Green’s function based space-charge field solver for electron source simulations

We present a theoretical formulation for calculating the electromagnetic space-charge fields within a simplified electron source geometry using time-dependent Green’s function methods. The source geometry is assumed to be comprised of a flat cathode along with a pipe of arbitrary but uniform cross section. Under the assumption that the beam currents are parallel to the pipe axis, we derive exact solutions for the electromagnetic potentials in the Lorentz gauge. In addition, for the special case of a pipe with rectangular cross section, we present the exact solutions of the electromagnetic potentials for arbitrary beam currents. Finally, we show the results of an analytical benchmark study in which the electromagnetic fields that are solved using the Green’s function method are in excellent agreement (<1% error) with the benchmark fields.