  Despite recent progress in nonlinear optics in wavelength-scale resonators,
there are still open questions on the possibility of parametric oscillation in
such resonators. We present a general approach to predict the behavior and
estimate the oscillation threshold of multi-mode subwavelength and
wavelength-scale optical parametric oscillators (OPOs). As an example, we
propose an OPO based on Mie-type multipolar resonances, and we demonstrate that
due to the low-Q nature of multipolar modes in wavelength-scale resonators,
there is a nonlinear interaction between these modes. As a result, the OPO
threshold, compared to the single-mode case, can be reduced by a factor that is
significantly larger than the number of interacting modes. The multi-mode
interaction can also lead to a phase transition manifested through a sudden
change in the parametric gain as well as the oscillation threshold, which can
be utilized for enhanced sensing. We establish an explicit connection between
the second-harmonic generation efficiency and the OPO threshold. This allows us
to estimate the OPO threshold based on measured or simulated second-harmonic
generation in different classes of resonators, such as bound states in the
continuum and inversely designed resonators. Our approach for analyzing and
modeling miniaturized OPOs can open unprecedented opportunities for classical
and quantum nonlinear photonics.
