Space-Charge Limitations on the Focus of Electron Beams

A calculation has been made of the effect of space charge in the region between the lens and the focal point on the focus of electron beams of rectangular and circular cross section under conditions of zero velocity of electron emission and with the lenses free from spherical aberration. The results show that for an electron beam of circular cross section having a given current and voltage, and included in a cone of a given initial angle, there is a resultant minimum beam diameter at the focal point a given distance from the lens which cannot be reduced by changing the radial force, or focusing component, of the lens. The value of the minimum beam diameter is nowhere zero and it increases more rapidly than the distance between the lens and focal point. Thus, for the production of a television picture with a given angle of deflection the definition should improve as the screen approaches the lens. For a beam of circular cross section the factors determining this minimum spot size at a given distance from the lens are the initial beam radius R
o
, the beam velocity V
d
, and the current I. The spot size may be reduced only by increasing R
o
, increasing V
d
, or decreasing I. For a rectangular beam with one dimension infinite, the minimum beam thickness depends upon the perpendicular force or focusing component supplied by the lens.