We theoretically study the controllable optical response in an optomechanical system with membrane-in-the-middle geometry, where the cavity mode is coupled to the square of the position of the membrane. When the optical cavity is driven by a strong control field and the movable membrane is excited by a weak coherent mechanical driving field, the optical response of the system can be detected by applying a weak probe field to the optical cavity. Due to the additional mechanical driving field, more complex interference exists in this optomechanical system. Under the two-phonon resonance condition, the probe transmission can be larger than unity or suppressed to be zero because of the interference effect, which is dependent on the phase difference of the applied fields. We show that the transmission coefficient and group delay of the probe field can be controlled flexibly by the power of the control field, the amplitude and phase of the mechanical driving field, and the environment temperature. Our results provide a flexible route to control light propagation based on the quadratically coupled optomechanical system.

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