  State-of-the-art optical oscillators employing cryogenic reference cavities
are limited in performance by the Brownian thermal noise associated with the
mechanical dissipation of the mirror coatings. Recently, crystalline
Al$_{1-x}$Ga$_{x}$As/GaAs coatings have emerged as a promising candidate for
improved coating thermal noise. We present measurements of the frequency noise
of two fully crystalline cryogenic reference cavities with
Al$_{0.92}$Ga$_{0.08}$As/GaAs optical coatings. We report on previously
unmeasured birefringent noise associated with anti-correlated frequency
fluctuations between the polarization modes of the crystalline coatings, and
identify variables that affect its magnitude. Comparing the birefringent noise
between the two cryogenic reference cavities reveals a phenomenological set of
scalings with intracavity power and mode area. We implement an interrogation
scheme that cancels this noise by simultaneous probing of both polarization
modes. The residual noise remaining after this cancellation is larger than both
cavities thermal noise limits, but still lower than the instabilities
previously measured on equivalent resonators with dielectric coatings. Though
the source of these novel noise mechanisms is unclear, we demonstrate that
crystalline coatings can provide stability and sensitivity competitive with
resonators employing dielectric coatings.
