  By combining analytical and numerical approaches, we study resonantly
enhanced second-harmonic generation (SHG) by individual high-index dielectric
nanoparticles made of centrosymmetric materials. Considering both bulk and
surface nonlinearities, we describe second-harmonic nonlinear scattering from a
silicon nanoparticle optically excited in the vicinity of the magnetic and
electric dipolar resonances. We discuss the contributions of different
nonlinear sources, and the effect of the low-order optical Mie modes on the
characteristics of the generated far-field. We demonstrate that the multipolar
expansion of the radiated field is dominated by dipolar and quadrupolar modes
(two axially symmetric electric quadrupoles, an electric dipole, and a magnetic
quadrupole), and the interference of these modes can ensure directivity of the
nonlinear scattering. The developed multipolar analysis can be instructive for
interpreting the far-field measurements of the nonlinear scattering, and it
provides prospective insights into a design of CMOS-compatible nonlinear
nanoantennas fully integrated with silicon-based photonic circuits, as well as
new methods of nonlinear diagnostics.
