  Arising from modulation instability, Turing rolls in optical Kerr
microresonators have been used in the generation of optical frequency combs and
the synthesis of microwave and terahertz frequencies. In this work, by applying
electro-optic modulation on terahertz-frequency Turing rolls, we implement
electro-optic frequency division with a microcomb to synthesize variable
low-noise microwave signals. We also actively stabilize the terahertz
oscillations to a microwave reference via intracavity power modulation,
obtaining fractional frequency instabilities that are better than those of the
free-running situation by up to six orders of magnitude. This study not only
highlights the extraordinary spectral purity of Turing roll oscillations but
also opens the way for bidirectional terahertz-to-microwave links with hybrid
optical frequency comb techniques.
