## 2 Simulations of 5 th-Harmonic Multiplication with Parameters of a 250-GHz High-Power Gyrotron
\(\omega_{r}^{c}(Z)=u_{mn,n_{0}e}/R(z)\) describes the axial dependence of the cut-off frequency, \(R(z)\) is the profile of the gyrotron cavity which includes the cutoff neck, the regular part with \(R(z)=R_{0}\), and the collector widening, \(I_{n}=64\frac{eI_{h}}{m_{e}c^{3}}\frac{\overline{B}_{||0}\beta_{\perp 0}^{2(n_{ 0}-4)}}{\gamma_{0}}s_{n}^{3}\ \left(\frac{s_{n}^{2}}{2^{n}s_{n}! }\right)^{2} \frac{I_{mn-n_{0}e}^{2}\left(v_{mn,n_{0}e}R_{h}/R_{0}\right)}{(v_{n}^{2}-m_{n} ^{2})\overline{I}_{mn}\left(v_{mn,n_{0}e}\right)}\) is the field excitation factor for the beam with an injection radius of \(R_{b}\) and a current of \(I_{b}\). \(\sigma_{n}=4\beta_{||0}^{2}\beta_{\perp 0}^{-4}Q_{n}^{-1}s_{n}^{2}\) is the parameter of Ohmic losses, \(Q_{n}\) is the Ohmic Q-factor. The radiation power in each mode can be found based on the following relation: \(P_{n}[\delta W]=5\mathrm{i}\mathrm{L}\mathrm{/}\mathrm{\overline{6}}\cdot \left[A\right]\cdot\left(\gamma_{0}\beta_{\perp 0}^{2}/G_{n}\right)\cdot\mathrm{Im}(a_{n} \cdot\partial\alpha_{n}^{\prime}/\partial Z)\right|_{Z=L}\) where \(L=\beta_{\perp 0}^{2}\omega_{H}^{0}/2\beta_{||0}c\) is the normalized length of the interaction space. The motion equations in the system (3) have to be supplemented with standard boundary conditions corresponding to electrons uniformly distributed over the cyclotron rotation phases: \(p(Z=0)=p_{0}\exp(i\theta_{0})\), \(\theta_{0}\in[0,2\pi)\). For the mode amplitudes at the left (cathode) edge of the system \(Z=0\), we use zero boundary conditions corresponding to the cutoff neck. At the output cross-section \(Z=L\) located in the collector widening outside the region of interaction with electrons, we apply non-reflection boundary conditions:  
\[a_{n}(\tau,L)+\frac{1}{\sqrt{i\tau}s_{n}}\int\limits_{0}^{\tau}\frac{e^{-i \delta_{n}^{-1}(i\Delta_{n}+i\delta_{n}(L)+\sigma_{n})(\tau-\tau^{\prime})}}{ \sqrt{\tau-\tau^{\prime}}}\frac{\partial a_{n}(\tau^{\prime},L)}{\partial Z}d \tau^{\prime}=0 \tag{4}\]  
This condition means matching with the output waveguide, where all modes propagate far from the cutoff frequencies. The results of simulations are presented in Figure 2. Figure 2a shows the operation zones for the operating \(\mathrm{TE}_{19,8}\) mode, the most dangerous parasitic \(\mathrm{TE}_{-17,9}\) mode at the 1 st cyclotron harmonic and for the \(\mathrm{TE}_{95,37}\) mode at the 5 th cyclotron harmonic. One can see that the zone optimal for excitation of the \(\mathrm{TE}_{95,37}\) mode is partially overlapped by the zone of parasitic \(\mathrm{TE}_{-17,9}\) mode generation. However, there is a region of magnetic fields in which effective excitation of the 5 th CH is possible. Figure 2b demonstrates the establishment of the regime of stationary generation for the intensity of the magnetic field of 98.4 kOe. It should be noted that in this simulation, the level of initial azimuthal modulation, which is the same for all competing modes, is maintained in the beam during the entire time of the onset of oscillations. Nevertheless, the operating mode \(\mathrm{TE}_{19,8}\) (blue curve) wins in the competition, despite the proximity of parasitic mode \(\mathrm{TE}_{-17,9}\) (green curve). The red curve corresponds to the level of CW excitation at the 5 th cyclotron harmonic. According to simulations, the power of radiation at the 5 th CH (i.e., at the frequency of 1.25 THz) can reach 8W. Taking into account the spread in velocities of about 15-20%, which is typical for magnetron-injector guns of gyrotrons, the radiation power is reduced (orange curve in Figure 2a). However, according to the simulation, it remains at the level of several Watts (\(K\approx 1.5\cdot 10^{-4}\)).