Effects of space charge and lens aberrations in the focusing of an electron beam by a solenoid lens

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Abstract
The effects of space charge and lens aberrations on a 5-keV, 80–200-mA electron beam focused by a solenoid lens are investigated theoretically and experimentally. A third-order radial trajectory equation is derived with space charge added in the form of discrete annular current elements. This equation is numerically solved for the trajectories of focused electrons in a short solenoid lens. Trajectories launched near the axis are ‘‘reflected’’ from the axis at the waist due to space charge, and trajectories launched farther from axis, cross the axis. The resulting discontinuity is best illustrated in a transverse phase-space plot where it may be described as a ‘‘tearing’’ of the electron distribution. Due to the combined action of the nonlinear lens forces and the space charge, the initially uniform radial current density distribution becomes markedly nonuniform (hollow or peaked depending on distance) downstream from the lens. Detailed experimental measurements of the density profiles at different positions are found to be in very good agreement with computations.