## 2 Simulations of 5 th-Harmonic Multiplication with Parameters of a 250-GHz High-Power Gyrotron  
The possibility of reaching the THz range based on a five-fold frequency multiplication is obviously associated with the use of higher frequency gyrotrons than in the experiment [21]. Simultaneously, since the conversion coefficient remains rather low, it is necessary to use high-power fundamental-harmonic gyrotrons. Currently, the most powerful MW-level radiation is generated by gyrotrons intended for plasma applications, which typically operate up to 170 GHz [8]. At the same time, gyrotrons with operating frequencies of about 250 GHz are in demand for controlled thermonuclear fusion installations of the DEMO type. Note here that such gyrotrons typically exploit very high-order transverse modes. Thus, conditions of the double resonance (1) should be met with greater accuracy. Further, we demonstrate in simulations that efficient 5 th CH multiplication can be obtained in the DEMO-prototype gyrotron [9] which has been recently developed at IAP RAS (Figure 1a). The gyrotron provides the output power of 200 kW in CW operation regime at an accelerating voltage of 55 kV and an electron beam current of 12.5 A (see Figure 1b). The operating TE\({}_{19,8}\) mode is excited at the fundamental cyclotron resonance. In accordance with the selection rules (1) and (2), the effective excitation of the 5 th cyclotron harmonic is expected at the mode TE\({}_{95,37}\). The accuracy of condition (1) in this case is \((v_{5m_{1},45}-5v_{m_{1},d_{1}})/5v_{m_{1},d_{1}}\approx 0.02\%\). For simulations, the time-domain multi-mode averaged model [20] was used. According to this model, the electric field of each mode in the interaction space (smoothly tapered cylindrical waveguide) is presented in the form\(\overset{\rightarrow}{E}_{n}=\text{Re}(A_{n}(z,t)\overset{\rightarrow}{E}_{ \perp}(r)\exp(is_{n}\omega_{H}^{0}t-im_{n}\varphi))\), where \(A_{n}(z,t)\) is the slowly varying amplitude of the \(n\)-th mode, \(\overset{\rightarrow}{E}_{\perp}(r)\) is its transverse structure, \(\varphi\) is the azimuthal angle. The electron-wave interaction is described by the following system with the parabolic equations standing for the evolution of mode amplitudes \(a_{n}\) and non-isochronous oscillator equations for the complex transverse momentum of electrons :  
\[ \begin{array}{c}i\frac{\partial^{2}a_{n}}{\partial Z^{2}}+s_{n}\frac{\partial a _{n}}{\partial\tau}+(i\Delta_{n}+i\delta_{n}(Z)+\sigma_{n})a_{n}=i\frac{I_{n}} {4\pi^{2}}\int\limits_{0}^{2\pi}\left\langle p^{s_{n}}\right\rangle_{0}e^{i(m _{n}-s_{n})\varphi}d\varphi,\\ \frac{\partial p}{\partial Z}+\frac{\overset{\rightarrow}{E}_{0}^{2}}{ \partial\tau}\frac{\partial p}{\partial\tau}+ip\big{(}|p|^{2}-1\big{)}=i \frac{\Gamma}{n}a_{n}(p^{*})^{s_{n}-1}e^{-i(m_{n}-s_{n})\varphi},\end{array}  \tag{3}\]  
Figure 1: (a) General view of the 250 GHz gyrotron with TE\({}_{19,8}\) operating mode. (b) Experimentally obtained dependence of the output power and efficiency on the value of the beam current. where \(a_{n}=\frac{eA_{n}I_{mn-nm}\left(v_{mn,n_{0}e}R_{h}/R_{0}\right)}{mc\omega_{H}^{0}} \frac{s_{n}^{n}}{2^{n}-s_{n}! }\frac{\beta_{\perp 0}^{n-4}}{\gamma_{0}}\), \(Z=\frac{\beta_{\perp 0}^{2}}{2\beta_{\parallel 0}}\frac{\omega_{l}^{0}}{c}z\), \(\tau=\frac{\beta_{\perp 0}^{4}}{8\beta_{\parallel 0}^{2}}\omega_{H}^{0}t\), \(\beta_{\perp 0}=V_{\perp 0}/c\) and \(\beta_{||0}=V_{||0}/c\) are values of transverse and axial electron velocities at the entrance of the interaction space, \(g_{0}=\beta_{\perp 0}/\beta_{||0}\), \(\Delta_{n}=\frac{8\beta_{||0}^{2}s_{n}}{\beta_{\perp 0}^{4}}\frac{s_{n} \omega_{H}^{0}-\overline{c}_{n}^{n}}{\omega_{n}^{n}}\), \(\delta_{n}(Z)=\frac{8\beta_{||0}^{2}s_{n}^{2}}{\beta_{\perp 0}^{4}}\frac{ \overline{c}_{n}^{n}-\omega_{H}^{0}(Z)}{\overline{c}_{n}^{n}}\) are cyclotron and geometrical detunings.