Second harmonic generation from tetragonal centrosymmetric crystals

We present a phenomenological theory of second harmonic generation (SHG) from tetragonal centrosymmetric crystals. It includes the lowest order bulk response, which is of electric quadrupole and magnetic dipole symmetry, and the lowest order surface response, which is of electric dipole symmetry. Based on the symmetry properties of the nonlinear response tensor, complete expressions are obtained for the SHG expected in reflection from a crystal with a (001) surface orientation. We suggest experimental configurations that can be used to extract particular tensor elements. We find that dichroism is present in SHG from tetragonal centrosymmetric crystals, although the crystal is not chiral.