On the use of structured light in nonlinear optics studies of the symmetry group of a crystal

We put forward a technique that allows to extract information about the symmetry group to which certain nonlinear crystals belong using a single illuminating beam. It provides such information by considering the outcome of a nonlinear optics process characterized by the electric nonlinear susceptibility tensor, whose structure is dictated by such symmetry group. As an example, we consider the process of spontaneous parametric down-conversion, when it is pumped with a special type of Bessel beam. The observation of the spatial angular dependence of the lower-frequency generated light provides direct information about the symmetry group of the crystal. We should stress that the choice of the appropriate illumination is of paramount importance for unveiling the sought-after information.