  In this paper we study and design quasi-2D optomechanical crystals,
waveguides, and resonant cavities formed from patterned slabs. Two-dimensional
periodicity allows for in-plane pseudo-bandgaps in frequency where resonant
optical and mechanical excitations localized to the slab are forbidden. By
tailoring the unit cell geometry, we show that it is possible to have a slab
crystal with simultaneous optical and mechanical pseudo-bandgaps, and for which
optical waveguiding is not compromised. We then use these crystals to design
optomechanical cavities in which strongly interacting, co-localized
photonic-phononic resonances occur. A resonant cavity structure formed by
perturbing a "linear defect" waveguide of optical and acoustic waves in a
silicon optomechanical crystal slab is shown to support an optical resonance at
wavelength 1.5 micron and a mechanical resonance of frequency 9.5 GHz. These
resonances, due to the simultaneous pseudo-bandgap of the waveguide structure,
are simulated to have optical and mechanical radiation-limited Q-factors
greater than 10^7. The optomechanical coupling of the optical and acoustic
resonances in this cavity due to radiation pressure is also studied, with a
quantum conversion rate, corresponding to the scattering rate of a single
cavity photon via a single cavity phonon, calculated to be 292 kHz.
