  Thin-film lithium niobate (TFLN) in the form of x- or z-cut
lithium-niobate-on-insulator (LNOI) has recently popped up as a very promising
and novel platform for developing integrated optoelectronic (nano)devices and
exploring fundamental research. Here, we investigate the coherent interaction
length $l_{c}$ of optical second-harmonic (SH) microscopy in such samples, that
are purposely prepared into a wedge shape, in order to elegantly tune the
geometrical confinement from bulk thicknesses down to $\approx$ 50 nm. SH
microscopy is a very powerful and non-invasive tool for the investigation of
structural properties in the biological and solid-state sciences, especially
also for visualizing and analyzing ferroelectric domains and domain walls.
However, unlike bulk LN, SH microscopy in TFLN is largely affected by
interfacial reflections and resonant enhancement that both rely on film
thickness and substrate material. In this paper we show that the dominant SHG
contribution measured in back-reflection, is the co-propagating phase-matched
SH signal and \textit{not} the counter-propagating SH portion as is the case
for bulk LN samples. Moreover, $l_{c}$ dramatically depends also on the
incident pump laser wavelength (sample dispersion) but even more on the
numerical aperture of the focussing objective in use. These experimental
findings on x- and z-cut TFLN are excellently backed up by our advanced
numerical simulations.
