  Fiber Fabry-Perot cavities, formed by micro-machined mirrors on the
end-facets of optical fibers, are used in an increasing number of technical and
scientific applications, where they typically require precise stabilization of
their optical resonances. Here, we study two different approaches to construct
fiber Fabry-Perot resonators and stabilize their length for experiments in
cavity quantum electrodynamics with neutral atoms. A piezo-mechanically
actuated cavity with feedback based on the Pound-Drever-Hall locking technique
is compared to a novel rigid cavity design that makes use of the high passive
stability of a monolithic cavity spacer and employs thermal self-locking and
external temperature tuning. Furthermore, we present a general analysis of the
mode matching problem in fiber Fabry-Perot cavities, which explains the
asymmetry in their reflective line shapes and has important implications for
the optimal alignment of the fiber resonators. Finally, we discuss the issue of
fiber-generated background photons. We expect that our results contribute
towards the integration of high-finesse fiber Fabry-Perot cavities into compact
and robust quantum-enabled devices in the future.

Atomic Physics;Optics;Quantum Physics