  The optoelectronic properties of nanoscale systems such as carbon nanotubes
(CNTs), graphene nanoribbons and transition metal dichalcogenides (TMDCs) are
determined by their dielectric function. This complex, frequency dependent
function is affected by excitonic resonances, charge transfer effects, doping,
sample stress and strain, and surface roughness. Knowledge of the dielectric
function grants access to a material's transmissive and absorptive
characteristics. Here we introduce the dual scanning near field optical
microscope (dual s-SNOM) for imaging local dielectric variations and extracting
dielectric function values using a mathematical inversion method. To
demonstrate our approach, we studied a monolayer of WS$_2$ on bulk Au and
identified two areas with differing levels of charge transfer. Our measurements
are corroborated by atomic force microscopy (AFM), Kelvin force probe
microscopy (KPFM), photoluminescence (PL) intensity mapping, and tip enhanced
photoluminescence (TEPL). We extracted local dielectric variations from s-SNOM
images and confirmed the reliability of the obtained values with spectroscopic
imaging ellipsometry (SIE) measurements.
