ELECTROSTATIC ELECTRON-OPTICS 



11 



where the inverse focal term is 





2 y' 



2V2z; 



dz 



or on integration by parts 



^ _ 1 



F~ 2 



\ V2z;/2 



^^v 



+ 



^ r(i'')H2i')- 



.72 



■'iHz. 



(33) 



(34) 



The substitution in equation 32 of the values for Ti and T2 as given 

 by equation 24 now gives the lens equation 



■yjlvi _ ^J2vi _ 1 

 ~dr ~d7 ~ F' 



(35) 



This equation is analogous to the equation for a thin optical lens 



M2 

 d2 



di ~ f' 



(36) 



bounded on its two sides by media with different refractive indices 

 ixi and /X2, the ^2v corresponding to refractive index. 



Electron rays parallel to the axis do not come to a focus at a distance 

 F from an electron lens ; in other words, F is not a principal focal 

 distance. There are, in general, two principal focal points on opposite 

 sides of an electron lens. Their principal focal distances /i and fi 

 are found by setting first dx, and then d^, equal to infinity in equation 

 35. This gives 



fi= - V2^F, h = V2^^ (37) 



as the two principal focal distances. It may be shown from equation 

 33 that these principal focal distances really involve the voltages 

 only in the form of the ratio i'2/f i- By substituting them in the lens 

 equation 35, it may be written in the convenient form 



-+-= 1, 

 di d\ 



(38) 



which likewise involves the voltages only in the form of a ratio. 



There are two types of electron lenses that deserve special con- 

 sideration. The first is a small aperture in a thin plate separating 

 two uniform fields of different intensities — as a special case one of the 

 fields may be zero. An example of such a lens is illustrated in Fig. 6. 



