The harmonic operation in a terahertz gyrotron traveling-wave amplifier (gyro-TWA) permits a reduced magnetic field, whereas a harmonic multiplying gyro-TWA enables magnetic field reduction and frequency multiplication. This study comparatively analyzes the 400-GHz fourth-harmonic gyro-TWAs and fourth-harmonic multiplying gyro-TWAs with axis-encircling electron beams. This property of the gyro-TWAs with axis-encircling electron beams enables us avoid the appearance of the majority of the competing modes. According to the simulation results obtained using multi-mode codes, the attenuating severs suppress the remaining competing modes. In the case of a harmonic gyro-TWA containing a sever section, the copper-section start-oscillation length is significantly influenced by a decrease in the sever-section radius when compared with an increase in the sever-section length. Furthermore, a stable fourth-harmonic TE41-mode gyro-TWA containing sever sections is proposed. We propose fourth-harmonic TE41-mode multiplying gyro-TWAs, where the drive stages operate in the fundamental-harmonic TE11 mode to reduce the frequency of the drive wave. We subsequently develop a stable fourth-harmonic multiplying gyro-TWA with sever sections in the drive and amplified stages to avoid the competing modes and enhance the output power. The stable high-gain fourth-harmonic gyro-TWA can yield a peak output power of 2.7 kW at 400.6 GHz with a saturated gain of 75 dB and a bandwidth of 0.7 GHz for a 75-kV and 2-A electron beam with an axial velocity spread of 3%. Furthermore, a peak output power of 1.7 kW can be obtained by the fourth-harmonic multiplying gyro-TWA at 400.4 GHz with a saturated gain of 57 dB and a bandwidth of 0.4 GHz. Subsequently, we calculate the power and gain scaling for the harmonic gyro-TWA and harmonic multiplying gyro-TWA.
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