Two-dimensional features of virtual cathode and microwave emission

A two-dimensional model for the electron flow in the presence of a virtual cathode is presented. The model allows for electron reflexing and velocity distribution spread. Solutions with substantial radial flow agree closely with high-power microwave emission observed in experiments. The radial motion is the result of the self-magnetic field of the beam, which bends some of the electrons from propagating to side walls to reflex between real and virtual cathodes. The resulting plasma frequency solution agrees with the measured microwave frequency, which scales linearly with the diode current instead of the square-root scaling for a one-dimensional flow. Scaling laws are derived and agree with experimental measurements.