  Optically levitated mechanical sensors promise isolation from thermal noise
far beyond what is possible using flexible materials alone. One way to access
this potential is to apply a strong optical trap to a minimally supported
mechanical element, thereby increasing its quality factor $Q_m$. Current
schemes, however, require prohibitively high laser power ($\sim10$ W), and the
$Q_m$ enhancement is ultimately limited to a factor of $\sim$ 50 by
hybridization between the trapped mode and the dissipative modes of the
supporting structure. Here we propose a levitation scheme taking full advantage
of an optical resonator to reduce the circulating power requirements by many
orders of magnitude. Applying this scheme to the case of a dielectric disk in a
Fabry-Perot cavity, we find a tilt-based tuning mechanism for optimizing both
center-of-mass and torsional mode traps. Notably, the two modes are trapped
with comparable efficiency, and we estimate that a 10-micron-diameter,
100-nm-thick Si disc could be trapped to a frequency of $\sim$ 10 MHz with only
$30$ mW circulating in a cavity of (modest) finesse 1500. Finally, we simulate
the effect such a strong trap would have on a realistic doubly-tethered disc.
Of central importance, we find torsional motion is comparatively immune to
$Q_m$-limiting hybridization, allowing a $Q_m$ enhancement factor of $\sim$
1500. This opens the possibility of realizing a laser-tuned 10 MHz mechanical
system with a quality factor of order a billion.
