  Thermal noise in optical cavities imposes a severe limitation in the
stability of the most advanced frequency standards at a level of a few 10^(-16)
(s/t)^(1/2) for long averaging times t. In this paper we describe two schemes
for reducing the effect of thermal noise in a reference cavity. In the first
approach, we investigate the potential and limitations of operating the cavity
close to instability, where the beam diameter on the mirrors becomes large. Our
analysis shows that even a 10 cm short cavity can achieve a thermal noise
limited fractional frequency instability in the low 10^(-16) regime. In the
second approach, we increase the length of the optical cavity. We show that a
39.5 cm long cavity has the potential for a fractional frequency instability
even below 10^(-16), while it seems feasible to achieve a reduced sensitivity
of <10^(-10)/g for vibration-induced fractional length changes in all three
directions.
