  We present an experimental study of an optomechanical system formed by a
vibrating thin semi-transparent membrane within a high-finesse optical cavity.
We show that the coupling between the optical cavity modes and the vibrational
modes of the membrane can be tuned by varying the membrane position and
orientation. In particular we demonstrate a large quadratic dispersive
optomechanical coupling in correspondence with avoided crossings between
optical cavity modes weakly coupled by scattering at the membrane surface. The
experimental results are well explained by a first order perturbation treatment
of the cavity eigenmodes.
