Conical quasi-phase-matching second harmonic generation in a 2D photonic crystal with hybrid Ewald geometric method

The direction variation of the fundamental wave in the same nonlinear photonic crystal would cause different pattern of harmonics generation. In a 2D/3D crystal with dense reciprocal lattice vectors, there will be large numbers of conical harmonic beams evolving with direction change of the fundamental wave. By rearranging the Ewald sphere and superposing it into the Ewald shell, we have a hybrid Ewald construction. It becomes a simple but useful geometric method to comprehensively depict the distribution of these quasi-phase-matching second harmonics and their conical form evolution. It presents conical second harmonic beams by their related reciprocal lattice vectors and simplifies the beams' distribution according to spatial arrangement of those reciprocal lattice vectors. It finds that the conical beams will create, annihilate, or get enhanced in specific order when fundamental waves change incident directions. We applied the method on a periodically poled 2D LiTaO