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 cavitys transmission, reflection, and finesse as a function of the membranes 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 membranes 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 membranes period of oscillation. Lastly, we present calculations of quantum non-demolition measurements of the membranes phonon number in the low signal-to-noise regime where the phonon lifetime is comparable to the QND readout time.
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