  Bloch waves in 1D periodic systems carry Zak phase, which plays a key role in
determining the band topology. In general, for systems that possess inversion
symmetry, the Zak phase of an isolated band is quantized as 0 or Pi and is
associated with the spatial field symmetries at the Brillouin zone center and
boundary. The phase is Pi if the field symmetries are different but is 0 when
they are the same. Since the radiation losses from leaky systems are strongly
associated with the Bloch waves, one may probe the far-field continuum to
determine the Zak phases. Here, we formulate the diffractions from photonic
systems at the zone center and boundary and find their spectral profiles reveal
the Bloch wave symmetries and thereby the corresponding Zak phase. The field
symmetries also generalize the occurrence of bound states in the continuum at
high symmetry points. For verification, we have studied the Zak phases of
one-dimensional TM plasmonic and TE photonic crystals by electrodynamic
simulations and measuring the optical properties of plasmonic crystals using
Fourier space diffraction spectroscopy and common path interferometry. In
addition, a topological protected interface state is demonstrated when two 0
and Pi systems are joined together. The results prove our method provides a
simple way for characterizing the band topology of non-Hermitian systems via
far-fields.
