Second harmonic generation on the yellow 1S exciton in Cu2O in symmetry-forbidden geometries

We observe optical second harmonic generation (SHG) on the $1S$ exciton resonance of the yellow exciton series in ${\mathrm{Cu}}_{2}\mathrm{O}$ in four crystal directions ([001], $[\overline{1}10],\phantom{\rule{0.28em}{0ex}}[11\overline{2}]$, and [111]). For light with the $k$-vector parallel [001] and $[\overline{1}10]$, SHG is symmetry forbidden in an ideal crystal according to band structure. We explain the observation of SHG signals in the forbidden geometries by local strain inducing splitting and mixing of the $1S$ components. Experimental data and a microscopic theory confirm that this effect observed on the $1S$ exciton requires a relatively long exciton lifetime. The SHG signals of excitons with principal quantum number $n\ensuremath{\ge}2$ are compliant with selection rules derived by group theory from the symmetry of an unstrained crystal.