## 2 Design and Simulation
### A 0.22 THz BWO Based on Dual-Gridded CNT Cold Cathode Electron Gun
At the operating frequency, the velocity of the electron beam is synchronous with the phase velocity of the backward wave, which is approximately equal to the slope of the beam line. A completely symmetric, disk-loaded waveguide SWS structure is modeled to study the reflection and transmission characteristics of the terahertz VED. 20 periods are considered. The \(S\)-parameters are calculated by the transient solver in the CST Microwave Studio. Figure 6d shows the magnitudes of the calculated \(S\)-parameters. There are 19 resonant frequencies--corresponding to the 0-mode, \(\pi\)/18-mode, 2\(\pi\)/18-mode, 3\(\pi\)/18-mode, 4\(\pi\)/18-mode, 5\(\pi\)/18-mode, 6\(\pi\)/18-mode, 7\(\pi\)/18-mode, 8\(\pi\)/18-mode, 9\(\pi\)/18-mode, 10\(\pi\)/18-mode, 11\(\pi\)/18-mode, 12\(\pi\)/18-mode, 13\(\pi\)/18-mode, 14\(\pi\)/18-mode, 15\(\pi\)/18-mode, 16\(\pi\)/18-mode, 17\(\pi\)/18-mode, and \(\pi\)-mode--observed in both \(S_{11}\) and \(S_{12}\). As per frequency relation formula [29]:  
\[f=\sum a_{m}\cos(m\beta L), \tag{2}\]  
these frequencies correspond to a phase shift \"\(\beta L\)\" of 0, \(\pi\)/18, 2\(\pi\)/18, 3\(\pi\)/18, 4\(\pi\)/18e, 5\(\pi\)/18, 6\(\pi\)/18, 7\(\pi\)/18, 8\(\pi\)/18, 9\(\pi\)/18, 10\(\pi\)/18, 11\(\pi\)/18, 12\(\pi\)/18, 13\(\pi\)/18, 14\(\pi\)/18, 15\(\pi\)/18, 16\(\pi\)/18, 17\(\pi\)/18, and \(\pi\), respectively. The frequency passband of the disk-loaded waveguide BWO is largely discrete. As can be seen, \(S_{11}\) remains at about \(-\)38 dB around the operating frequency whilst \(S_{21}\) remains at approximately 0 dB, suggesting that the disk-loaded waveguide SWS is indeed well-suited for terahertz BWO applications. Figure 6: Electric field distribution of the TM\({}_{01}\) mode on the cross section at the center of (**a**) transverse-plane and (**b**) longitudinal-plane. (**c**) The TM\({}_{01}\) mode dispersion curve as a function of the normalized wavenumber. (**d**) Transmission characteristic. The nonlinear interaction between the electron beam and electromagnetic wave in the disk-loaded waveguide BWO is simulated. Some typical parameters, including output power and operating frequency, are obtained. To make the BWO more practical, a simple matched output circular waveguide is designed, as shown in Figure 5. Unlike the traveling-wave tube (TWT), the BWO does not require an input signal. The output signal is produced by the electron beam during transit. In this structure, the electron beam has a radius of 0.2 mm and an emission current of 50 mA, with an electron beam radius of 0.18 mm having been obtained upon optimization. Typical particle-in-cell (PIC) results of the BWO are shown in Figure 7, with the operating beam voltage set to 21 kV. As is shown in Figure 7a, after stable oscillation, a 32 W, 220 GHz output wave is obtained. Figure 7b,c presents the output signal spectrum showing mode purity and electron beam bunching. As an ordinary type (O-type) VED, BWO employs a magnetically focused electron beam. With the beam-wave interaction between the electron beam and the specify frequency of electromagnetic wave, electron bunching occurs in the SWS, as shown in Figure 7c. The energy extracted from the slowing electrons is transferred to the electromagnetic wave.