Simple vibration insensitive cavity for laser stabilization at the 10^{-16} level

We present the design and realization of two reference cavities for ultra-stable lasers addressing narrow transitions in mixed-species (In+ / Yb+) Coulomb crystals. With a simple setup, we achieve a fractional frequency instability close to the thermal noise limit of a 12 cm long cavity, reaching 4.7 * 10^{-16} at 10 s with a linear drift of 53 mHz/s. We discuss the individual instability contributions and show that in a setup with a lower thermal noise floor and vibration sensitivity, an instability of 1 * 10^{-16} can be reached. To achieve this, we implement a vibration insensitive design for a 30 cm long cavity mounted horizontally and conduct first tests that show a sensitivity of 1.8 * 10^{-11} / ms^{-2} to vertical accelerations. This is about a factor of 20 less than the value observed for the short cavity. Mechanical tolerances and ways to further reduce the sensitivity are discussed.