Open waveguides in doubly periodic junctions of domains with different limit dimensions

Considering the spectral Neumann problem for the Laplace operator on a doubly periodic square grid of thin circular cylinders (of diameter $\varepsilon\ll1$) with nodes, which are sets of unit size, we show that by changing or removing one or several semi-infinite chains of nodes we can form additional spectral segments, the wave passage bands, in the essential spectrum of the original grid. The corresponding waveguide processes are localized in a neighborhood of the said chains, forming $\mathrm I$-shaped, $\mathrm V$-shaped, and $\mathrm L$-shaped open waveguides. To derive the result, we use the asymptotic analysis of the eigenvalues of model problems on various periodicity cells.