  The interaction between a high-frequency dilational mode of a thin dielectric
film and an optical cavity field is studied theoretically in the
membrane-in-the-middle setup. A derivation from first principles leads to a
multi-mode optomechanical Hamiltonian where multiple cavity modes are coupled
by the thickness variation of the membrane. For membrane thicknesses on the
order of one micron, the frequency of this dilational mode is in the GHz range.
This can be matched to the free spectral range of the optical cavity, such that
the mechanical oscillator will resonantly couple cavity modes at different
frequencies. Furthermore, such a large mechanical frequency also means that the
quantum ground state of motion can be reached with conventional refrigeration
techniques. Estimation of the coupling strength with realistic parameters
suggests that optomechanical effects can be observable with this dilational
mode. It is shown how this system can be used as a quantum limited optical
amplifier. The dilational motion can also lead to quantum correlations between
cavity modes at different frequencies, which is quantified with an
experimentally accessible two-mode squeezing spectrum. Finally, an explicit
signature of radiation pressure shot noise in this system is identified.
