PHYSICAL LIMITATIONS IN ELECTRON BALLISTICS 



307 



where it is 2, as in Fig. 1. Our lens may be converging or diverging; strong 

 or weak. In the analogous electric case, however, tiie potential throughout 

 the lens space must satisfy Laplace's equation, and this means that if it is 

 specified along the axis it is known everywhere. We can easily see this by 

 writing down Laplace's equation for an axially symmetrical field. 



1 d 

 r dr 



('?) 



+ 



d-v 



= 



(4) 



f^ 





LIGHT-CONDITIONS OFF 

 AXIS NOT FIXED BY 

 CONDITIONS ON AXIS 





ELECTRIC FIELD-FIELD 

 OFF AXIS SPECIFIED BY 

 POTENTIAL ON AXIS 



v=^ / f(5+ir cose)de 



Fig. 1 — Contrast between optical and electric focussing conditions. 

 The field near the axis may be expanded in powers of / 



dr 



Substituting this into (4), 



-d-v 



= ar -\- 



(5) 



i ^ (ar') =2a =~ 

 r or 



dV -Id'V 



dz^ 



dr 



2 dz" 



(6) 



As a matter of fact, the potential V {z,r) remote from the axis can be 

 expressed in terms of the potential Vo{z) on the axis as 



V = - I Fo(c + ir cos d) dd 



T Jq 



(7) 



If we could introduce charges into our lens, Laplace's equation would no 

 longer hold and we would have more freedom of design. The methods 

 proposed for the introduction of charges comprise the use of free charges 

 (space charge) which are largely uncontrollable, and the use of curved grids, 

 which do more damage than good. In other w'ords, the cures are worse 

 than the disease. 



