M 



270 THE DIFFRACTION THEORY OF MICROSCOPY 



If the source of illumination is Lambertian, one may set 



/'"'V°^ , = 1. (10.30) 



no" - Vo - Qo 



The following formulation is equivalent to Eq. 10.29 and is more con- 

 ^^enient for some theoretical and practical purposes. It may be derived 

 from certain transformations which exist among the various diffraction 

 integrals of phase microscopy. These transformations are not included 

 in order to conserve space. Let 



Kpo, qo) = -^ 5 o' (10.31) 



^0 - Po~ - Qo" 



a quantity which may be set equal to unity with Lambertian sources. 

 Then it can be shown that 



v H s t 



in which the integers v, p., s, and / range from — co to + "^ ; in which 

 /^,^ is conjugate to /y.^L,; in which 



Q..i..s.t=Jj I(Pihqo)B,,i,Bsjdpodqo; (10.33) 



and in which the integration with respect to dpo dqo extends over the 

 optical direction cosines of the rays incident upon the object plane. 



It can be shown from Eqs. 10.31-10.35 that the resolution formulas 

 10.26 and 10.27 hold irrespective of the source function I{po, qo) and of 

 the pupil function P(p, q), provided only that the source function is 

 continuous and that P(p, q) is nowhere equal to zero within the circle 

 -p^ + g" = PnC'- In a phase microscope the numerical aperture of the 

 condenser is equal to the product noPom = n^ sin t?om where §om is the 

 angle made with the optical axis by the most steeply inclined ray which 

 is incident upon the object specimen, no is the refractive index of the 

 object space. If, for example, the opening in the condenser diaphragm 

 is annular, the numerical aperture of the condenser corresponds to the 

 numerical aperture of the outermost edge of the annular opening. The 

 physical limit of resolution is therefore somewhat larger with the phase 

 microscope than with the ordinary microscope in viewing periodic 



