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A single-polarization (SP) coupler based on air-core photonic bandgap fibers (PBFs) is proposed and numerically simulated. The physical mechanism is the decoupling phenomenon occurring at special separations between the cores of the dual-core PBFs called decoupling points. The coupling length of one polarization in the coupler tends to be infinite while the coupling length of other polarization remains in normal level. The coupling ratio for the primary polarization can vary by adjusting the length of the coupler since the other polarization is always decoupled out at any length of the coupler. When this novel SP coupler is incorporated into a PBFs based fiber ring resonator, its unique polarization property is theoretically simulated. In this full-PBFs configuration, the SP coupler functions as the power splitter and the polarizer simultaneously. By the finite element method, the feasibility of the SP coupler to filter the secondary eigenstate of polarization (ESOP) propagating in the resonator is proved. And the performance of the SP coupler to suppress the temperature-related polarization fluctuation in the resonant fiber optic gyro is promised to be more significant to the in-line polarizer integrating in the resonator. Furthermore, the mechanism of decoupling phenomenon in the SP coupler and the influence of polarization-axis angular misalignment are discussed. In conclusion, an optimized SP coupler can couple the primary ESOP properly, and a polarization extinct ratio larger than 30 dB can be achieved within the angular misalignment of 0.9 degree. The aforementioned unique properties make this novel SP coupler attractive for the resonant fiber optic gyro.