  We present a theoretical study of the characteristics of the nonlinear
spin-orbital angular momentum coupling induced by second-harmonic generation in
plasmonic and dielectric nanostructures made of centrosymmetric materials. In
particular, the connection between the phase singularities and polarization
helicities in the longitudinal components of the fundamental and
second-harmonic optical fields and the scatterer symmetry properties are
discussed. By in-depth comparison between the interaction of structured optical
beams with plasmonic and dielectric nanostructures, we have found that
all-dielectric and plasmonic nanostructures that exhibit magnetic and electric
resonances have comparable second-harmonic conversion efficiency. In addition,
mechanisms for second-harmonic enhancement for single and chiral clusters of
scatterers are unveiled and the relationships between the content of optical
angular momentum of the incident optical beams and the enhancement of nonlinear
light scattering is discussed. In particular, we formulate a general angular
momenta conservation law for the nonlinear spin-orbital angular momentum
interaction, which includes the quasi-angular-momentum of chiral structures
with different-order rotational symmetry. As a key conclusion of our study
relevant to nanophotonics, we argue that all-dielectric nanostructures provide
a more suitable platform to investigate experimentally the nonlinear
interaction between spin and orbital angular momenta, as compared to plasmonic
ones, chiefly due to their narrower resonance peaks, lower intrinsic losses,
and higher sustainable optical power.
