  The development of nanoscale solid-state devices exploiting the promising
topological surface states of topological insulator materials requires careful
device engineering and improved materials quality. For instance, the
introduction of a substrate, device contact or the formation of oxide layers
can cause unintentional doping of the material, spoiling the sought-after
properties. In support of this, nanoscale imaging tools can provide useful
materials information without the need for complex device fabrication. Here we
study Bi$_2$Se$_3$ nanoribbons suspended across multiple material stacks of
SiO$_2$ and Au using infrared scattering scanning near-field optical
microscopy. We validate our observations against a multilayer finite dipole
model to obtain quantitative imaging of the local Bi$_2$Se$_3$ properties that
vary depending on the local environment. Moreover, we identify experimental
signatures that we associate with quantum well states at the Bi$_2$Se$_3$
surfaces. Our approach opens a new direction for future engineering of
nanoelectronic devices based on topological insulator materials.
