## 2 Design and Simulation
### High-Compression-Ratio CNT Electron Gun Based on Dual-Gridded System  
There remain many challenges in the realization of terahertz BWOs based on CNT cold cathodes. Here, a high-compression-ratio CNT electron gun based on the mentioned dual-gridded system above is designed and investigated to overcome many of these limitations. The dual-gridded CNT cold cathode electron gun has been simulated herein using commercially available 3D simulation software (CST). The basic field emission process from such semi-metals is well described by the generalized Fowler-Nordheim equation, in the following form:  
\[J=AE^{2}\exp(-B/E). \tag{1}\]  
Here, \(A\) and \(B\) are the approximate Fowler-Nordheim constants, which are estimated by numerical fitting based on the previous experimental results [26]. Applying our fitting procedure (R\({}^{2}\) = 0.9897), we find \(A\) = 5 \(\times\) 10\({}^{-8}\) A/V\({}^{2}\) and \(B\) = 1.62 \(\times\) 10\({}^{7}\) V/m. In our previous experimental study [26], we have investigated field emission (FE) of a vertically aligned, well-ordered carbon nanotube (CNT) mesh (VACM) by growing the CNTs in a network with hexagonal holes as shown in the inset (a) of Figure 3. The synthesized carbon nanotubes are uniform, dense, well aligned, and perpendicular to the substrate as shown in the inset (b) of Figure 3. The FE properties of the emitters were determined using a simple diode configuration in a vacuum chamber at a pressure of 1 \(\times\) 10\({}^{-6}\) Pa. Emission area of 16 mm\({}^{2}\) was used. A maximum operating emission current density of 1.5 A/cm\({}^{2}\) was obtained at a field of less than 4.5 kV/mm. In this paper, we adopt the VACM as cathode emitters on the smaller substrate surface of 3.91 mm\({}^{2}\) at a higher vacuum degree of 1 \(\times\) 10\({}^{-7}\) Pa. Figure 3 shows typical experimental data and the resulting fitted emission characteristics. It indicates that the dual-gridded electron gun exhibits a maximum emission current of 50 mA at the grid voltage of 0.5 kV (the electric field on the per cathode unit surface is 4.17 kV/mm); the corresponding emission current density \(J\) is 1.5 A/cm\({}^{2}\) when the effective emission area \(S_{e}\) is 3.33 mm\({}^{2}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbols** & **Parameters** & **Values and Units** \\ \hline \(r_{c}\) & Cathode substrate radius & 1.116 mm \\ \(h_{1}\) & Mesh hole of the FSG & 0.12 mm \\ \(h_{2}\) & Mesh hole of the SSG & 0.13 mm \\ \(w_{1}\) & Wire line of the FSG & 0.02 mm \\ \(w_{2}\) & Wire line of the SSG & 0.01 mm \\ \(t_{1}\) & Thickness of the FSG & 0.02 mm \\ \(t_{2}\) & Thickness of the SSG & 0.02 mm \\ \(d_{1}\) & Distance between CNT and SSG & 0.1 mm \\ \(U_{c}\) & Cathode voltage & 0 \\ \(U_{2}\) & SSG voltage & 0.5 kV \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of dual-gridded field emission structure based on CNT cold cathode. Figure 2: Surface electric field intensity as a function of first separating grid mesh hole \(h_{1}\). (a) Electric field intensity at square cathode surface. Inset: Schematic drawing of dual-gridded system. (b) Expanded view of the central unit CNT cathode surface electric field intensity. As shown in Figure 4a, the CNT electron gun comprises a dual-gridded system, a focusing electrode, and an anode. The dimensions of dual-gridded system are listed in Table 1. The focusing electrode is connected to the SSG, and the distance between the cathode and anode is 8 mm. In order to obtain a high-compression-ratio electron beam, a suitable axial magnetic field is required. Figure 4a shows the simulated beam trajectories under a field of 0.95 T operating at 21 kV/50 mA. A beam transparency of 100% is obtained, as shown in Figure 4b. Figure 4 e shows the profile of the magnetic field intensity along the central axis.