## 3 PIC Simulation and Analysis  
A dual-beam, dual-frequency EIO hot test model was established using simulation software CST studio and is being researched for its beam-wave interaction using particle-in-cell (PIC) simulation. Taking into account machining precision and ohmic losses, the metal material is set to copper with a conductivity of 2.8 \(\times\) 10\({}^{7}\) S/m. To simplify the simulation model, the cathode emission surface was simplified as a rectangular metallic object, and a DC emission mode is used with an electron beam size of 5 \(\times\) 0.5 mm and an injection current of 3 A at a voltage of 34 kV at 94 GHz; at 140 GHz, the dimensions were 1.5 \(\times\) 0.2 mm, with an injection current of 3 A and a voltage of 56 kV.  
Due to the structural characteristics of the dual-electron-beam channel, in addition to the co-directional emission of electron beams used in this study, counter-directional emission can also be adopted. Regardless of the emission method, the dual electron beams are effectively modulated within the interaction cavity, achieving the bunching phenomenon. The following Figure 5 presents the particle simulation results for both emission methods.  
The key difference between the two lies in their transmission directions: in the co-directional emission, both electron beams travel along the +z direction, are continuously  
Figure 4: The \(Q_{e}\) of 2\(\pi\) mode with different values of the coupling slot ax.  
Figure 5: Output signal of (**a**) co-direction emission and (**b**) counter-direction emission.  
modulated, and achieve effective bunching at the end of the circuit, converting their energy into high-frequency field energy. In the counter-directional emission, the electron beams travel along the +z and \(-\)z directions, respectively, and are modulated along their respective transmission paths. Effective bunching occurs at the front and end of the circuit, where energy is then converted into high-frequency field energy.  
Additionally, the counter-directional emission poses a greater challenge to the pre-stage electron optical system due to the opposite emission directions of the electron beams. However, regardless of the position in the circuit where the electron beams transfer their energy to the high-frequency field, the electromagnetic field working in the cavity as a whole is ultimately amplified.  
The output power signals of EIO operating independently at 94 GHz and 140 GHz are shown in the inset of Figures 6 and 7, with the signal stabilizing and amplifying within 140 ns. The corresponding spectrum graph is shown, where the output signal spectrum is clean, with no competing mode spectra observed.  
Figures 8 and 9 show the electron energy distribution of electron bunching and the energy of the sheet beam along the z direction at the 60 ns. It can be seen that in the latter part of the EIO, the modulated electron beam experiences a decrease in electron velocity, resulting in bunching, and releasing energy. The electron bunching diagram shows that the electrons are well modulated as they travel within the cavity, indicating good beam-wave interaction. From the electric field distribution diagram, it can be seen that in the interaction gap, the electric field distribution within the electron beam channel is fairly uniform.  
The curves of the circuit output characteristics as a function of voltage are shown in Figures 10 and 11. The 94 GHz and 140 GHz electron beams can both operate stably within voltage ranges of 32-35 kV and 46-58 kV, respectively. Simulation results indicate that the output power of the interaction circuit generally increases with the increase in beam voltage. Specifically, the 94 GHz circuit output power increases from 4.1 kW to 6.7 kW, and the beam-wave interaction efficiency rises from 3.8% to 6.6%. For the 140 GHz circuit, the output power increases from 10 kW to 42 kW, and the beam-wave interaction efficiency  
Figure 6: Output power and spectral analysis at 94 GHz.  
Figure 7: Output power and spectral analysis at 140 GHz.