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We explore the optical response of a multimode optomechanical system with quadratic coupling to a weak probe field, where the cavity is driven by a strong control field and the two movable membranes are, respectively, excited by weak coherent mechanical driving fields. We study the two cases that the two movable membranes are degenerate and nondegenerate. For the degenerate case, it is shown that only one transparency window occurs and the transition between optomechanically induced transparency and Fano resonance can be realized by tuning the cavity-control field detuning. For the nondegenerate case, two transparency windows are observed and the absorption spectrum can switch between a single Fano resonance and double Fano resonances. Furthermore, we show that the output probe field can be greatly amplified or completely suppressed due to the complex interference effect by tuning the amplitude and phase of the mechanical driving fields. Our results can be extended to the optomechanical system with multiple membranes, which enables us to control the light propagation more flexibly.

Optomechanically induced transparency, amplification, and Fano resonance in a multimode optomechanical system with quadratic coupling