ELECTROSTATIC ELECTRON-OPTICS 



17 



which is one form of the equation relating the conjugate focal distances 

 of a lens. This equation may be converted into a more useful form 

 by the following considerations. 



A combination of the three preceding relations gives 



-J- {di — Q!2) = X (^1 ~ '^l)' 

 ^2 CLl 



where 



a =/i(l - ^i), 



^2 = /2(1 — ^2). 



(57) 



(58) 



To interpret this equation, we erect two imaginary planes as shown 

 in Fig. 10. The first plane is located at a distance ai from Zi. If the 



Fig. 10 — The principal planes. 



path of the incident ray is projected it intersects this plane at some 

 radial distance R\. The second plane is erected at a distance 0:2 from 

 22. The path of the exit ray intersects it at a radial distance Ri. 

 The equation says — from simple geometry — that the two radial 

 distances Ri and R2 are equal. The path of an electron through the 

 lens is therefore the same as if the electron proceeded in a straight line 

 to the first plane, passed parallel to the axis to the second plane, and 

 then proceeded again in a straight line to the second conjugate focal 

 point. These two planes are called the first and second principal 

 planes of the lens. The action of a thick lens is the same as if the 

 space between the principal planes were non-existent, leaving them 

 in coincidence, and a thin lens were located at the plane of coincidence. 



The principal planes of a lens may lie either inside or outside of 

 the lens. In most convergent lenses, ax is positive and a^ negative, 

 and the two planes both lie inside the lens. 



The first conjugate focal distance measured from the first principal 

 plane is designated as Di, and the second conjugate focal distance 

 measured from the second principal plane is designated as D^. When 

 they are measured in this manner, the two conjugate distances are 



Di = d, 



Oil, 



D2 — di — a2. 



(59) 



