Dynamics and Stability of an Optically Levitated Mirror

We analyse the dynamics of a one-dimensional vertical Fabry-P\'erot cavity, where the upper mirror levitates due to intra-cavity radiation pressure force. A perturbative approach is used based around separation of timescales, which allows us to calculate the physical quantities of interest. Due to the dynamics of the cavity field, we find that the upper mirror's motion will always be unstable for levitation performed using only a single laser. Stability can be achieved for two lasers, where one provides the trapping potential and the other a damping effect, and we locate and characterise all parameter regimes where this can occur. Finally we analyse photothermal effects due to heating of the mirror substrate. We show that this can stabilise the system, even with only a single input laser, if it acts to increase the optical path length of the cavity. This work serves as a foundation for understanding how levitated optical cavity schemes can be used as stable metrological platforms.