In this article, design of a high-order-mode (TE 20 -mode) staggered double vane (SDV) traveling wave tube amplifier (TWTA) with two pencil electron beams is presented. Since, the dimensions are larger when compared to fundamental mode TWTAs, the high-power capacity can be achieved. Meanwhile, several ways to suppress the fundamental mode competition and improve the stability of our high-order-mode TWTA are studied thoroughly, such as limiting the gain of each stage and using metal or wave-absorbing material columns. A three-stage TE 20 -mode SDV TWTA with aluminum nitride column at the center of vanes is simulated to get an over 145-W peak power with a corresponding gain of 32.5 dB. The frequency spectrum is pure at 220 GHz and the output signal is a stable TE 20 -mode signal. The bandwidth reaches about 38 GHz. The results show that high-order-mode operation can develop the high-power capacity vacuum electron devices at terahertz band.