  We investigate theoretically in detail the non-linear effects in the response
of an optical/microwave cavity coupled to a Duffing mechanical resonator. The
cavity is driven by a laser at a red or blue mechanical subband, and a probe
laser measures the reflection close to the cavity resonance. Under these
conditions, we find that the cavity exhibits optomechanically induced
reflection (OMIR) or absorption (OMIA) and investigate the optomechanical
response in the limit of non-linear driving of the mechanics. Similar to linear
mechanical drive, an overcoupled cavity the red-sideband drive may lead to both
OMIA and OMIR depending on the strength of the drive, whereas the blue-sideband
drive only leads to OMIR. The dynamics of the phase of the mechanical resonator
leads to the difference between the shapes of the response of the cavity and
the amplitude response of the driven Duffing oscillator, for example, at weak
red-sideband drive the OMIA dip has no inflection point. We also verify that
mechanical non-linearities beyond Duffing model have little effect on the size
of the OMIA dip though they affect the width of the dip.
