  We present the results of theoretical and experimental studies of
dispersively coupled (or "membrane in the middle") optomechanical systems. We
calculate the linear optical properties of a high finesse cavity containing a
thin dielectric membrane. We focus on the cavity's transmission, reflection,
and finesse as a function of the membrane's position along the cavity axis and
as a function of its optical loss. We compare these calculations with
measurements and find excellent agreement in cavities with empty-cavity
finesses in the range 10^4 to 10^5. The imaginary part of the membrane's index
of refraction is found to be approximately 10^(-4). We calculate the laser
cooling performance of this system, with a particular focus on the
less-intuitive regime in which photons "tunnel" through the membrane on a time
scale comparable to the membrane's period of oscillation. Lastly, we present
calculations of quantum non-demolition measurements of the membrane's phonon
number in the low signal-to-noise regime where the phonon lifetime is
comparable to the QND readout time.
