The Zak phase for lossy 1D photonic structures

We consider the topological aspects of wave propagation in 1D photonic crystals. It was shown by Zak that in 1D structures, bands could be characterized by means of a geometric phase, provided the structure possesses an inversion symmetry, that is the potential V is symmetric with respect to some point. This phase is defined as an integral over the Brillouin zone. We propose another view on the Zak phase, based on a dynamical system approach, that allows to identify the topological properties with the presence of poles of a meromorphic function. This allows to extend the notion to lossy systems. Numerical examples are given in the case of 1D structure whose basic period comprises two slabs filled with a homogeneous material.