A basic plasmonic system, consisting of a stub metal-insulator-metal (MIM) waveguide coupled with a ring resonator, is presented to realize Fano resonance and electromagnetically induced transparency-like (EIT-like) effect, which are numerically calculated by the finite element method (FEM). Meanwhile, the formation mechanism of Fano resonance is analyzed according to numerical simulations. Besides, the coupled mode theory (CMT) and the standing wave theory are used for explaining the Fano and EIT-like resonances phenomenon. Based on this system, an inner ring cavity is connected to the ring resonator by a slot and another ring cavity is later introduced under the stub resonator in order to constitute a new coupled plasmonic resonator system, providing quadruple Fano resonances and double EIT-like responses finally. In addition, the Fano and EIT-like resonances can be independently tuned by adjusting the structural parameters, which makes the design of highly integrated photonic circuits more flexible. The main contribution of this paper is that the proposed structure has a relatively good sensitivity of 1600 nm/RIU and an ultra-high FOM value of $1.2\times 10^{6}$ as a refractive index nanosensor. Moreover, it can serve as an all-optical switch with a high on/off extinction ratio of about 43 dB. Additionally, its maximum group delay time and group index are about 1.49 ps and 221, indicating that the proposed system has a pretty good slow light effect. Therefore, the proposed structures are believed to have significant applications in high-performance nanosensors, switches, slow light devices and nonlinear areas in highly integrated plasmonic devices.
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