THE OBJECT FUNCTION 259 



approximation 



R = R^- ^»^- + ^"^" ; (8.4) 



Rv 



,2^ _ e^"'-"^" _2^,„^ XqX, + ypy. 



i^ Rv R'p 



(8.5) 



With reference to Fig. VII. G, the rays originating at point Pv are incident 

 upon the object plane along the direction P^Oq as Z^ approaches infinity. 

 Let po, qo be the optical direction cosines of rays which are parallel to 

 the direction PiOq. Then from the trigonometric meaning of a direction 

 cosine 



Po = —^ — ; 9o = -^^0^- (8.6) 



Let po, 50 from Eq. 8.6 be substituted into Eq. 8.5. We learn, ac- 

 cordingly, that the plane wave originating at point P,. induces upon the 

 object plane XqYq the amplitude and phase distribution 



Rv 



e 



2iri(poa;o+g(M/o) 



(8.7) 



in which -p^, Qq are the optical direction cosines of the normals to the 

 wave front. If fixo, ijq, po, qo) denotes the change produced in the 

 amplitude and phase transmission of the object plane by the presence 

 of the object specimen, we see that the wavelets from point P„ emerge 

 from the object plane with an amplitude and phase distribution 

 x{xo, yo, Po, Qo) which is given by 



'ZirinoRv 



Xixo, yo, Po, qo) = ^^— e^'-'(^"^»+^°^«V(xo, yo, Po, qo)- (8.8) 



It will be noted that Eq. 8.8 agrees with Eq. 8.1 and specifies the complex 

 number C. 



As in Eq. 7.1, let F{x, y, po, qo) denote the amplitude and phase 

 distribution produced in the image plane by the incidence upon the 

 object plane of a wave front whose normals have the optical direction 

 cosines po, qo- Then 



2irinoRv 



Fix, y, Po, qo) = s — ^o(^-, y, Po, qo) (8.9) 



where 



Foix, y, Po, go) - r f e2--(^o-o+^o^o) 



«y — CO t/ — 00 



X/(a:o, yo, Po, qo)U{x - Mxo, y - Myo) dxo dyo. (8.10) 



