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In this study, a new refractive index sensor based on a metal–insulator–metal waveguide coupled with a notched ring resonator and stub is designed. The finite element method is used to study the propagation characteristics of the sensor. According to the calculation results, the transmission spectrum exhibits a typical Fano resonance shape. The phenomenon of Fano resonance is caused by the coupling between the broadband spectrum and narrowband spectrum. In the design, the broadband spectrum signal is generated by the stub, while the narrowband spectrum signal is generated by the notched ring resonator. In addition, the structural parameters of the resonators and the structure filled with media of different refractive indices are varied to study the sensing properties. The maximum achieved sensitivity of the sensor reached 1071.4 nm/RIU. The results reveal potential applications of the coupled system in the field of sensors.