  At the example of two coupled waveguides we construct a periodic second order
differential operator acting in a Euclidean domain and having spectral gaps
whose edges are attained strictly inside the Brillouin zone. The waveguides are
modeled by the Laplacian in two infinite strips of different width that have a
common interior boundary. On this common boundary we impose the Neumann
boundary condition but cut out a periodic system of small holes, while on the
remaining exterior boundary we impose the Dirichlet boundary condition. It is
shown that, by varying the widths of the strips and the distance between the
holes, one can control the location of the extrema of the band functions as
well as the number of the open gaps. We calculate the leading terms in the
asymptotics for the gap lengths and the location of the extrema.
