  We develop a non-overlapping domain decomposition method (DDM) for scalar
wave scattering by periodic layered media. Our approach relies on robust
boundary-integral equation formulations of Robin-to-Robin (RtR) maps throughout
the frequency spectrum, including cutoff (or Wood) frequencies. We overcome the
obstacle of non-convergent quasi-periodic Green functions at these frequencies
by incorporating newly introduced shifted Green functions. Using the latter in
the definition of quasi-periodic boundary-integral operators leads to
rigorously stable computations of RtR operators. We develop Nystr\"om
discretizations of the RtR maps that rely on trigonometric interpolation,
singularity resolution, and fast convergent windowed quasi-periodic Green
functions. We solve the tridiagonal DDM system via recursive Schur complements
and establish rigorously that this procedure is always completed successfully.
We present a variety of numerical results concerning Wood frequencies in two
and three dimensions as well as large numbers of layers.
