## 2 Design and Simulation
### A 0.22 THz BWO Based on Dual-Gridded CNT Cold Cathode Electron Gun  
Backward wave oscillator (BWO) is one of the most established VED sources of THz radiation. A BWO is a VED that is used to generate microwaves up to the THz frequency range. By adjusting the beam voltage, it tunes fully and rapidly over a very wide band of frequencies [1]. The basic idea of the BWO is to use the electron beam interacting along their path with a periodic slow wave structure (the first space harmonic). When the electron beam passes the periodic structure, it initiates a wave propagating in backward direction. If the total phase delay in the loops is \(\theta_{n}=n\cdot 2\pi\), a backward wave oscillation can start [27]. When the phase velocity of the normal mode of the SWS equals that of the drifting electrons, each electron feels an accelerating or decelerating force, depending on the local phased of the wave. As a consequence, the beam bunches, but there is no net exchange of energy, because equal numbers of electrons speed up and slow down. If the electrons are initially somewhat faster than the wave, however, more electrons are decelerated than accelerated, and a resonant transfer of energy to the wave occurs in addition to the bunching. In BWO of this type, the exchange of energy causes the wave to grow in amplitude, and the bunching enforces the coherence of the radiation [28]. Figure 5 shows the scheme of the inner structure of the proposed CNT terahertz BWO device. The BWO consists of a dual-gridded CNT cold cathode electron gun, a beam-wave interaction slow wave structure (SWS), a collector, an output circular waveguide, an output window, and magnetic circuit. The beam-wave interaction SWS is a disk-loaded waveguide. The operating parameters of the BWO are listed in Table 2.  
Firstly, electrons are emitted from the CNT cold cathode under the action of the applied field between the second separating grid and cold cathode. Electrons will be compressed and as a result form an electron beam under the electric and magnetic fields, while they are accelerated by the anode into the beam tunnel. Here, the magnetic field will gradually increase and electric field will not change, resulting in magnetic adiabatic compression of the beam. After the electron beam is compressed in the beam tunnel, it enters the beam-wave interaction SWS, where it initiates and interacts with the TM\({}_{01}\) mode thusly generating electromagnetic radiation. The disk-loaded waveguide structure, as a promising SWS, has attracted wide attention for terahertz VEDs, since it is geometrically suitable to be realized by modern microfabrication processes. The CST eigenmode simulation in Microwave Studio was applied to study the high-frequency characteristics of the disk-loaded waveguide BWO by setting the electric field and periodic boundary to the transverse and longitudinal direction, respectively. Here, the TM\({}_{01}\) mode has been selected as an operating mode, as the cylindrical axial electric field \(Ez\) of the TM\({}_{01}\) mode is concentrated in the center of the SWS, as shown in Figure 6a,b. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbols** & **Parameters** & **Values and Units** \\ \hline \(d\) & Circular waveguide inner diameter & 1.054 mm \\ \(t\) & Disk inner diameter & 0.4 mm \\ \(w\) & Disk width & 0.076 mm \\ \(p\) & Period length & 0.3044 mm \\ \(f\) & Operating frquency & 0.22 THz \\ \(U\) & Operating voltage & 21 kV \\ \(l\) & Operating current & 50 mA \\ \hline \hline \end{tabular}
\end{table} 
Table 2: BWO operating parameters. Figure 5: Cross-sectional illustration of dual-gridded CNT, cold-cathode, terahertz, disk-loaded waveguide backward wave oscillator (BWO). The dispersion curve of the TM\({}_{01}\) mode is plotted as a function of the normalized wavenumber, as shown in Figure 6c. The 21 kV beam line (dashed red line) is superimposed onto the dispersion curve, intercepting the dispersion curve at an operating frequency of 220 GHz.