Nonlinear optics determination of the symmetry group of a crystal using structured light

We put forward a technique to unveil to which symmetry group a nonlinear crystal belongs, making use of nonlinear optics with structured light. We consider as example the process of spontaneous parametric down-conversion. The crystal, which is illuminated with a special type of Bessel beam, is characterized by a nonlinear susceptibility tensor whose structure is dictated by the symmetry group of the crystal. The observation of the spatial angular dependence of the lower-frequency generated light provides direct information about the symmetry group of the crystal.