  We consider the planar magnonic waveguide with a periodic sequence of
antidots forming zig-zag pattern, where two neighboring antidots are shifted
towards the opposite edges of the waveguide. This system has a complex base
with two antidots in one unit cell. The Brillouin zone is here two-times
narrower than the Brillouin zone for the waveguide without displacement of
antidots. We have shown that for dispersion relation folded into narrower
Brillouin zone, new frequency gap can be opened and their width can be
controlled by the shift of the antidots. We found that, the different strength
of spin wave pinning at the edges of the periodic waveguide (and their
antidots)determines the dependence of the width of gap on the shift of
antidots. For the systems with completely free or ideally pinned magnetization,
these dependencies are qualitatively different. We have found an optimum shift
of antidot for maximzing the width of the gap for the system with pinned
magnetization. More interestingly, we notice that for this kind of geometry of
the structure, majority of the modes are doubly degenerate at the edge of
Brillouin zone and have a finite group velocity at the very close vicinity of
the edge of Brillouin zone, for larger values of antidot shift. This empowers
us to design magnonic waveguide to steer the spin waves.
