Optically accessible high-finesse millimeter-wave resonator for cavity quantum electrodynamics with atom arrays

Cavity quantum electrodynamics (QED) is a powerful tool in quantum science, enabling preparation of non-classical states of light and scalable entanglement of many atoms coupled to a single field mode. While the most coherent atom-photon interactions have been achieved using superconducting millimeter-wave cavities coupled to Rydberg atoms, these platforms so far lack the optical access required for trapping and addressing individual atomic qubits. We present a millimeter-wave Fabry-P\'erot cavity with finesse $5.8(1) \times 10^7$ at a temperature of 1 K providing generous transverse optical access (numerical aperture 0.56). Conflicting goals of strong atom-photon coupling and optical access motivate a near-confocal geometry. Close to confocality, however, post-paraxial corrections to the cavity spectrum introduce unexpected degeneracies between transverse modes, leading to excess cavity loss. Modeling these corrections allows for tuning the cavity geometry to evade this loss, producing a high finesse that will enable cavity QED experiments with trapped atoms deep in the strong coupling regime.