Generalized plane wave expansion method for non-reciprocal discretely modulated waveguides

In this manuscript we investigate one-way wave propagation in spatio-temporal phononic wave-guides in which the modulation of material properties is provided piecewise in space and time. Non-reciprocal dispersion diagram is computed using a generalized plane wave expansion method, whose formulation, provided in the paper, is able to describe how a generic discrete or continuous unit cell is able to break the mirror symmetry in the momentum space, as already shown in other works by means of more expensive and less accurate computational tools. The proposed method is then used to verify a physical interpretation we provided for bandgap formation by means of wave superposition in spatio-temporal modulated waveguides.