  We present the results of a systematic micromagnetic study of the effect of
the Dzyaloshinskii-Moriya interaction (DMI) on the spin wave band structure of
two one-dimensional magnonic crystals (MCs), both with the same periodicity
p=300 nm, but different implementation of the DMI modulation. In the first
system the artificial periodicity was achieved by modulating the interfacial
DMI constant D, while in the second system also the sample morphology was
modulated. Due to the folding property of the band structure in the dispersion
relations of the magnonic crystals it is possible to extend the sensitivity of
Brillouin light scattering towards weak DMI strength (D in the range from 0 to
0.5 mJ/m2), by measuring the frequency splitting of folded modes in high-order
artificial Brillouin zones, since the splitting increases almost linearly with
the band index. For relatively large values of the DMI (D in the range from 1.0
to 2.0 mJ/m2) the spin waves dispersion relations present flat modes for
positive wavevectors, separated by forbidden frequency gaps whose amplitude
depend on the value of D. These frequency gaps are more pronounced for the
sample with morphology modulation. The non-reciprocal, localised, spatial
profiles of these modes in both MCs are discussed with reference to spin waves
in plain films and in isolated stripes of the same thickness.
