  Frequency combs from continuous-wave-driven Kerr-nonlinear microresonators
have evolved into a key photonic technology with applications from optical
communication to precision spectroscopy. Essential to many of these
applications is the control of the comb's defining parameters, i.e.,
carrier-envelope offset frequency and repetition rate. An elegant and
all-optical approach to controlling both degrees of freedom is the suitable
injection of a secondary continuous-wave laser into the resonator onto which
one of the comb lines locks. Here, we study experimentally such sideband
injection locking in microresonator soliton combs across a wide optical
bandwidth and derive analytic scaling laws for the locking range and repetition
rate control. As an application example, we demonstrate optical frequency
division and repetition rate phase-noise reduction to three orders of magnitude
below the noise of a free-running system. The presented results can guide the
design of sideband injection-locked, parametrically generated frequency combs
with opportunities for low-noise microwave generation, compact optical clocks
with simplified locking schemes and more generally, all-optically stabilized
frequency combs from Kerr-nonlinear resonators.
