Determination of the Zak phase of one-dimensional diffractive systems with inversion symmetry via radiation in Fourier space

Bloch waves in one-dimensional periodic systems carry the Zak phase, which plays a key role in determining the band topology. In general, for a system that possesses inversion symmetry, the Zak phase of an isolated band is quantized as 0 or $\ensuremath{\pi}$ and is associated with the spatial-field symmetries of the Bloch waves at the Brillouin-zone center and boundary. 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 radiations from photonic systems in Fourier space at the zone center and boundary and find they reveal the field symmetries and thereby the corresponding Zak phase. For verification, we have studied the Zak phases of TM plasmonic and TE photonic crystals by electrodynamic simulations and measuring the topologically properties of plasmonic crystals using Fourier-space optical spectroscopy and common-path interferometry. In addition, a topologically protected interface state is demonstrated when two topologically trivial and nontrivial systems are joined together. Our results provide a simple way for characterizing the band topology of photonic systems via far fields.