# Analysis of Injected Electron Beam Propagation in a Planar Crossed-Field Gap
## 1 Introduction  
High power microwave crossed-field tubes such as magnetron oscillators [1] and crossed field amplifiers (CFA) [2] are used in many applications, including radars, communication systems, and material processing. Improvements of these devices in terms of power density, phase-locking or control, and faster startup times are of particular interest. Magnetrons have very high efficiency and are also applied in radar and communication systems if phase control can be implemented properly. A recent simulation study [3] showed the feasibility of controlling the phase of a magnetron by modulating the electron injection. Simulating these approaches requires validated electron transport models and an understanding of the stability of crossed-field electron transport under high current density and magnetic field tilt. Electron transport in a microwave crossed-field diode is most commonly explained by assuming mono-energetic electron emission. Additional assumptions, such as zero electric field on the cathode surface, are sometimes introduced [4, 5, 6]. These idealized assumptions allow simultaneous, steady-state solutions to Poisson's equation, the force law, and the continuity equation to be obtained. Prior studies investigated one-dimensional (1-D), crossed-field devices to understand the development of noise [4, 7, 8, 9, 10, 11, 12] and instability [13, 14, 15]; however, beyond comparison to 1-D particle-in-cell(PIC) simulations, the theoretical analysis of current transport in a diode type crossed-field device is limited [4; 16]. The anode current depends on the magnitude of the applied magnetic field \(B\) and the space charge density. The anode current quickly approaches zero once \(B\) approaches and exceeds Hull cutoff, \(B_{H}\)[17], which corresponds to the condition when an electron emitted from the sole just reaches the anode before turning around. The maximum injected beam current before the electron propagation destabilizes for a \(B>B_{H}\) cannot be deduced theoretically and must be calculated numerically; its existence has been corroborated in 1D particle-in-cell (PIC) simulations [5; 6; 12]. The critical current density is defined by the current density beyond which the crossed magnetic field fails to insulate the anode [7]. Another crucial parameter of the crossed-field device is magnetic field misalignment [11]. Several studies have shown that tilting the magnetic field significantly reduces noise compared to a perfectly crossed-field (PCF) gap [18; 19; 20]; however, this tilt also introduces an instability [1] in beam propagation that has not been rigorously studied in a 3-D model. Moreover, these theories are often limited to planar 1-D geometries, not more realistic 3-D. Recent theoretical studies [21] applied variational calculus (VC) and conformal mapping to derive the first exact, closed form solutions to the 1-D space-charge limited current density in spherical [21] and cylindrical [21; 22] coordinates; such an approach may ultimately provide a means to solve for instabilities in crossed-field devices with more realistic geometries. However, despite considerable progress, the perturbation problem in magnetrons [23; 24] remains incompletely solved. Validating electron transport theory with simulation in crossed-field configurations is an essential requirement to correlate to these 1D models to a 3D, practical model. Our research plan includes developing an injected beam, planar, crossed-field device to study electron transport and stability criteria. The first step entails assessing the validity of a 1-D theory by using a 3-D simulation of a practical, feasible geometry. Our study of electron propagation through a planar crossed field gap with different current densities and magnetic field intensities will be used to elucidate stability conditions and to validate the simulations and theoretical models.