  Extraneous thermal motion can limit displacement sensitivity and radiation
pressure effects, such as optical cooling, in a cavity-optomechanical system.
Here we present an active noise suppression scheme and its experimental
implementation. The main challenge is to selectively sense and suppress
extraneous thermal noise without affecting motion of the oscillator. Our
solution is to monitor two modes of the optical cavity, each with different
sensitivity to the oscillator's motion but similar sensitivity to the
extraneous thermal motion. This information is used to imprint "anti-noise"
onto the frequency of the incident laser field. In our system, based on a
nano-mechanical membrane coupled to a Fabry-P\'{e}rot cavity, simulation and
experiment demonstrate that extraneous thermal noise can be selectively
suppressed and that the associated limit on optical cooling can be reduced.
