This paper reports the design and cold test of the cavity beam position monitor (CBPM) for SX-FEL to fulfill the requirement of beam position measurement resolution of less than 1{mu}m, even 0.1{mu}m. The CBPM was optimized by using a coupling slot to damp the TM010 mode in the output signal. The isolation of TM010 mode is about 117dB, and the shunt impedance is about 200{Omega}@4.65GHz with the quality factor 80 from MAFIA simulation and test result. A special antenna was designed to load power for reducing excitation of other modes in the cavity. The resulting output power of TM110 mode was about 90mV/mm when the source was 6dBm, and the accomplishable minimum voltage was about 200{mu}V. The resolution of the CBPM was about 0.1{mu}m from the linear fitting result based on the cold test.
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