  The generation of spectrally pure high-frequency microwave signals is a
critical functionality in fundamental and applied sciences, including metrology
and communications. The development of optical frequency combs has enabled the
powerful technique of optical frequency division (OFD) to produce microwave
oscillations of the highest quality. The approaches for OFD demonstrated to
date demand multiple lasers with space- and energy-consuming optical
stabilization and electronic feedback components, resulting in device
footprints incompatible with integration into a compact and robust photonic
platform. Here, we demonstrate all-optical OFD on a single photonic chip driven
with a single continuous-wave laser. We generate a dual-point frequency
reference using the beat frequency of the signal and idler fields from a
microresonator-based optical parametric oscillator (OPO), which achieves high
phase stability due to the inherently strong signal-idler frequency
correlations. We implement OFD by optically injecting the signal and idler
fields from the OPO to a Kerr-comb microresonator on the same chip. We show
that the two distinct dynamical states of Kerr cavities can be passively
synchronized, allowing broadband frequency locking of the comb state, which
transfers the stability of the OPO frequencies to the repetition rate of the
Kerr comb. A 630-fold phase-noise reduction is observed when the Kerr comb is
synchronized to the OPO, which represents the lowest noise generated on the
silicon-nitride platform. Our work demonstrates a simple, effective approach
for performing OFD and provides a pathway toward chip-scale devices that can
generate microwave frequencies comparable to the purest tones produced in
metrological laboratories. This technology can significantly boost the further
development of data communications and microwave sensing.
