THE OBJECT FUNCTION 257 



do with the rays which enter the objective in the object space. If the 

 source obeys Lambert's law and is uniformly bright, the intensity of the 

 source is proportional to cos d^- For simplicity of presentation, we 

 take the combination of proportionality factors as unity and assert that 

 with Lambertian sources of uniform brightness 



Gi^, y) =JJ k'X-r, y, Po, go) I" ^Ipo dqo (7.5) 



in which the integral extends over the optical direction cosines po, go of 

 the rays which are incident upon the object plane. Except when it is 

 demonstrable that the more general expression of Eq. 7.3 can or should 

 be used, Ecj. 7.5 suffices for the purpose of computing the variation in 

 the total energy density over the image plane of a phase microscope or 

 of an ordinary microscope. 



A remarkable simplification occurs when the opening in the diaphragm 

 of the substage condenser is small. In this case the variation in pq, Qq is 

 so small over the restricted range of po, q^ that the integrand in Eq. 7.3 

 remains sensibly constant. Then 



G{x, y) = —, 7, ^ \F{x, y, po, qo)\ j j dpo dqo 



"d" — Pn~ — f/o^ 'J'J 



Sipo, qo) 

 Lr{x, y) = —^ 



so that 



G(x, y) = K\F{x, y, po, qo)\^ (7.6) 



wherein K is a parameter which depends on the size and location of the 

 opening in the condenser diaphragm. With Lambertian sources of 

 uniform brightness K will depend only upon the size of the opening in 

 the condenser diaphragm. Of course, K increases with the strength of 

 the source of light. Equation 7.6 applies to the types of illumination 

 which were treated by Abbe. More specifically, it applies to narrow 

 cones of either axial or oblique illumination. 



8. THE DIFFRACTION INTEGRALS IN TERMS OF THE OBJECT FUNC- 

 TION /(.ro, yo, po, qo) 



There exists a broad class of object specimens for which the function 

 x(^Oj yo, Po, qo) of the last section can be written in the form 



xi-i-o, yo, Po, qo) = Ce^-'^^«-"+'^»^"y(.i-o, yo, Po, qo). (8.1) 



C is a complex number not depending on (.ro, yo, Po, qo)- The expo- 

 nential is a phase factor due to the incidence of an inclined wave front. 

 The object function /(.ro, yo, Po, qo) may be interpreted as the change 

 produced in the amplitude and phase transmission of the object plane 



