In these two papers, we theoretically developed the multipolar selection rules for second harmonic generation by nanostructures. In the first paper, analytical computations of SHG from a dielectric sphere are conducted. It was shown, that the maximal enhancement is achieved, when both ω and 2ω are resonant but also the generation of the particular multipole is not prohibited by the selection rules. In the second paper, the selection rules were generalized for the case of an arbitrary nanoparticle shape with the help of representation theory. 

The symmetry of the nanostructure [1, 3, 4] or metasurface [2] dictates, how the eigenmodes should look like, and what types of eigenmodes exist in a particular structure. The symmetry behavior of vector spherical harmonics (multipoles) dictates the multipolar content of each mode. Modes with the same symmetry can interact with each other, and with the different can not [1, 3]. 

These considerations help us to explain many different phenomena in multipolar terms, such as BICs in nanostructures [1] or metasurfaces [2], exceptional points and conditions for their appearance [3], and even help to make the computations much faster [4]. 

Circular dichroism can exist even if the nanostructure is not chiral! One just needs to take into account the nonlinear response. 

In the first paper, this was discovered for AlGaAs dimers using the mode hybridization approach and checked experimentally by collaborators in ANU. 

In the second one, using a different theoretical approach, we generalized the "selection rules" for the second harmonic circular dichroism for arbitrary nanostructure and lattice symmetry.

In the third one, we managed to generalize it even further, for arbitrary high harmonic and vector beams. After all, the very general condition is written by one simple formula.