PUPIL FUNCTIOx\ P{p, q) IN PHASE MICROSCOPY 247 



wave converges upon and since this wave is a wave front in the par- 

 axial region, TFo(0, 0) = 0. Tip, q) shall be normalized such that 

 T{0, 0) = 1. It has already been assumed in stating the primary dif- 

 fraction integral that the objective satisfies Abbe's sine condition. 

 When the bundle of axial rays is converged in accordance with Abbe's 

 sine condition, there is introduced over the spherical reference wave 

 (Osterberg and Wilkins, 1949) an amplitude variation k{p) given by 



k(p, q) = k{p) = ^ , ; (3.1) 



[1 - {nMp/riQyy 



P = ; (3.1a) 



n 



in which /?o and n are the refractive indices in the object and image 

 space, respectively, and M is the magnification ratio between the object 

 and image planes. Hence 



A,(p,q) = T(p,q)k{p,q) (3.2) 



is the normalized amplitude variation over the spherical reference wave 

 in the absence of coating material upon the diffraction plate. 27rlFo(p, q) 

 is the corresponding phase variation in radians over the spherical 

 reference wave. 

 Let 



,(p,,).di(M)f!!!!!!^. (3.3) 



(n — p'' — q ) 



Then the pupil function P{p, q) is given by 



P(p, q) = c(p, q)rP{p, q) (3.4) 



in which c(p, q) is a complex coating function which specifies the ampli- 

 tude and phase transmission of the coating material on the diffraction 

 plate. Let c{p, q) be written in the form 



c{p,q) = \c{p,q)\e'^'^'^^''^\ (3.4a) 



The amplitude transmission, \c{p, q)\, is to be taken along the direction 

 in which the axial rays pass through the coating material. The phase 

 angle, arg c(p, q), is to be determined from the optical path along the 

 normal to the coating material when the diffraction plate is perpendic- 

 ular to the optical axis. 



It is often expedient to replace the optical direction cosines p, q by 

 the polar angles ^ and the azimuthal angles 4>. It follows from the 



