  Resonant enhancement of nonlinear photonic processes is critical for the
scalability of applications such as long-distance entanglement generation. To
implement nonlinear resonant enhancement, multiple resonator modes must be
individually tuned onto a precise set of process wavelengths, which requires
multiple linearly-independent tuning methods. Using coupled auxiliary
resonators to indirectly tune modes in a multi-resonant nonlinear cavity is
particularly attractive because it allows the extension of a single physical
tuning mechanism, such as thermal tuning, to provide the required independent
controls. Here we model and simulate the performance and tradeoffs of a
coupled-resonator tuning scheme which uses auxiliary resonators to tune
specific modes of a multi-resonant nonlinear process. Our analysis determines
the tuning bandwidth for steady-state mode field intensity can significantly
exceed the inter-cavity coupling rate if the total quality factor of the
auxiliary resonator is higher than the multi-mode main resonator. Consequently,
over-coupling a nonlinear resonator mode to improve the maximum efficiency of a
frequency conversion process will simultaneously expand the auxiliary resonator
tuning bandwidth for that mode, indicating a natural compatibility with this
tuning scheme. We apply the model to an existing small-diameter triply-resonant
ring resonator design and find that a tuning bandwidth of 136 GHz ~ 1.1 nm can
be attained for a mode in the telecom band while limiting excess scattering
losses to a quality factor of 10^6. Such range would span the distribution of
inhomogeneously broadened quantum emitter ensembles as well as resonator
fabrication variations, indicating the potential for the auxiliary resonators
to enable not only low-loss telecom conversion but also the generation of
indistinguishable photons in a quantum network.
