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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 indoi:10.1088/1367-2630/10/9/095008 fatcat:ue3xgzf45bhlnpqdikjrloehnu