  Optomechanical systems using a membrane-in-the-middle configuration can
exhibit a long-range type of interaction similar to how atoms show collective
motion in an optical potential. Photons bounce back and forth inside a
high-finesse Fabry-P\'{e}rot cavity and mediate the interaction between
multiple membranes over a significant distance compared to the wavelength.
Recently, it has been demonstrated that light coupling the center-of-mass mode
of the membranes can cause coherent mechanical noise cancellation. Long-range
interactions involving the breathing mode of the membranes have to date not
been experimentally demonstrated, however. In our experiment, a double-membrane
system inside a Fabry-P\'{e}rot cavity resonantly enhances the cavity field,
resulting in a stronger optomechanical coupling strength from the increased
radiation pressure. The resonance condition is first identified by analyzing
the slope of the dispersion relation. Then, the optomechanical coupling is
determined at various chip positions over one wavelength range. The optimum
coupling conditions are obtained and enhancement is demonstrated for double
membrane arrays with three different reflectivites, reaching nearly four-fold
enhancement for the breathing mode of $R=65\%$ double membranes. The cavity
losses at the optimum coupling are also characterized and the potential of
reaching the single-photon strong coupling regime is discussed.
