  We consider an optical and mechanical mode interacting through both linear
and quadratic dispersive couplings in a general cavity-optomechanical set-up.
The parity and strength of an intrinsic quadratic optomechanical coupling (QOC)
provides an opportunity to control the optomechanical (OM) interaction. We
quantify this interaction by studying normal-mode splitting (NMS) as a function
of the QOC's strength. The proposed scheme exhibits NMS features equivalent to
a hybrid-OM system containing either an optical parametric amplifier (OPA) or a
Kerr medium. Such a system in reality could offer an alternative platform for
devising state-of-art quantum devices with requiring no extra
degrees-of-freedom as in hybrid-OM systems.
