## 3 Results and Discussion
### The Results of the Triode Field Emission Structure Simulation
As shown in Figure 3c, linear function fitting was performed, and these results showed that the current of each electrode decreased with increasing grid wire distance. The linear fitting curve in Figure 3d revealed that the electron transmittance remained between 65% and 70% with a slight increase. The simulation results indicate that although the aperture ratio remained constant, the current in each electrode decreased, and the electron transmittance slightly increased. In addition, the simulation results of the electron beam spot were analyzed. Figure 4a shows the simulation results. The voltage was kept constant, and the width of the grid wire was set at 20 \(\upmu\)m. The distance between the grid wires was gradually increased, and grids with different aperture ratios were simulated. A particle monitor was set up on the gate surface near the cathode, which then calculated the electron beam spot size in the x and y directions. When the aperture ratio increased from 44% to 90%, the electron beam spot size in both the x and y directions increased. Specifically, the electron beam spot size in the x direction increased from 1.9413 mm to 2.0042 mm, while the electron beam spot size in the y direction increased from 1.9405 mm to 2.0134 mm. As the aperture ratio of the grid increased, the size of the electron beam spot gradually increased. The electric field  
Figure 3: Simulation results of cathode currents, gate currents, anode currents, and electron transmittance with different grid wire distances. **(a)** Cathode currents, gate currents, and anode currents as the grid wire distance increases from 10 \(\upmu\)m to 600 \(\upmu\)m. **(b)** Variations in electron transmittance as the grid wire distance increases from 10 \(\upmu\)m to 600 \(\upmu\)m. **(c)** Cathode currents, gate currents, and anode currents for different grid wire distances at an aperture ratio of 75%. **(d)** Variations in electron transmittance at an aperture ratio of 75%. intensity on the tip of the cathode was higher, and the ability of electrons to emit from the side was enhanced, increasing the electron beam spot size. At last, the width of the grid wire was set to 20 \(\upmu\)m, and the distance between the grid wires was set to 50 \(\upmu\)m. We set up a particle monitor on the surface of the anode. Figure 4b,c show the relationship between the size of electron beam spots and voltage when only increasing the gate voltage and anode voltage. The simulation results showed that when the gate voltage increased from 1.4 kV to 2.2 kV, the width of the electron beam spot in the x direction increased from 11.5686 mm to 13.6816 mm and the width of the electron beam spot in the y direction increased from 11.5973 mm to 13.7069 mm. As the gate voltage increased, the field emission capability of the cathode increased and more electrons were emitted, leading to an increase in the electron beam spot size. When the anode voltage increased from 1.5 kV to 8.0 kV, the width of the electron beam spot in the x direction decreased from 19.3699 mm to 12.6113 mm and the width of the electron beam spot in the y direction decreased from 21.37351 mm to 12.7133 mm. The variation trend of the electron beam spot in the x and y directions remained consistent. An increase in anode voltage shortened the time of flight of the electrons, thereby reducing the electron beam spot size. Therefore, changing the gate voltage and anode voltage affected the size of the electron beam spot.