  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.
