  Optical fibers play a key role in many different fields of science and
technology. In particular, fibers with a diameter of several micrometers are
intensively used in photonics. For these applications, it is often important to
precisely know and control the fiber radius. Here, we demonstrate a novel
technique to determine the local radius variation of a 30-micrometer diameter
silica fiber with sub-\AA ngstr\"om precision with axial resolution of several
tens of micrometers over a fiber length of more than half a millimeter. Our
method relies on taking an image of the fiber's whispering-gallery modes
(WGMs). In these WGMs, the speed of light propagating along the fiber axis is
strongly reduced. This enables us to determine the fiber radius with a
significantly enhanced precision, far beyond the diffraction limit. By exciting
different axial modes, we verify the precision and reproducibility of our
method and demonstrate that we can achieve a precision better than 0.3 \AA. The
method can be generalized to other experimental situations where slow light
occurs and, thus, has a large range of potential applications in the realm of
precision metrology and optical sensing.
