A mechanically stable and tunable cryogenic Fabry-Perot microcavity

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.