  State-of-the-art optical oscillators based on lasers frequency stabilized to
high finesse optical cavities are limited by thermal noise that causes
fluctuations of the cavity length. Thermal noise represents a fundamental limit
to the stability of an optical interferometer and plays a key role in modern
optical metrology. We demonstrate a novel design to reduce the thermal noise
limit for optical cavities by an order of magnitude and present an experimental
realization of this new cavity system, demonstrating the most stable oscillator
of any kind to date. The cavity spacer and the mirror substrates are both
constructed from single crystal silicon and operated at 124 K where the silicon
thermal expansion coefficient is zero and the silicon mechanical loss is small.
The cavity is supported in a vibration-insensitive configuration, which,
together with the superior stiffness of silicon crystal, reduces the vibration
related noise. With rigorous analysis of heterodyne beat signals among three
independent stable lasers, the silicon system demonstrates a fractional
frequency stability of 1E-16 at short time scales and supports a laser
linewidth of <40 mHz at 1.5 \mu m, representing an optical quality factor of
4E15.
