Optomechanically induced transparency in optomechanics with both linear and quadratic coupling

We explore the optomechanically induced transparency (OMIT) of a compound optomechanical system, which contains two cavities and a membrane. Due to the linear and quadratic coupling between the membrane and the cavity, i.e., ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{1}^{\ifmmode\dagger\else\textdagger\fi{}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{1}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{1}^{\ifmmode\dagger\else\textdagger\fi{}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{1}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}}^{2}$, OMIT can be observed near $\ensuremath{\delta}={\ensuremath{\omega}}_{m}$ and $\ensuremath{\delta}=2{\ensuremath{\omega}}_{m}$ simultaneously. The transmission rate of the probe field near $\ensuremath{\delta}=2{\ensuremath{\omega}}_{m}$ has the same order of magnitude with the transmission rate near $\ensuremath{\delta}={\ensuremath{\omega}}_{m}$. This is in contrast to the second-order sideband OMIT effect which is caused by the intrinsic nonlinear optomechanical coupling. We also find that the positions of the two transmission windows depend on the power of the control field. By modulating the equilibrium position of the membrane, the transmission rate near $\ensuremath{\delta}={\ensuremath{\omega}}_{m}$ can be controlled. The group delay of the transmitted light near $\ensuremath{\delta}={\ensuremath{\omega}}_{m}$ is more sensitive to the equilibrium position of the membrane than the transmitted light near $\ensuremath{\delta}=2{\ensuremath{\omega}}_{m}$. So we can separate the two transmitted lights by modulating the equilibrium position of the membrane. Our investigation provides a method to steer the performance of OMIT devices.