# Design Studies of Re-Entrant Square Cavities
## 3 Simulation Results  
After calculating various dimensional parameters of the preliminary design of the RF section, we go for cold simulation using the Eigenmode solver of the CST microwave studio. The advanced Krylov Subspace (AKS) method is used to find the lowest order Eigenmodes for the specified re-entrant boundary condition of our RRSCs. Figure 3a,c show the electric field patterns of the operating mode inside the RRSCs. This mode is favorable for velocity modulation because the electric field maxima occur at the cavity's center, along the beam axis. However, there are fringing fields near the cavity gap but the majority of the field remains parallel to the axis. The corresponding magnetic fields are confined in the orthogonal plane, forming closed loops following the right-hand rule, as shown in Figure 3b,d. The side walls of the cavity are the region of an enormous magnetic field and a small electric field. In contrast, the gap is a region of tremendous electric and small magnetic fields. Thus, the gap is equivalent to a capacitor, while the cavity walls provide inductance. As the resonant RF field oscillates, the electric energy converts to magnetic energy, and vice versa. The conduction current flows along the inner wall and completes the loop through the displacement current flowing, employing a time-varying electric field through the gap. In a PIC simulation, an electron beam with its energy corresponding to the applied beam voltage is launched into the R.F. Section. A velocity-modulated electron beam from the input cavity (due to the R.F. signal introduced through the input port) enters the second cavity and subsequent cavities. Because these cavities are unloaded (except for the electron beam) and have higher Q-factors, the beam-induced R.F. field is stored inside the cavities for several resonant cycles, which amplifies the velocity modulation of the beam. The velocity modulation of the beam is gradually increased. The velocity modulation at different time steps (represented by frames) is recorded by phase space monitors in the z-direction (across the beam propagation direction), shown in Figure 4.  
After the cold simulation, the performance of the RRSC Klystron is evaluated by the particle in cell (PIC) solver of CST microwave studio [23]. This solver computes the non-linear beam-field interaction in vacuum electronics devices. Copper (Cu) is used as cavity material for high thermal and electrical conductivity. The cylindrical electron beam radius \(r_{b}\) = 0.15 mm is determined according to Equation (8), considering 20 KeV energy, and 0.2 A current with beam tunnel radius \(r\) = 0.22 mm. Figure 2: Perspective view of RRSC Klystron in CST microwave studio suite. cavities, transforming the continuous electron beam into electron bunches. At the output cavity, this bunched beam induces a field of the same frequency as the input RF signal. This induced field grows in amplitude at the expense of the beam's kinetic energy, which is ensured by the proper placement of the output cavity. The simulation result predicts 256 W peak power at the output cavity, corresponding to a gain of 27.17 dB. Figure 6 shows how the power gain changes as the number of cavities increases. The RF signal power gain increases proportionally with the rise in cavity numbers. With two cavities, the RRSC Klystron exhibits attenuation of the input signal represented by a negative gain of \(-5.39\) dB; however, with a total of eleven cavities, it exhibits 27.14 dB gain. We scanned frequencies ranging from 60 to 60.25 GHz with a step size of 0.025 GHz. The stable output signal power for each frequency point for a specific frequency range  
Figure 5: RF input signal and amplified output signal at the input and output cavity ports, respectively. Figure 6: Plot of output power gain vs total number of cavities. is shown in Figure 7. At 60.1 GHz, the highest output signal power is 256 W, and the saturated gain is 27 dB with a 3 dB bandwidth of more than 150 MHz.