  We consider the nonlinear Schr\"odinger equation on the half-line with a
given Dirichlet boundary datum which for large $t$ tends to a periodic
function. We assume that this function is sufficiently small, namely that it
can be expressed in the form $\alpha g_0^b(t)$, where $\alpha$ is a small
constant. Assuming that the Neumann boundary value tends for large $t$ to the
periodic function $g_1^b(t)$, we show that $g_1^b(t)$ can be expressed in terms
of a perturbation series in $\alpha$ which can be constructed explicitly to any
desired order. As an illustration, we compute $g_1^b(t)$ to order $\alpha^8$
for the particular case that $g_0^b(t)$ is the sum of two exponentials. We also
show that there exist particular functions $g_0^b(t)$ for which the above
series can be summed up, and therefore for these functions $g_1^b(t)$ can be
obtained in closed form. The simplest such function is $\exp(i\omega t)$, where
$\omega$ is a real constant.
