomes more and more nearly 

 the normal at F^, and, for a small aperture, may be treated as the 

 normal in the estimation of distances. Thus, ultimately, F^R is the 

 ray distance from the wave front OF-^ to R. 



Let OP 2 be a diffracted ray of order i. By symmetry, OP2 is 

 in the plane OXY, since OP^ is in that plane. Let P^X = 6^. 

 Take the axial ray OP2 as the axis of ^2 in a set of axes Org, Os^, 

 Ot2, of which OS2 is in the plane OXY and 0^2 coincides with OZ. 

 Let the equation to the diffracted wave front passing through be 



_ 1 

 "~ 2 



S2S2^+W2S2t2+hT2t2' (4) 



Then, if F2R, parallel to OP2, cuts the diffracted wave front OF^ 

 in #2, the distance F2R is ultimately the ray distance from F2 to R. 

 We then have 



F2R = qd sin ^2 - i>52?^^^ cos^ 62 - W2qdz cos 62 - lT2Z^....{b) 



The optical condition is that F2R differs from F^R by qiX, where i 

 is a positive integer. We thus obtain 



F2R = FJl ± qiX. 



Since this holds for all values of z and all integral values of q, we 

 have, by (3) and (5), 



sin 62 = sin ^^ ± iX/d, (6) 



S2 = k^S„ T4^2 = '^^i, T2=T„ ..(7) 



where k = cos ^^/cos $2. Since 6^ and $2 both lie between — ^tt 

 and ^77 for a transmitted beam, k is positive. 



The direction of the axial ray of the diffracted beam is given 

 by (6) and is independent of the constants S^, W^, T^. Equations (7) 

 give the form of the diffracted wave front which passes through 0. 



If the deviation of the axial ray is a minimum, it follows from 

 Part I, § 6, or otherwise, that sin ^^ = — sin ^j. Since d-^ and 9^ 

 both lie between — ^tt and ^tt, cos ^2 = cos ^^, and thus k==\. 

 Hence, in the case of minimum deviation, the form of the wave 

 front is unchanged and the diffraction merely turns it through 2d 

 about OZ. The restriction stated in § 2 must be noted. 



If the planes of the principal sections of the incident wave 

 front are OXY and ZOP^, or, what is the same thing, the planes 

 Or^Si, Orjtj^, then W^ = 0. It follows, from (7), that IF2 = 0, and 

 thus the principal planes of the diffracted wave front are OXY 

 and ZOP2. 



When TFj = 0, the section of the incident front by the hori- 

 zontal plane ^^ = is r ^ = ^S^s^^, and the distance of the centre of 

 curvature of this section from is Sj~'^. The vertical focal line of 

 the beam passes through this centre of curvature. Similarly, the 

 horizontal focal line is at a distance T^~^ from 0. The distances 



