  In this paper we study the operator
$H_{\beta}=-\Delta-\beta\delta(\cdot-\Gamma)$ in $L^{2}(\mathbb{R}^{2})$, where
$\Gamma$ is a smooth periodic curve in $\mathbb{R}^{2}$. We obtain the
asymptotic form of the band spectrum of $H_{\beta}$ as $\beta$ tends to
infinity. Furthermore, we prove the existence of the band gap of
$\sigma(H_{\beta})$ for sufficiently large $\beta>0$. Finally, we also derive
the spectral behaviour for $\beta\to\infty$ in the case when $\Gamma$ is
non-periodic and asymptotically straight.
