A UNIFORM PARTICLE IN A LARGE FIELD OF VIEW 283 



over the plane of the image by the incidence upon the object plane of a 

 wave front whose normals have the optical direction cosines po, qq. For 

 the present F„ and Fd are merely definitions, but we shall see later that 

 they correspond, respectively, to the undeviated and deviated waves 

 which reach the plane of the image. 



Let U from Ec]. 2.19 be introduced into Eq. 11.2. Then 



^" "//X X X =0^^^' g)e2-i(P-+</2/)g2.i[-o(PO-Mp)-f2,o(90-M,)l ^^.^ ^y^ ^^ ^^. 

 = lim ffp{p,q)e^'''^P='+iy^ 



X0 = 



sin [2TrXo(po-Mp) ] s\n[2Tryo(qo-Mq)] 



T{po-Mp) Tr(qo-Mq) ^ ^' 



Successive integration with respect to d{Mp) d{Mq) gives, in view of 

 Eq. 10.11, the result 



Fu{x, y, Po, go) = ^ e^^-/^^n^o-+^o-)p {fl'^)' ^11-^) 



Discontinuities in the pupil function are ignored because they appear at 

 isolated loci which do not usually exceed three in number. We recall 

 that the points {p = po/M, q = qo/M) belong to the conjugate area of 

 the diffraction plate. The amplitude and phase of F^ is therefore 

 modulated by the coating function at the conjugate area. Hence 

 Fu must represent the wave which has passed through the conjugate 

 area and must in fact be the unde\'iated wave. This conclusion is 

 furthermore consistent with the fact that the phase multiplier of the 

 infinite integrals of Eq. 11.2 is the phase and amplitude transmission 

 factor /i = 1 of the surround. It is interesting to note that the un- 

 deviated wave of the more general theory is substantially the same as 

 the undeviated wave as given by Eq. 7.3 in Chapter II of the elementary 

 theory. However, Eq. 7.4 of Chapter II is far too simple to describe 

 aderiuately the deviated wave Fd. 



Introduce into Eq. 11.3 the change of variable 



r = a; - Mxo; v = y - My^. (11.6) 



Then 



Fd{x, y, Po, qo) = ^e(2-/^)(^«^+''"^'^)^e-(^^^'/^^^)(^»f+''o''^[y^(f,,7)dfrfr, 



(11.7) 

 in which the integration with respect to f/f dri extends over the area of 

 the geometrical image of the particle. 

 From Eqs. 11.1, 11.4, 11.5, and 11.7 



