256 THE DIFFRACTION THEORY OF MICROSCOPY 



mation. When x(-'"o, yo, Vo, 9o) has been found, the amphtude and 

 phase distribution F(x, y, po, qo) produced over the image plane by the 

 coherent hght in the incident wave front whose normals have the 

 optical direction cosines po, Qo is given according to the transport 

 property of the primary diffraction integral by the equation 



F{x, ij, Po, f/o) = / / xC^o, yo, Po, go) 



y.V{x- Ma-o, y - Mvq) dxQ dijQ (7.1 ) 



in which x is recjuired to be zero for points Xq yo which lie beyond the 

 field of view of the objective. Equation 7.1 will be most accurate for 

 points .r, y which fall near the optical axis. Consequently, the object 

 specimen should be placed upon or near the optical axis whenever its 

 imagery is to be interpreted with the aid of the diffraction integrals of 

 phase microscopy. 



Let E{x, y, pa, qo) denote the energy density produced over the image 

 plane by the light in the incident wave front whose normals have the 

 optical chrection cosines po, qo. We shall call E{x, y, po, qo) the partial 

 energy density. It is produced by the light from an infinitesimally small 

 portion of the source and is proportional to \F{x, y, po, qo)\- 



E{x,y,po,qo) ■ \F{x, y, po, qo)\^- (7.2) 



Let G{x, y) denote the total energy density produced in the image plane 

 by light from all the effective elements of area in the source. G{x, y) 

 can be obtained by integrating over the effective area of the source or 

 by integrating over the optical direction cosines po, qo of the total beam 

 of rays which are incident upon the object plane. We know that there 

 must exist a positive function S{po, qo) such that 



G{x, y) = ff /^P"^'^ ^ \F(x, y, po, 9o)|' dpo dqo (7.3) 

 J J 7io^ - Po - qo 



in which the limits of integration extend over the optical direction 

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

 plane. This family of rays, and hence po, qo, is limited by the opening 

 in the diaphragm of the substage condenser. Since S{po, qo) must be 

 proportional to the energy radiated in the po, qo direction, S(po, qo) 

 will be proportional to the intensity of the source. Furthermore, 

 S{po, qo) must be expected to be proportional to 



cos T?o = — (V - Po^ - qo^)' (7.4) 



Wo 



because the normal to the aperture of the objective makes the angle 



