THE PRIMARY DIFFRACTION INTEGRAL 



243 



the axial bundle as in Fig. VII. 2 and are limited by the numerical 

 aperture of the objective such that 



p- + q' ^n^ sin^ §^ = n'^pj (2.11) 



where ??,„ is defined together with p,,, in Fig. VII. 2. By means of 

 Eq. 2.10 the amplitude and phase distribution U{x — Mx^, y — MyQ, z) 

 is computed in terms of the axial data of the objective. The results 

 obtained by means of Eq. 2.10 will therefore be highly accurate for 



XqYo (object plane) 



XY (image plane) 



A 



Fig. VII.2. The axi:il, spherical wave of reference. 



object points .To, ?/o which fall in the paraxial region and will be less 

 accurate when .Tq, ?/n lies in the extra paraxial region. 



It is convenient to define the pupil function Pip, q) such that 



Ai(?), g)p-""'"'^''^' 



P(p, q) ^ Mp, r/).^-"'""-'') = 'y';'''\ ^. (2.12) 



in — p — q")' 



Then from Eqs. 2.12 and 2.10 

 U{x - Mxo, y - Myo, z) 



= fO'iP, 5)e2Tib(x-M^o)+9(2/-MOT)+«] ^p ^q. (2.13) 



p' + q'^ rrpj; 



s = (n~ — p~ — q )\ 



(2.13a) 

 (2.136) 



Finally, it is convenient and significant to define the afocal pupil 

 function Pz(p, q) such that 



P,(p, q) ^ P(p, q)e^-i^(n2-p2-a2)\ 



(2.14; 



