254 THE DIFFRACTION THEORY OF MICROSCOPY 



It follows from a criterion which is often attributed to Airy that two 

 object points are just resolvable when their separation in the object 

 space is equal to r/|il/| wavelengths as determined by the classical 

 Eq. 5.10. To those who have not had the opportunity of studying 

 Luneberg's derivation of the primary diffraction integral, Eqs. 5.7, 

 5.8, and 5.10 will serve to demonstrate that the primary diffraction 

 integral includes the classical behavior of diffraction images. 



Airy-type objectives may become important in adapting the phase 

 microscope to the measurement of the properties of an object particle. 

 It is pointed out that an objective not normally of the Airy type may be 

 rendered of the Airy type by placing upon the diffraction plate or upon 

 an adjacent plate a second coating material whose auxiliary coating 

 function a{p, q) is related to the aberration function i/'(p, q) by the 

 equation 



a{v,q)Hv,q) = 1- ^5.11) 



Then 



P(p, q) = c(p, q)a{v, q)xl^(p, q) = dp, q). (5.12) 



6. A TRANSPORT PROPERTY OF THE PRIMARY DIFFRACTION IN- 

 TEGRAL 



Suppose that the object plane has been illuminated in any suitable 

 way w^hich causes Huygens' wavelets to leave elements of area dxo dyo 

 of the object plane with a coherent amplitude and phase distribu- 

 tion described by some "civilized" but otherwise arbitrary function 

 x(.ro, yo) dxodyo. The function x(-^'o, Vo) is then equivalent to the 

 existence of x unpolarized dipoles of unit strength per unit area. Since 

 each unpolarized dipole produces the amplitude and phase distribution 

 U{x—Mxo, y — Myo) in the conjugate image plane, the assembly of 

 X unpolarized dipoles in the element of area dxQ dyo produces the ampli- 

 tude and phase distribution 



xixo, yo)U{x - Mxq, y - Myo) dxo dyo (6.1) 



in the conjugate image plane. The physical meaning of this amplitude 

 and phase distribution is subject to the reservations described at the 

 end of Section 2. We may say that the phase and amplitude 

 x(^o, ^o) dxo dyo of the wavelets which leave an element of area 

 dxo dyo of the object plane is transported to the image plane as the 

 product of x(-i'o, Vo) dxo dyo and the primary diffraction integral. 



The coherent distributions (Eq. 6.1) which are produced from differ- 

 ent elements of area dxo dyo of the object plane overlap at any point .r, y 

 of the image plane. The resultant amplitude and phase so produced at 

 point X, y is therefore the sum or integral of the products (Eq. 6.1) over 



