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In this paper, a stacked dual beam electron gun and the associated electron optical system are proposed. The stacked dual beam electron gun includes two compact focusing electrodes which help to achieve dual sheet beams. As an application of this dual beam electron gun, a 340 GHz integrated dual beam traveling wave tube (TWT) based on the staggered dual vane slow-wave structure (SWS) is also put forward. In order to reduce the length of the TWT, a novel input/output coupler is introduced. The overall transmission characteristics of the SWS structure together with the input/output couplers show a wide bandwidth covering a frequency range of 306 GHz to 360 GHz. Based on the parameters obtained for the integrated TWT, a stacked dual-beam electron gun with dual focusing electrodes is designed to achieve a beam current of 43 mA, a beam voltage of 21.4 kV, and a cross-sectional size of each beam of 0.3 mm × 0.08 mm. A uniform magnetic field of 0.52 T is utilized to focus the dual electron beams, and a beam transmission efficiency of 97.1% is achieved over a length of 50 mm. Finally, particle in cell simulation results show that the integrated dual-beam TWT can generate an output power of 5 W over the frequency range of 315 GHz to 350 GHz, with the maximum output power of 24.5 W at 330 GHz. [ABSTRACT FROM AUTHOR]