260 THE DIFFRACTION THEORY OF MICROSCOPY 



The object function /(xq, ijq, po, go) is required to be zero beyond the 

 field of view of the objective. All distances are dimensionless because 

 they are to be measured in wavelength numbers. As we shall see 

 presently, Fo{x, y, po, go) is the function of most significance to the 

 theory of microscopy. We shall call Fo{x, y, po, go) the amplitude and 

 phase distribution of the image plane even though Fq differs from F by 

 the factor of Eq. 8.9. 



The partial energy density E{x, y, po, Qo) is given by 



E{x, y, Po, qo) = \F(.x, y, Po, qo)\'^ = ^ \Fq{x, y, po, qo)\'^. (8.11) 



Let 



S{po, qo) = Six^,, ?/^,, Xo, yo); xq = yo ^ (8.12) 



be an energy function which is proportional to the energy radiated from 

 point Pi, (Fig. VH.(j) in the po, qo direction and which is subsequently 

 accepted by the projected area of the objective. If G{x, y) denotes the 

 total energy distribution in the image plane, 



Gi.^, ?/) = // '" D 9 °' \Po{x, y, Po, qo)\^ dxy dy, (8.13) 



in which the integral extends over the luminous area of the X^Y^, plane, 

 that is, over that portion of the A',,}',, plane which is occupied by the 

 image of the opening in the diaphragm of the substage condenser. 

 Xo, yo in Eq. 8.13 refers to the particular point Xq = 0, yo = 0, with 

 reference to which po and go are determined by Eq. 8.6. 

 From Eq. 8.6 



_ _^ Zvpo/np 



Xy — Po — / 2 2 2\' 



Wo (aio - Po - qo)' 



Rv Zi,qo/no 



Vv = qo = 



(8.14) 



'i'J / 2 2 2\'- 



no (no - Po - qo ) 



Hence 



J ("^ = . ^\ , (8.15) 



\po, qo/ 



2 9 2 



no - Po" - qo 



where ./ is the Jacobian of the indicated variables. If, therefore, the 

 variables .t„, y^ in Eq. 8.13 are transformed to the variables po, qo in 

 accordance with Eqs. 8.14, it follows that 



Sjpo, qo) 

 no" - Po^ - qo' 



G{x, y) = ff /^Po>/o^ \Foix, y, po, qo) Y dpo dqo (8.16) 



J J Tin — Vn — Qn 



