  The main task in this paper is to prove that the perfectly matched layers
(PML) method converges exponentially with respect to the PML parameter, for
scattering problems with periodic surfaces. In [5], a linear convergence is
proved for the PML method for scattering problems with rough surfaces. At the
end of that paper, three important questions are asked, and the third question
is if exponential convergence holds locally. In our paper, we answer this
question for a special case, which is scattering problems with periodic
surfaces. The result can also be easily extended to locally perturbed periodic
surfaces or periodic layers. Due to technical reasons, we have to exclude all
the half integer valued wavenumbers. The main idea of the proof is to apply the
Floquet-Bloch transform to write the problem into an equivalent family of
quasi-periodic problems, and then study the analytic extension of the
quasi-periodic problems with respect to the Floquet-Bloch parameters. Then the
Cauchy integral formula is applied for piecewise analytic functions to avoid
linear convergent points. Finally the exponential convergence is proved from
the inverse Floquet-Bloch transform. Numerical results are also presented at
the end of this paper.
