Symmetry and phase determination of second-harmonic reflection from calcite surfaces

We perform second harmonic reflection (SHR) on the calcite surface perpendicular to the c axis. First, we record the signal as a function of the azimuthal angle of the sample, which clearly evidences the $C_{3v}$ symmetry of this surface, and we show that the nonvanishing background of the p-polarized SHR is an indication of a complex-valued nonlinear susceptibility. Then, we measure the relative phases of the susceptibility tensor components with a method that was used recently to study thin films. We vary the state of polarization of the fundamental beam with a rotating quarter waveplate which introduces some phase difference between the different polarization components. This method appears to be an easy and efficient way to measure the phase of SHR on a crystalline surface. Furthermore, comparison of both sets of experiments shows very good quantitative agreement, even though the origin of this phase difference is not perfectly clear.