20 BELL SYSTEM TECHNICAL JOURNAL 



A more convenient expression for magnification is now obtained by 

 combining the two preceding expressions to give 



and from equation 61 



^=ftg-: (^^) 



M = - xlTT^-Ff- (6/) 



The magnification is not in general equal to the ratio of the image 

 distance to the object distance, as it is for an optical lens in air. It 

 is only equal to that ratio when the voltage is the same on both 

 sides of the lens. 



Section III — Aberration in a Lens 

 Returning to the first section, we see that the general expression 

 for focal distance is 



1 ^ + 5^2 + Cr* • • • 



^ V2^ + ^',2+^V 



(20) 



The exact focal distance of an electron thus depends on its radial 

 coordinate r, and a ray passing through a lens at a distance from the 

 axis does not come to the same focus as a ray near the axis. A precise, 

 general theory for rays at a distance from the axis could — in theory 

 at least — be derived by solving the differential equations for as many 

 of the higher coefficients as desired and substituting them in the 

 above equation. Such a general solution would, however, be very 

 difficult indeed, and one is content — as he usually is in optics — to 

 treat the performance of a lens in a much more restricted manner. 



The equation for focal distance can be simplified to some extent by 

 noting that its denominator is the velocity component z. With the 

 aid of the energy equation 6, this component can be written in the form 



= 42v 



^ + (!)?"• 



In most lenses r is small compared to d, and the last factor in the 

 above equation may be approximately set equal to unity. This 

 approximation is accurate to one per cent even for a lens with an 

 angular aperture corresponding to F3.5 — an F2 lens is a very fast 

 camera lens. With this approximation the inverse focal distance is 



1 A ^- Br'' + Cr^ + ■•• 



2-- VI (*'> 



