When does nonlinear circular dichroism appear in achiral dielectric nanoparticles?

Sometimes it appears, sometimes it does not. We present a theoretical study of circular dichroism in the second-harmonic signal of single dielectric nanostructures with different symmetries and noncentrosymmetric materials. We show that this effect is not fully identified only by macroscopic and lattice symmetries, but their relative orientation as well. Based on the cumbersome symmetry and modal analysis, we provide a general and very simple formula to determine whether the dichroism exists, and a table with the most significant cases.