  Thin film lithium niobate is of great recent interest and an understanding of
periodically poled thin-films is crucial for both fundamental physics and
device developments. Second-harmonic (SH) microscopy allows for the
non-invasive visualization and analysis of ferroelectric domain structures and
walls. While the technique is well understood in bulk lithium niobate, SH
microscopy in thin films is largely influenced by interfacial reflections and
resonant enhancements, which depend on film thicknesses and the substrate
materials. We present a comprehensive analysis of SH microscopy in x-cut
lithium niobate thin films, based on a full three dimensional focus
calculations, and accounting for interface reflections. We show that the
dominant signal in back-reflection originates from a co-propagating
phase-matched process observed through reflections, rather than direct
detection of the counter-propagating signal as in bulk samples. We can explain
the observation of domain structures in the thin film geometry, and in
particular, we show that the SH signal from thin poled films allows to
unambiguously distinguish areas, which are completely or only partly inverted
in depth.
