10 BELL SYSTEM TECHNICAL JOURNAL 



Thin Lenses 

 Approximate solutions of the differential equation 16 for A are 

 obtained more clearly by first changing the space variables to time 

 variables. This is done by using relations 24 and 25, which transform 

 the equation to 



or 



The new equation tells how the focal time T varies with time as an 

 electron moves along. ^^ 



A thin lens is defined as a region of non-uniform field extending 

 over such a short distance along the axis that an electron traverses 

 it in a period of time small compared to the focal times involved : 

 the thickness of the lens is small compared to the conjugate focal 

 distances. By taking the origin of time t at the middle of an electron's 

 period of transit through a lens, / in the lens is not greater than half 

 the period of transit, and / may therefore be neglected in comparison 

 to r in a thin lens. With this approximation in equation 31, it 

 reduces to 



d /\\ v" 



dt\TJ 2 (^^) 



In integrating this equation through a lens we choose two points 

 Zi and Zi at the approximate boundaries of the non-uniform field, that 

 is, the points where v" substantially drops to zero as illustrated in 

 Fig. 5. Then, remembering that dt is dz/yJ2v, an integration from Zi 



k 



z 



to Z2 gives 



Fig. 5 — Conjugate focal distances, thin lens 

 11 1 



T. T, F' ^^^^ 



" The period of time that a train requires to reach its terminal point also varies 

 with time as the train moves along. 



