I. Introduction
Relativistic magnetrons are among the most powerful and compact sources of microwave radiation, and are continuing to receive considerable attention by researchers [1]. In our earlier work, using particle-in-cell (PIC) simulations, we proposed the transparent cathode to replace the traditional solid cathode with cylindrical profile in magnetrons to provide faster start of oscillations, higher efficiency, and when combined with the appropriate value of magnetic field to eliminate mode competition [2]. These results were validated in experiments with the A6 magnetron with radial extraction [3]. Subsequently, in PIC simulations, we investigated using the transparent cathode in a magnetron with diffraction output (MDO) and demonstrated record efficiency of about 70% [4]. In recent MDO experiments, 63% efficiency was demonstrated until end-cap failure occurred; further experiments are planned [5]. The transparent cathode emerged from our desire to allow the RF electric field to reach the axis of the interaction space of the magnetron as opposed to going to zero on the surface of a solid cathode, so that the radial drift velocity of electrons, which is the velocity that transfers electron energy to the RF field, would be much stronger. The MDO with virtual cathode (VC) goes one step further by completely eliminating a physical cathode from the interaction space. This makes the MDO with VC immune from electron bombardment, pulse shortening, decreasing electron efficiency, and frequency shift attributed to expanding cathode plasma.