PLATE-LIKE OBJECT PARTICLES 261 



in which the integral extends over the range of optical direction 

 cosines po, qo of the rays which are incident upon the object plane. 

 This range is limited by the opening in the diaphragm of the substage 

 condenser. 



In summary, the total energy density G{x, y) in the image plane is 

 given by Eq. 8.16 in which Fq(x, y, Pq, go) is determined by Eq. 8.10. 

 Furthermore 



OO t/ — 00 



(8.17) 

 P{p, 9) = 



when 



P~ + q"" > pJ\ (8.18) 



P(p, q) = P(p) = 7- ,, 



[1 - (.l/p//lo)1* 



when 



^ p' + ?' = p' ^ pJ- (8.19) 



For out-of -focus image planes P(p, g) is to be replaced by the afocal 

 pupil function Pz{p, q) as in Eq. 3.25. 



The properties of the optical system which are peculiar to phase 

 microscopy are contained in the modification of the pupil function 

 P{p, q) by the coating function c{p, q) and in the restriction of the 

 range of the optical direction cosines po, 9o by the choice of opening in 

 the diaphragm of the substage condenser. 



The problem of integrating Eq. 8.16 becomes formidable since 

 Eqs. 8.10 and 8.17 have to be combined with Eq. 8.16. 



For reasons already stated in Section 7, G{x, y) is given with good 

 approximation by 



G{x, y) =JJ\Fq{x, y, po, ?o)|' dpo dqo (8.20) 



when the source of light is Lambertian. If the opening in the diaphragm 

 of the substage condenser is very small, Eq. 7.6 applies again, provided 

 that Fq is determined by Eq. 8.10. 



9. THE OBJECT FUNCTION /(xn, jn, po, qo) FOR PLATE-LIKE OBJECT 

 PARTICLES 



If the object particle is of uniform thickness and refractive index, 

 the incident rays which have like optical direction cosines po, qo pass 



