Second harmonic-multiplying gyrotron traveling -wave amplifier The second harmonic TE02 gyrotron traveling wave amplifier (gyro-TWT) is a high-power, broadband, millimeter-wave amplifier with a low applied magnetic field. A multi-stage gyro-TWT design with lossy section, severed section and mode-selective circuit is proposed to stabilize the amplification. This study presents a nonlinear analysis of typical oscillations, including absolute instability and gyrotron backward oscillation (gyro-BWO). The multi-stage gyro-TWT seem to increase effectively the start-oscillation currents of the absolute instability and gyro-BWO. The multi-stage gyro-TWT is predicted to yield a peak output power of ~330 kW at 180 GHz, a saturated gain of ~70 dB and a bandwidth of 5.0 GHz for a 100 kV, 15 A electron beam with an axial velocity spread Δvz/vz=5%. Meanwhile, a mode-selective circuit with a transverse sliced interaction structure is employed for suppressing absolute instabilities in the gyro-TWA. Since the wall surface current for the TE0n mode is in the azimuthal direction, the TE01 and TE02 waves are almost unaffected as it travels through the transverse slice. In contrast, the competing modes with axial surface currents are significantly affected.