  We study the defocusing nonlinear Schr\"odinger equation in the quarter plane
with asymptotically periodic boundary values. By studying an associated
Riemann-Hilbert problem and employing nonlinear steepest descent arguments, we
construct solutions in a sector close to the boundary whose leading behaviour
is described by a single exponential plane wave. Furthermore, we compute the
subleading terms in the long time asymptotics of the constructed solutions.
