ELECTROSTATIC ELECTRON-OPTICS 21 



The presence of the terms in f causes a diffusion of the focus in a 

 lens, and a clearer picture of this diffusion is obtained by expressing 

 it as lateral aberration. So we now proceed to derive an expression 

 for this aberration, and the meaning of the term becomes apparent 

 from the derivation. For this purpose we consider electrons entering 

 a lens as if they all came from a point source at a distance di from the 

 first side of the lens. We are at liberty to set the higher coefficients 

 equal to zero at that side of the lens, and this gives 



Ai = --7—' ('0) 



di 



B,= Ci^ •■■ = 0. 

 At the exit side of the lens, the focal distance is 



^ = - (^2 + Br~ + C.r' +•••), (71) 



d2 



where the coefficients are solutions of their differential equations 

 subject to the initial conditions 70. The focal distance do for rays 

 near the axis is given by 



^'' = - A,. (72) 



d. 



The difference between the focal distance d2 of a ray leaving the lens 

 at a distance r from the axis and the focal distance do of a ray near 

 the axis is 



d2-do^^ (B^r' + dr' +•••)• (73) 



^I2V2 



This difference is called the longitudinal aberration of the lens. It is 

 the distance that the focal point is diffused along the axis, when the 

 lens is limited by an exit diaphragm or radius r. 



If a screen is placed at a distance do from the lens, rays near the 

 axis will come to a point focus on the screen ; but rays leaving the 

 lens at a distance r from the axis will strike the screen along a circular 

 line. The radius 5 of this circle is called the lateral aberration of the 

 lens. It follows rather simply, from the value of the longitudinal 

 aberration, that the lateral aberration is 



(B2r' -\- C2r' ■•■). (74) 



This is the radius of the diffuse image of a point source, when the lens 

 is limited by a diaphragm of radius r. 



