Abstract: High-frequency relativistic klystron amplifier is one of the research hotspots in the field of high power microwave in recent years, and its development is mainly limited by mode competition, phase jitter and low efficiency. This paper presents the design of a novel Ku-band radial-line klystron amplifier, which consists of an input cavity, two groups of double-gap bunching cavities and a three-gap extraction cavity. By comparing the coupling coefficient of single-gap bunching cavity with that of non-uniform double-gap bunching cavities, it is proved that non-uniform double-gap bunching cavity has stronger modulation ability to the electron beam. The working mode of non-uniform double-gap bunching cavity with a TEM reflector is TM₀₁-π mode, which has a large Q value and benefit from energy isolation between resonant cavities. When the injection power is only 10 kW, the modulation depth of fundamental current is about 110% by cascading two groups of double-gap cavities. PIC simulationshows that this device has high efficiency. When electron beam voltage is 400 kV, beam current is 5 kA and magnetic field is only 0.4 T, high power microwaves with frequency of 14.25 GHz and output power of 825 MW are obtained.