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A highly sensitive refractive index sensor based on surface plasmon resonance in a side-polished low-index polymer optical fiber is proposed for biosensing. Benefitting from the low refractive index of the fiber core, the sensitivity of the device can reach ~44567 nm/RIU theoretically for aqueous solutions, at the expense of a lowered upper detection limit that is down to ~1.340. The sensor is fabricated by coating 55-nm-thick Au-film on the polished surface of a graded-index perfluorinated polymer optical fiber. Results show that the sensor exhibits a sensitivity of ~22779 nm/RIU at 1.335 with a figure of merit of 61.2. When employed for glucose sensing, the sensor presents an averaged sensitivity of 24.50 nm/wt%, or 0.46 nm/mM. This device is expected to have potential applications in cost-effective bio- and chemical-sensing.