  Periodically structured materials can sustain both optical and mechanical
excitations which are tailored by the geometry. Here we analyze the properties
of dispersively coupled planar photonic and phononic crystals: optomechanical
crystals. In particular, the properties of co-resonant optical and mechanical
cavities in quasi-1D (patterned nanobeam) and quasi-2D (patterned membrane)
geometries are studied. It is shown that the mechanical Q and optomechanical
coupling in these structures can vary by many orders of magnitude with modest
changes in geometry. An intuitive picture is developed based upon a
perturbation theory for shifting material boundaries that allows the
optomechanical properties to be designed and optimized. Several designs are
presented with mechanical frequency ~ 1-10 GHz, optical Q-factor Qo > 10^7,
motional masses meff 100 femtograms, optomechanical coupling length LOM < 5
microns, and a radiation-limited mechanical Q-factor Qm > 10^7.
