  Corbino-geometry has well-known applications in physics, as in the design of
graphene heterostructures for detecting fractional quantum Hall states or
superconducting waveguides for illustrating circuit quantum electrodynamics.
Here, we propose and demonstrate a photonic Kagome lattice in the
Corbino-geometry that leads to direct observation of non-contractible loop
states protected by real-space topology. Such states represent the "missing"
flat-band eigenmodes, manifested as one-dimensional loops winding around a
torus, or lines infinitely extending to the entire flat-band lattice. In finite
(truncated) Kagome lattices, however, line states cannot preserve as they are
no longer the eigenmodes, in sharp contrast to the case of Lieb lattices. Using
a continuous-wave laser writing technique, we experimentally establish finite
Kagome lattices with desired cutting edges, as well as in the Corbino-geometry
to eliminate edge effects. We thereby observe, for the first time to our
knowledge, the robust boundary modes exhibiting self-healing properties, and
the localized modes along toroidal direction as a direct manifestation of the
non-contractible loop states.
