^50 THE DIFFRACTION THEORY OF MICROSCOPY 



The phase factor p-'*'^-''-' - will l)e recognized as the classical zonal 

 phase factor associated with an observation plane which is displaced z 

 wavelengths from the conjugate image plane. 



In summary, the primary diffraction integral V {x — .l/.ro, y — Myo) 

 is given by 



U{x - Mxo, y - ^[yo) 



P(p, g)(.-^^'lp(x-.U.ro) + ,(i/-3/^o)) ^^ ^^ ^322) 



=/:/- 



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



Pip, 9) = (3.23) 



when 



and by 

 when 



p- + q~ > pm~ (3.23a) 



P{p, q) = P{p)- iV^I ^ ^^-2^^ 



[1 - (J/p /((,)-]' 



^ p2 ^ r = P- ^ pj. (3.24a) 



The coating function c(p, q) = c(p) specifies in accordance with the 

 convention of Eq. 3.4a the amplitude and phase transmission of the 

 coating material of the diffraction plate. T{p, q) — T{p) is the am- 

 plitude transmission along the axial rays in the absence of coating 

 material on the dift'raction plate. 27riro(p, q) = 27riro(p) radians is 

 determined from the integral over the lateral spherical aberration of the 

 axial rays in the aV)sence of coating material on the diffraction plate. 

 Woip) is to be considered as negative when it represents a reduction of 

 the optical path with respect to the spherical reference wave, no and 

 n are, respectively, the refractive indices of the object and image space 

 with n = 1. M is the magnification ratio between the conjugate oliject 

 and image planes. When /( = \, p and q are the direction cosines of 

 the normals to the spherical reference wave of the image space. Further- 

 more, p and q are related to the polar angle d and to the azimuthal angle 

 (/) of these normals in accordance with Eqs. 3.5. All distances are to be 

 measured as numbers of wavelengths. Equations 3.22 and 3.24 con- 

 tain the assumption that the objective satisfies the Abbe sine condition. 

 The primary diffraction integral of Eq. 3.22 applies to the conjugate 

 object and image planes but holds also for obser\'ation planes w^hich are 

 displaced by z wavelengths from the conjugate image plane when the 

 pupil function P(p, q) is replaced by the afocal pupil function 



Pzip, q) = Pz^p) - e-'^'-'P(p)e-'^^-'''^)/-. (3.25) 



