Title: Cryogenic sapphire optical reference cavity with crystalline coatings at $\mathrm{ 1 \times 10^{-16}}$ fractional instability

Abstract: The frequency stability of a laser locked to an optical reference cavity is fundamentally limited by thermal noise in the cavity length. These fluctuations are linked to material dissipation, which depends both on the temperature of the optical components and the material properties. Here, the design and experimental characterization of a sapphire optical cavity operated at 10 K with crystalline coatings at 1069 nm is presented. Theoretical estimates of the thermo-mechanical noise indicate a thermal noise floor below $\mathrm{4.5\times10^{-18}}$. Major technical noise contributions including vibrations, temperature fluctuations, and residual amplitude modulation are characterized in detail. The short-term performance is measured via a three-cornered hat analysis with two other cavity-stabilized lasers, yielding a noise floor of $1\times10^{-16}$. The long-term performance is measured against an optical lattice clock, indicating cavity stability at the level of $2\times10^{-15}$ for averaging times up to 10,000 s.