  Second harmonic generation (SHG) spectroscopy ubiquitously enables the
investigation of surface chemistry, interfacial chemistry as well as symmetry
properties in solids. Polarization-resolved SHG spectroscopy in the visible to
infrared regime is regularly used to investigate electronic and magnetic orders
through their angular anisotropies within the crystal structure. However, the
increasing complexity of novel materials and emerging phenomena hamper the
interpretation of experiments solely based on the investigation of hybridized
valence states. Here, polarization-resolved SHG in the extreme ultraviolet
(XUV-SHG) is demonstrated for the first time, enabling element-resolved angular
anisotropy investigations. In non-centrosymmetric LiNbO$_3$, elemental
contributions by lithium and niobium are clearly distinguished by energy
dependent XUV-SHG measurements. This element-resolved and symmetry-sensitive
experiment suggests that the displacement of Li ions in LiNbO$_3$, which is
known to lead to ferroelectricity, is accompanied by distortions to the Nb ion
environment that breaks the inversion symmetry of the NbO$_{6}$ octahedron as
well. Our simulations show that the measured second harmonic spectrum is
consistent with Li ion displacements from the centrosymmetric position by
$\sim$0.5 Angstrom while the Nb-O bonds are elongated/contracted by
displacements of the O atoms by $\sim$0.1 Angstrom. In addition, the
polarization-resolved measurement of XUV-SHG shows excellent agreement with
numerical predictions based on dipole-induced SHG commonly used in the optical
wavelengths. This constitutes the first verification of the dipole-based SHG
model in the XUV regime. The findings of this work pave the way for future
angle and time-resolved XUV-SHG studies with elemental specificity in condensed
matter systems.
