  We analyze third-harmonic generation from high-index dielectric nanoparticles
and discuss the basic features and multipolar nature of the parametrically
generated electromagnetic fields near the Mie-type optical resonances. By
combining both analytical and numerical methods, we study the nonlinear
scattering from simple nanoparticle geometries such as spheres and disks in the
vicinity of the magnetic dipole resonance. We reveal the approaches for
manipulating and directing the resonantly enhanced nonlinear emission with
subwavelength all-dielectric structures that can be of a particular interest
for novel designs of nonlinear optical antennas and engineering the magnetic
optical nonlinear response at nanoscale.
