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

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