Raman heterodyne interference: Symmetry analysis

The Raman heterodyne interference phenomenon observed recently in the impurity-ion crystals ${\mathrm{Pr}}^{3+}$:${\mathrm{YAlO}}_3$ and ${\mathrm{Pr}}^{3+}$:${\mathrm{LaF}}_3$ is examined from the viewpoint of symmetry analysis of the nonlinear susceptibility tensor. General Raman heterodyne signals are derived for all crystal symmetries and for all optical-frequency and radiofrequency polarizations. An example of the utility of this approach resolves a controversy over the crystal symmetry of ${\mathrm{LaF}}_3$, establishing that it is trigonal rather than hexagonal.