## 2 Design and Simulation  
### Dual-Gridded Field Emission Structure Based on CNT Cold Cathode  
In previous work on conventional gridded triode with planar CNT cold cathodes, non-uniform electric field distributions often appear on the cathode surface [25]. Such devices exhibit large emission current densities immediately beneath the grid, owing to the higher local electric field, with a reduced emission current in the center of grid holes due to the lower effective electric field. This, as a result, allows for greater current interception, and hence losses. Experimental results have shown that low beam transparency may lead to small emission current, high grid losses, arcing, and unwanted grid deformation [23; 25]. In response, here we report on the spatial topology of a dual-gridded CNT field emission system. The dual-gridded system is composed of a cathode substrate, a CNT electron emitter, a first separating grid (FSG), a second separating grid (SSG), and an insulator, as depicted in Figure 1. The CNTs are synthesized directly on the cathode substrate surface. The FSG is attached to the cathode surface to mold the cathode array. The SSG is coaxial with the FSG and controls electron extraction from the CNT electron emitters. The two separating grids are connected with each other coaxially through the insulator. The grid mesh holes are square. The design parameters of the specific dual-gridded system are listed in Table 1. The blue line on the surface of the square array (inset Figure 2a) denotes the probed surface under study. The mesh-hole \(h_{1}\) of the FSG is a dominant factor influencing the electric field distribution at the cathode surface in the present optimization. For brevity's sake, we consider six practically viable \(h_{1}\) values (0.10, 0.11, 0.12, 0.13, 0.14, and 0.15 mm) in the present geometry. Figure 2 shows the electric field distribution along the blue line as function of \(h_{1}\). Correspondingly, the SSG mesh-hole \(h_{2}\) values are 0.11, 0.12, 0.13, 0.14, 0.15, and 0.16 mm, ensuring that the SSG is coaxial with the FSG (\(h_{1}+w_{1}=h_{2}+w_{2}\)). It is evident that in this field emission structure the electric field distribution at the effective emission cathode surface is relatively smooth without significant edge effects. The difference in electric field intensity between the maximum and the minimum is less than 0.5 kV/mm, which is \(<\)11% variation in the electric field from the center field intensity. The simulation results also illustrate that the surface electric field intensity increases as \(h_{1}\) decreases. However, the smaller \(h_{1}\) causes a rise in the electric field at the center of the square cathode surface, as well as reducing the emission cathode area. The larger unit square cathode area causes a drop in the electric field at the center of the square cathode surface. The rise or drop in electric field at the center of the square cathode surface is, nevertheless, a disadvantage to the production of a uniform emission current. Therefore, an optimal \(h_{1}\) is significant to the current design. Our studies suggest an optimal \(h_{1}\) of 0.12 mm, which is used herein to ensure that the electric field distribution on the emission cathode surface remains relatively uniform (\(<\)2.3% variation).  
Figure 1: Schematic of the dual-gridded field emission structure based on carbon nanotube (CNT) cold cathode. SSG: second separating grid; FSG: first separating grid.