ELECTROSTATIC ELECTRON-OPTICS 



15 



A plane located at a distance ai from the point Zi is the approximate 

 first principal plane of the lens ; and a plane located at a distance a^ 

 from the point Z2 is the approximate second principal plane of the lens. 

 In the lens equation, di — ai and di — a^ are the conjugate focal 

 distances measured from the principal planes. If the focal distances 

 measured in this manner are designated as Di and D2 respectively, 

 the lens equation assumes the simpler form 



\hj% _ -shji _ 1 

 _, 



D, 



Dx 



(48) 



An electron lens frequently has both a positive and a negative 

 maximum of v" , and the preceding approximation cannot be applied 

 to the lens as a whole. There is, however, necessarily a point between 

 the two maxima where v" is zero and by taking this as a division 

 point, the lens can be separated into two components. The approxi- 

 mation can then be separately applied to each component, and the 

 whole lens treated as a combination of two lenses. 



The General Theory of Thick Lenses ■ 

 The equation for the coefficient A, 



dA A^ = ^" 



dz J2i 



2V2y 



is a Racciti equation and, with the exception of special cases, it has 

 no exact solution in the usual sense. Particular solutions can be 

 obtained only by integration in series. It is, however, possible to 

 express the general solution of a Racciti equation in terms of any two 

 particular solutions, and this property enables us to develop the 

 general theory of a thick lens in terms of its principal focal distances. 



Fig. 9 — Paths corresponding to X and Y. 



In considering a thick lens, two points Zi and S2 are again taken at 

 the substantial boundaries of the non-uniform field constituting the 



