High-order homogenisation of the time-modulated wave equation: non-reciprocity for a single varying parameter

Laminated media with material properties modulated in space and time in the form of travelling waves have long been known to exhibit non-reciprocity. However, when using the method of low frequency homogenisation, it was so far only possible to obtain non-reciprocal effective media when both material properties are modulated in time, in the form of a Willis-coupling (or bi-anisotropy in electromagnetism) model. If only one of the two properties is modulated in time, while the other is kept constant, it was thought impossible for the method of homogenisation to recover the expected non-reciprocity since this Willis-coupling coefficient then vanishes. Contrary to this belief, we show that effective media with a single time-modulated parameter are non-reciprocal, provided homogenization is pushed to the second order. This is illustrated by numerical experiments (dispersion diagrams and time-domain simulations) for a bilayered modulated medium.