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Bloch waves in 1D periodic systems carry Zak phase, which plays a key role indetermining the band topology. In general, for systems that possess inversionsymmetry, the Zak phase of an isolated band is quantized as 0 or Pi and isassociated with the spatial field symmetries at the Brillouin zone center andboundary. The phase is Pi if the field symmetries are different but is 0 whenthey are the same. Since the radiation losses from leaky systems are stronglyassociated with the Bloch waves, one may probe the far-field continuum todetermine the Zak phases. Here, we formulate the diffractions from photonicsystems at the zone center and boundary and find their spectral profiles revealthe Bloch wave symmetries and thereby the corresponding Zak phase. The fieldsymmetries also generalize the occurrence of bound states in the continuum athigh symmetry points. For verification, we have studied the Zak phases ofone-dimensional TM plasmonic and TE photonic crystals by electrodynamicsimulations and measuring the optical properties of plasmonic crystals usingFourier space diffraction spectroscopy and common path interferometry. Inaddition, a topological protected interface state is demonstrated when two 0and Pi systems are joined together. The results prove our method provides asimple way for characterizing the band topology of non-Hermitian systems viafar-fields.

Determination of the Zak phase of one-dimensional photonic systems via far-field diffraction