  We employ structured light for the second-harmonic generation from
subwavelength AlGaAs nanoparticles that support both electric and magnetic
multipolar Mie resonances. The vectorial structure of the pump beam allows
addressing selectively Mie-resonant modes and control the strength of the
generated nonlinear fields. We observe experimentally the enhancement of the
second-harmonic generation for the azimuthally polarized vector beams near
magnetic dipole resonance, and match our observations with the numerical
decomposion of the Mie-type multipoles for the fundamental and generated
second-harmonic fields
