## 1 Introduction  
Multiple applications, including high-resolution spectroscopy, remote sensing, communications, medicine, and security [1, 2, 3], require continuous-wave (CW), compact and powerful radiation sources operating in the terahertz gap. Lack of powerful radiation sources is especially critical in the 0.5 to 3 THz band, which is \"too low\" for quantum electronic devices. Among vacuum electronics devices, which include backward wave oscillators, klystrons [3, 4], vircators [5, 6], etc., gyrotrons [7, 8, 9, 10, 11, 12, 13, 14] are reported to successfully operate at 0.1 to 1 THz [10], and even beyond with high output power. This is due to the fact that gyrotrons are based on interaction of electrons rotating in homogeneous magnetic fields with fast waves excited in smooth cylindrical waveguides. Gyrotrons are usually energized by weakly relativistic electron beams; therefore, they are much more compact than, for example, free-electron lasers. In gyrotrons, a helical electron beam excites TE\({}_{m,q}\) modes of a cylindrical waveguide at the frequencies close to the cutoff ones defined as \(\omega_{c}=v_{m,q}/R\) (\(v_{m,q}\) is the \(q\)th root of the equation \(J^{\prime}_{m}(x)=0\), \(J_{m}(x)\), \(J^{\prime}_{m}(x)\) are the \(m\)th order Bessel function and its first derivative, \(m\) and \(q\) are azimuthal and radial indexes of the operating mode). Thus, the condition of cyclotron resonance has the form \(\omega_{c}\approx s\omega_{H}\), where \(\omega_{H}=eH/mc\gamma\) is the gyrofrequency, \(\gamma\) is the relativistic mass factor of electrons, \(s\) is the number of a radiated cyclotron harmonic (CH). Accordingly, operation of weakly relativistic (\(\gamma\sim 1\)) gyrotrons at 1 THz with excitation of the 1 st cyclotron harmonic (\(s=1\)) requires a magnetic field of about 36 T achievable only in pulsed field magnets. State-of-the art steady-state magnets (cryomagnets [15]) produce the magnetic field of 10-15 T; thus, high harmonic operation (\(s>1\)) appears to be the sole solution for producing terahertz radiation in CW regime. It should be noted that there are two approaches for harmonic operation in gyrotrons, namely, selective excitation of a resonant mode at a given cyclotron harmonic and the effect of frequency multiplication. The first method makes it possible to provide a sufficiently high efficiency of the electron-wave interaction. However, in this case, the increase in frequency is limited by strong competition with modes at the fundamental cyclotron resonance. As a result, at frequencies above 100 GHz, selective excitation at the 2 nd CH was achieved in conventional CW gyrotrons [11; 13] with frequencies up to 0.5 THz, while so-called large-orbit gyrotrons (LOGs) provide CW operation at the 3 rd harmonic [14] with frequencies up to 0.4 THz. The effect of frequency multiplication [16; 17; 18; 19; 20] arises due to non-linear properties of the beam of rotating electrons; each of them represents a non-isochronous (non-linear) cyclotron oscillator. The non-isochronism of rotation is caused by the relativistic dependence of the gyrofrequency on the electron energy. As a result, the bunching of electrons in the field of a low-frequency (LF) wave leads to the emergence of high-frequency (HF) components of the electron current at every harmonic of this wave. Thus, LF and HF components radiate simultaneously from the interaction space of a gyrotron. In this case, no problem of mode competition exists as the HF mode is forcefully excited by the bunched electrons, while the total beam current is lower than the starting value. However, efficient multiplication requires the electrodynamic system of the gyrotron, namely, a slightly irregular cylindrical waveguide, to be simultaneously resonant to two TE\({}_{m,q}\) modes, one at the fundamental and one at the \(s\)-th CH.