Quadrupolar second-harmonic generation by helical beams and vectorial vortices with radial or azimuthal polarization

Abstract

We study the optical second-harmonic radiation (SHG) generated by scattering from a homogeneous centrosymmetric thin composite material illuminated by higher-order Gaussian laser beams. The induced second-order source polarization is taken as of quadrupolar type $(\mathbf{E}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\nabla})\mathbf{E}$, which depends on the inhomogeneity of the incident electric field $\mathbf{E}$. This nonlinear source has the same form as that responsible of the SH signal observed in a composite made of Si nanocrystals embedded uniformly in a SiO${}_{2}$ matrix and that calculated for a thin disordered array of nanospheres. We calculate the SH radiation angular patterns generated by several incident combinations of spatial modes and states of polarizations. In particular, excitation with radially and azimuthally polarized doughnut modes and helical beams carrying orbital angular momentum with linear or circular polarization are considered. We found that this quadrupolar SHG depends sensitively on the transverse structure and polarization of the driving field. The response to $\ensuremath{\nabla}\mathbf{E}$ introduces a factor $\mathbf{E}(\mathbf{E}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{K})$ in the Fourier component of the SH scattering amplitude, absent in electric-dipole-allowed SHG, that can give additional nodal lines or rings in the SH angular patterns, changes of the state of polarization, or additional azimuthal phases in the harmonic radiation. For circularly polarized beams with helical phase wave front, we found a selection rule according to which the nonlinear scattering of an optical vortex with charge ${l}_{\ensuremath{\omega}}$ and spin $\ensuremath{\sigma}=\ifmmode\pm\else\textpm\fi{}1$ induces a SH vortex field with a spin-dependent charge doubling ${l}_{2\ensuremath{\omega}}=2{l}_{\ensuremath{\omega}}+\ensuremath{\sigma}$. These features may be useful to identify SHG processes of quadrupolar nature and suggest a way to produce scattered SH radiation with a desired angular pattern and state of polarization.