  Bound states in the continuum (BICs), an emerging type of long-lived
resonances different from the cavity-based ones, have been explored in several
classical systems, including photonic crystals and surface acoustic waves.
Here, we reveal symmetry-protected mechanical BICs in the structure of
slab-on-substrate optomechanical crystals. Using a group theory approach, we
identified all the mechanical BICs at the $\Gamma$ point in optomechanical
crystals with $C_{4v}$ and $C_{6v}$ symmetries as examples, and analyzed their
coupling with the co-localized optical BICs and guided resonances due to both
moving boundary and photo-elastic effects. We verified the theoretical analysis
with numerical simulations of specific optomechanical crystals which support
substantial optomechanical interactions between the mechanical BICs and optical
resonances. Due to the unique features of high-$Q$, large-size mechanical BICs
and substrate-enabled thermal dissipation, this architecture of
slab-on-substrate optomechanical crystals might be useful for exploring
macroscopic quantum mechanical physics and enabling new applications such as
high-throughput sensing and free-space beam steering.
