Nonlinear Optical Spectroscopy as a possible probe of Hidden Order in URu$_2$Si$_2$

The symmetry of the primary order parameter underlying the hidden order (HO) phase of URu$_{2}$Si$_{2}$ is a topic of ongoing debate. Although high-rank multipole orderings are proposed, these are challenging to resolve experimentally using diffraction based probes. We investigate the possibility of using the nonlinear optical response of URu$_{2}$Si$_{2}$ to measure electronic multipolar ordering. We report temperature and wavelength dependent nonlinear optical harmonic generation measurements on micron scale areas of URu$_{2}$Si$_{2}$ single crystals using a novel low temperature rotational anisotropy technique\footnote{D. H. Torchinsky \emph{et al.}, \textbf{Rev. Sci. Instrum.} 85, 083102 (2014)}. We identify a nonlinear optical susceptibility tensor that couples to HO whose implications on the symmetry of the HO parameter will be discussed.