  High-finesse, open-geometry microcavities have recently emerged as a
versatile tool for enhancing interactions between photons and material systems,
with a range of applications in quantum optics and quantum information science.
However, mechanical vibrations pose a considerable challenge to their operation
within a closed-cycle cryostat, particularly when spatial tunability and
free-space optical access are required. Here, we present the design and
characterization of a system that can achieve $\sim$16 pm-rms passive
mechanical stability between two high-finesse mirrors while permitting both
three-dimensional positioning of the cavity mode and free-space confocal
imaging. The design relies on two cascaded vibration isolation stages connected
by leaf springs that decouple axial and lateral motion, and incorporates
tuned-mass and magnetic damping. Furthermore, we present a technique for
quantifying cavity length displacements similar to or larger than the cavity
linewidth, allowing in-situ measurement of vibrations with and without active
feedback. Our results facilitate operation of a tunable, high-finesse cavity
within a closed-cycle cryostat, representing an enabling technology for cavity
coupling to a variety of solid-state systems.
