244 THE DIFFRACTION THEORY OF MICROSCOPY 



Hence 



9 

 P 



It should be noted that Pzip, q) = P(p, q) when z = 0. 



For purposes of analysis it is frequently convenient to require that 



Pz(p,q) = (2.16) 



when p^ -\- q^ > trp,„" and that Pz(p, q) is given by Eq. 2.14 when 

 V~ -\- <f = n^p,n^. Equation 2.16 is simply the statement that the pupil 

 function is zero for rays that are blocked by the aperture of the objective. 

 It follows from Eqs. 2.15 and 2.16 that V{X, V, z) and P^ip, q) are 

 related as the pair of Fourier transforms 



/ P,{p,q)e~'''^P^+'^'^dpdq; (2.17) 



/ r(r, 77,2)p-2-(^f+'"'>rffrf^;, (2.18) 



■ 00 ty — 00 



in which z is regarded as a parameter. 



The presentation of the theory of phase microscopy will be simplified 

 with the aid of the following observation. Suppose that one has con- 

 structed a theory which involves operations upon the primary diffraction 

 integral U{^, -q, 0) = ('(f, 17) given by 



U{x - Mxo, tj - Myo) 



/OO pX 



/ F(p, g)e2'^^'^^^-^^^«^+^^^-^^«^^ dpdg. (2.19) 



Then the theory for out-of-focus planes which are located z wavelengths 

 away from the conjugate XY plane can be obtained from the theory 

 based on Fa\. 2.19, in which z = 0, by replacing P(p, q) by Pziv, q)- 

 For simplicity of presentation we shall therefore state the laws of phase 

 microscopy in terms of the primary diffraction integral as given by Eq. 

 2.19 with the understanding that P(p, q) is to be replaced by Pzip, q) 

 whenever the plane of observation is displaced from the conjugate image 

 plane by z wavelengths. A summary of the primary diffraction integral 

 and of the manner in which the pupil function Pip, q) is to he computed 

 will be given in the next section. 



The following properties of the primary diffraction integral ha'V'e been 

 demonstrated by Luneberg (1944) and will be restated here in order to 



