COATING FUNCTION c(p, q) L\ PHASE MICROSCOPY 251 



The functions P, T, W^, and c are expressed in terms of the direction 

 cosines p, q when the optical system is not symmetrical axially but can 

 be expressed in terms of the zonal numerical aperture p of the image 

 space when the optical system does possess axial symmetry. Finally, 

 the inequalities of K(\h. 3.23 and 3.24, which serve as limits of integration 

 in Eq. 3.22, have been written explicity for objectives having circular 

 apertures. These inequalities are easily modified for application to ob- 

 jectives having rectangular apertures. 



i. SPECIFICATION OF THE COATING FUNCTION c(p,q) IN PHASE 

 MICROSCOPY 



The following conventions will be used in specifying the coating 

 function c(p, q) in a Zernike system of phase microscopy. In Zernike's 

 method the conjugate and complementary areas of the diffraction plate 

 are coated differently with substantially uniform coatings. 



Let Cofp, q) and c^ {p, q) denote the amplitude and phase transmission 

 of the coating material on the conjugate and complementary areas, 

 respectively. We set 



Co(p, g) = ho(p,q)e'''; 



ci(p,q) = h,(p,q)e'^K (4.1) 



The amplitude transmissions ho and hi are to V)e specified along the 

 direction in which the axial rays pass through the coating material. 

 The phase transmissions Sq and 5i are to be determined from the optical 

 paths along the normal to the coating material. 6o and 5i may be taken 

 as independent of p, q because the amount of sphei'ical aberration as- 

 sociated with the passage of the axial rays through the extremely thin 

 coating materials of the diffraction plate is secondary and therefore 

 negligible. 



In a conventional microscope ^ p^ -\- q^ ^ p,ri^ with p„, < 0.04. If 

 the diffraction plate is located well away from the object space of the 

 objective, the range in p, q becomes so small that ho(p, q) and hxip, q) 

 are substantially constant. Whenever this is true, we let 



Ci(p,q) hi 



h is physically the ratio of the amplitude transmission of the coating 

 material on the conjugate area to the amplitude transmission of the 

 coating material on the complementary area. 5 is determined from the 

 optical path difference in radians between the conjugate and com- 

 plementary areas and is considered as positive when the optical path 



