  Multimode interference and multipolar interplay govern functionalities of
optical nanoresonators and nonlinear nanoantennas. Recently, excitation of the
high-quality supercavity modes (quasi-BIC states) in individual subwavelength
dielectric particles has been predicted to boost the nonlinear frequency
conversion at the nanoscale. Here, we put forward the multipolar model which
captures the physics behind linear and nonlinear response driven by such
high-$Q$ modes in nanoresonators. We show that formation of the quasi-BIC state
in the AlGaAs nanodisk can be understood through multipolar transformations of
coupled leaky modes. In particular, the hybridized axially symmetric
TE-polarized modes can be viewed as superpositions of multipoles, with a basis
of parent multipoles constituted mainly by magnetic dipoles and octupole. The
quasi-BIC point in the parameter space appears where dipolar losses are totally
suppressed. The efficient optical coupling to this state is implemented via
azimuthally polarized beam illumination matching its multipolar origin. We
establish a one-to-one correspondence between the standard non Hermitian
coupled-mode theory and multipolar models that enables transparent
interpretation of scattering characteristics. Using our approach, we derive the
multipolar composition of the generated second-harmonic radiation from the
AlGaAs nanodisk and validate it with full-wave numerical simulations.
Back-action of the second-harmonic radiation onto the fundamental frequency is
taken into account in the coupled nonlinear model with pump depletion. A hybrid
metal-dielectric nanoantenna is proposed to augment the conversion efficiency
up to tens of per cent and actualize the nonlinear parametric downconversion.
Our findings delineate the in-depth conceptual framework and novel promising
strategies in the design of functional elements for nonlinear nanophotonics
applications.
