## 3 Results and Discussion
### The Electron Transmittance Measurements  
The results of electron transmittance field emission experiments are analyzed in this section. The change trend of the electrode current is shown in Figure 5a, and the change trend of electron transmittance is shown in Figure 5b. As shown in Figure 5a, when the grid aperture ratio increased from 44% to 56%, there was a slight increase in cathode current. The increase in the aperture ratio enhanced the electric field at the cathode surface. However, the cathode current decreased continuously as the aperture ratio increased from 56% to 90%. As the distance between the gate filaments increased, the number of electrons intercepted by the gate filaments decreased, the gate current decreased, and the projected area of the gate filaments on the cathode decreased. As a result, the excitation ability of the electric field to the electrons was weakened, leading to a reduction in the field emission current. As the gate current decreased, the anode current increased, resulting in a continuous increase in electron transmittance. When the grid aperture ratio reached 60%, the anode current decreased and gradually became saturated. After the grid aperture ratio was greater than 75%, the current intercepted by the gate also gradually became saturated, and the electron transmittance increased significantly. Figure 5b shows the trend of the electron transmittance with the grid aperture ratio in both the experiments and simulation. The  
Figure 4: Simulation results of electron beam spot size for different aperture ratios, gate voltages, and anode voltages. (**a**) Electron beam spot widths in the x and y directions as the aperture ratio increases from 44% to 90%. (**b**) Electron beam spot widths in the x and y directions as the gate voltage increases from 1.4 kV to 2.2 kV. (**c**) Electron beam spot widths in the x and y directions as the anode voltage increases from 1.5 kV to 8.0 kV. error of electron transmittance between simulation and actual testing was due to the actual grid aperture ratio being greater than the design value. The grid wire may melt during laser processing, causing the actual width of the grid wire to be smaller than the design value, resulting in an actual grid aperture ratio greater than the design value. Therefore, the experimental electron transmittance was higher than that in simulation. Although there were some differences between experiment and simulation, the linear fitting curves revealed that the electron transmittance and the grid aperture ratio approximate a linearly increasing relationship. The currents of three electrodes at different anode voltages are shown in Figure 5c, and the corresponding electron transmittance is plotted in Figure 5d. The grid aperture ratio and gate voltage were kept constant. As shown in Figure 5c, both anode current and electron transmittance were minimal at a low anode voltage. The cathode current remained essentially constant as the anode voltage increased. However, as the anode voltage approached the gate voltage, a significant number of electrons were propelled through the grid hole by the high-voltage electric field and reached the anode, rapidly increasing both the electron transmittance and anode current. Upon reaching 3.7 kV, the anode current approached saturation, indicating that further increases in voltage had little impact on the field emission current of the cathode. Nevertheless, an increase in the anode voltage resulted in a higher anode current and electron transmittance. Overall, these findings demonstrated that when the gate voltage was held constant, the anode voltage had minimal influence on the emission current of the cathode field. However, a higher anode voltage could enhance the electron transmittance and anode current. Figure 5: Measurement results of cathode currents, gate currents, anode currents, and electron transmittance. (**a**) Cathode currents, gate currents, and anode currents as the grid aperture ratio increases from 44% to 90%.