Whenever a circularly polarized laser beam propagates along an axis of threefold symmetry in a piezo-electric crystal, the second-harmonic polarization is circularly polarized in the opposite sense. Rabin and Bey have discussed the theory of second-harmonic generation in crystals with natural optical activity. We have observed this circular polarization and also the difference in coherence lengths for opposite senses of circular polarization, with both fundamental and second-harmonic beams propagating along the [111] direction in single crystals of NaCl${\mathrm{O}}_3$ and NaBr${\mathrm{O}}_3$, belonging to the cubic class 23, for which the proper eigenmodes are circularly polarized waves. The nonlinear susceptibility of both NaCl${\mathrm{O}}_3$ and NaBr${\mathrm{O}}_3$ has been measured and compared with the nonlinear susceptibility of $\alpha$-quartz. The relative sign of the susceptibilities has been determined in second-harmonic interference experiments with the same laser beam traversing two crystals in succession. The interesting questions of the sign of the nonlinear susceptibility, piezoelectric constant, optical activity, and the absolute atomic configuration of the antipodes are discussed. The question of conservation of angular momentum is resolved by taking into account the crystal-line-field potential of threefold symmetry, which gives rise to a torque on the crystal lattice.

• Received 12 February 1968

DOI:https://doi.org/10.1103/PhysRev.171.1104