292 THE DIFFRACTION THEORY OF MICROSCOPY 



S denotes the portion of the area of the geometrical image which is 

 unoccupied by the inscribed circle of radius R. All dimensions are 

 to be measured as numbers of wavelengths. 

 From Eqs. 15.7 and 15.4 



Hr{x, y) = I - Joi^irpmR) + {he'^ - 1)[1 - Joi^irp^R)] (15.9) 



because 



/ ^^^ rdr= -- dWoiz)] = - [1 - JoiaR)]. 



Jo ar a" Jo a" 



With a large particle it will be possible to find an extended region of 

 points X, y for which R is so large that 



Jo(27rp„,R)-^0. (15.10) 



Moreover, as the numerical aperture of the objective is increased, this 

 region of points creeps toward the edge of the geometrical image of the 

 particle. With R so determined, it will also l)e possible to choose a 

 conjugate area (and hence pi) so small that 



1 - ./o(27rpi/?)->0. (15.11) 



Hence with a suitably chosen small conjugate area there will exist an 

 extended region A of points x, y for which 



HjiXx, y) = constant -^ 1. (15.12) 



This region of points x, y is an increasing function of the numerical 

 aperture of the objective and is a decreasing function of the numerical 

 aperture |^/|pi of the conjugate area. 

 From Eqs. 15.8 and 15.4 



H^{x,y) = 27rpJX 



'^ ^^-'"^) . de dr+ K (he- - l)f£ ^4^-^ r dO dr 



\_J Js 27rp,„r ' Pm^ ' JJs 2Tvpir 



in which 



(15.13) 



r = if + v^y > R- (15.14) 



The first double integral in the right-hand member will be negligible 

 when R satisfies Eq. 15.10. Except for a negligible term 



Hs{x,y) = ihe'^ - \)piffji(2Trpir)dddr. (15.15) 



whence 



\Hs{x,y)\<27rpi\he'^ - l| f Ji{2Tpir) 



dr 



(15.16) 



