Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals

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 Q_o > 10^7, motional masses m_(eff) ≈ 100 femtograms, optomechanical coupling length L_(OM) < 5 μm, and clampinig losses that are exponentially suppressed with increasing number of phononic crystal periods (radiation-limited mechanical Q-factor Q_m > 10^7 for total device size less than 30 μm).