Controllable optical response in a quadratically coupled optomechanical system with mechanical driving

We study the response of the output field from a quadratically coupled optomechanical system, where the optical cavity is driven by a strong control field and a weak probe field. In addition, a weak coherent mechanical driving field is applied to excite the mechanical mode and modify the optical response of the system. Under different driving conditions, we show that the transmission spectrum of the probe field can exhibit the phenomena of optomechanically induced transparency, amplification, and Fano resonance. Within the narrow transparency and amplification windows, the four-wave mixing (FWM) intensity can be resonantly enhanced. Moreover, it is shown that the FWM intensity can be further enhanced or suppressed by tuning the amplitude and phase of the mechanical driving, and the physical origin is explained in terms of interference effect. Our results may find potential applications in optical switching and frequency conversion.