  Topological phase transitions in condensed matter systems have shown
extremely rich physics, unveiling such exotic states of matter as topological
insulators, superconductors and superfluids. Photonic topological systems open
a whole new realm of research exhibiting a number of important distinctions
from their condensed matter counterparts. Photonic modes can couple to the
continuum of free space modes which makes it feasible to control and manipulate
scattering properties of the photonic structure via topology. At the same time,
the direct connection of scattering and topological properties of the photonic
states allows their probing by spectroscopic means via Fano resonances. Here we
demonstrate that the radiative coupling of modes supported by an all-dielectric
metasurface can be controlled and tuned under topological phase transitions due
to band inversion, correspondingly inducing a distinct switching of the quality
factors of the resonances associated with the bands. In addition, we develop a
technique to retrieve the topological properties of all-dielectric metasurfaces
from the measured far-field scattering characteristics. The collected
angle-resolved transmission and reflection spectra allow extracting the
momentum-dependent frequencies and lifetimes of the photonic modes. This
enables retrieval of the effective photonic Hamiltonian, including the effects
of a synthetic gauge field, and topological invariants~-- pseudo-spin Chern
numbers. Our results thus open a new avenue to design a new class of
metasurfaces with unique scattering characteristics controlled via topological
effects. This work also demonstrates how topological states of open systems can
be explored via far-field measurements.
