  We introduce transversely structured all-dielectric waveguides which exploit
the vectorial nature of light to achieve extreme sub-wavelength confinement in
high index dielectrics, enabling characteristic mode dimensions below
$\lambda_0^2$/1,000 without metals or plasmonics. We also derive the metric of
optical concentration and show its convenient usage in characterizing enhanced
linear and non-linear interactions at the nanoscale. This work expands the
toolbox of nanophotonics and opens the door to new types of ultra-efficient and
record performing linear and nonlinear devices with broad applications spanning
classical and quantum optics.
