In this paper, a new refractive index sensor based on Fano resonance is introduced. It is mainly composed of two parts: a metal–insulator–metal as a base waveguide and an annular resonant cavity with a double notch and a double convex circle as a coupling structure. The sensing characteristics of the design are investigated via finite element calculations. The influence of the remaining structure parameters on the system’s sensing performance and the relationship between the wavelength corresponding to the Fano trough and the refractive index are also investigated. Furthermore, the structure is applied to refractive index sensors, resulting in a type with a sensitivity of 2740 nm/RIU and a figure of merit of 52.69.