  Nonlinear optical generation from a crystalline material can reveal the
symmetries of both its lattice structure and underlying ordered electronic
phases and can therefore be exploited as a complementary technique to
diffraction based scattering probes. Although this technique has been
successfully used to study the lattice and magnetic structures of systems such
as semiconductor surfaces, multiferroic crystals, magnetic thin films and
multilayers, challenging technical requirements have prevented its application
to the plethora of complex electronic phases found in strongly correlated
electron systems. These requirements include an ability to probe small bulk
single crystals at the micron length scale, a need for sensitivity to the
entire nonlinear optical susceptibility tensor, oblique light incidence
reflection geometry and incident light frequency tunability among others. These
measurements are further complicated by the need for extreme sample
environments such as ultra low temperatures, high magnetic fields or high
pressures. In this review we present a novel experimental construction using a
rotating light scattering plane that meets all the aforementioned requirements.
We demonstrate the efficacy of our scheme by making symmetry measurements on a
micron scale facet of a small bulk single crystal of Sr$_2$IrO$_4$ using
optical second and third harmonic generation.
